A023200 -id:A023200

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A172112

Partial sums of A023200.

+20
2

3, 10, 23, 42, 79, 122, 189, 268, 365, 468, 577, 704, 867, 1060, 1283, 1512, 1789, 2096, 2409, 2758, 3137, 3534, 3973, 4430, 4893, 5380, 5879, 6492, 7135, 7808, 8547, 9304, 10073, 10896, 11749, 12608, 13485, 14368, 15275, 16212, 17179, 18188, 19275

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes in the partial sum begin: a(1) = 3, a(3) = 23, a(5) = 79, a(11) = 577, a(15) = 1283, a(17) = 1789, a(21) = 3137, a(27) = 5879. Of these, the smaller members of cousin prime pairs which appear among the partial sums of smaller member p of cousin prime pairs begin: 3, 79; which are the next in this subset?

LINKS

Table of n, a(n) for n=1..43.

FORMULA

a(n) = SUM[i=i..n] A023200(i) = SUM[i=i..n] {Primes p such that p and p + 4 are both primes}.

EXAMPLE

a(30) = 3 + 7 + 13 + 19 + 37 + 43 + 67 + 79 + 97 + 103 + 109 + 127 + 163 + 193 + 223 + 229 + 277 + 307 + 313 + 349 + 379 + 397 + 439 + 457 + 463 + 487 + 499 + 613 + 643 + 673 = 7808.

MATHEMATICA

Accumulate[Select[Prime[Range[250]], PrimeQ[#+4]&]] (* Harvey P. Dale, Oct 09 2023 *)

CROSSREFS

Cf. A000040, A023200, A046132.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 25 2010

EXTENSIONS

More terms from Max Alekseyev, Jan 31 2010

STATUS

approved

A172483

a(n) is the number of cousin primes between p^2 and p*(p+4) where p is the n-th cousin prime A023200(n).

+20
1

2, 1, 1, 2, 5, 4, 4, 2, 6, 4, 7, 7, 5, 9, 12, 13, 14, 14, 9, 12, 10, 11, 13, 20, 16, 15, 16, 15, 23, 19, 22, 26, 27, 28, 26, 22, 20, 27, 25, 27, 28, 26, 35, 29, 29, 29, 30, 45, 30, 36, 22, 30, 39, 39, 40, 44, 44, 43, 34, 38, 36, 48, 54, 43, 38, 43, 49, 45, 47, 53, 38, 51, 51, 62, 56

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If you graph the order of the consecutive cousin primes along the x-axis (i.e., first pair of cousin primes, second, third,...) and the number of cousin primes in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, greater are the number of consecutive cousin primes, on average, within the interval obtained. If one can prove that there's at least one consecutive cousin prime within each interval, this would imply that cousin primes are infinite. I suspect the number of consecutive primes within each interval will never be zero. Can you prove it?

REFERENCES

C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.

M. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

LINKS

J. S. Cheema, Table of n, a(n) for n = 1..1104 (2 prepended by Michael De Vlieger)

EXAMPLE

The 1st pair of cousin primes is (3, 7), between 3^2=9 and 3*7=21 there is 2 cousin primes: 13 and 19. So a(1) = 2.

The 2nd pair of cousin primes is (7, 11), between 7^2=49 and 7*11=77 there is 1 cousin prime: 67. So a(2) = 1.

PROG

(PARI) vcp(nn) = my(list=List(), p=3); listput(list, p); p=7; forprime(q=11, nn, if(q-p==4, listput(list, p)); p=q); Vec(list); \\ A023200

nbcp(p) = my(nb=0); forprime(q=p^2, p*(p+4), if (isprime(q+4), nb++)); nb;

lista(nn) = my(v=vcp(nn)); vector(#v, n, nbcp(v[n])); \\ Michel Marcus, Nov 02 2022

CROSSREFS

Cf. A023200, A046132, A087679.

KEYWORD

nonn

AUTHOR

Jaspal Singh Cheema, Feb 04 2010

EXTENSIONS

New name and a(1)=2 prepended by Michel Marcus, Nov 02 2022

STATUS

approved

A174046

Places n for which A001359(n) and A023200(n) is a twin prime pair.

+20
1

2, 3, 4, 6, 14, 16, 29, 356, 358, 359, 403, 446, 464, 485, 652, 655, 764, 861, 866, 1123, 1301, 1304, 1324, 1328, 1358, 1486, 1610, 2631, 2632, 3735, 3931, 3953, 3956, 3957, 4679, 4855, 4931, 5222, 5226, 5269, 5283, 5292, 5403, 5427, 5445

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Michel Marcus, Table of n, a(n) for n = 1..149

EXAMPLE

2 is in the sequence because A001359(2)=5 and A023200(2)=7 are twin primes.

PROG

(PARI) lista(nn) = {vp = primes(nn); va = select(x->isprime(x+2), vp); vb = select(x->isprime(x+4), vp); for (n=1, min(#va, #vb), if (vb[n] == va[n]+2, print1(n, ", ")); ); } \\ Michel Marcus, Jul 22 2017

CROSSREFS

Cf. A023200, A001359, A046132, A006512, A174042

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Mar 06 2010

EXTENSIONS

Terms beyond 29 from R. J. Mathar, Nov 03 2011

Edited by Michel Marcus, Jul 22 2017

STATUS

approved

A052351

First primes from A023200 where distance to the next 4-twin increases.

+20
0

7, 67, 19, 43, 163, 127, 397, 229, 769, 1489, 673, 9547, 1009, 1783, 1693, 2857, 11677, 23869, 499, 1093, 4003, 28657, 10459, 29383, 12487, 6043, 41647, 7039, 17029, 19207, 15073, 24247, 65839, 29629, 18583, 9883, 66697, 100699, 7243

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) is a "lesser of a 4-twin" prime whose distance to the next twin is 6n.

Both the smallest distance (A052380) and its increment for 4-twins is 6.

LINKS

Table of n, a(n) for n=1..39.

FORMULA

The prime a(n)=p is the first which determines a prime quadruple [p, p+4, p+6n, p+6n+4] and difference pattern of [4, 6n-4, 4].

EXAMPLE

a(1)=7 gives [7,11,7+6=13,17] with no primes between 11 and 13.

a(5)=163 specifies [163,167,163+30=191,193] with 4 primes between 167 and 193.

CROSSREFS

Cf. A023200, A053320, A052380, A052381.

KEYWORD

nonn

AUTHOR

Labos Elemer, Mar 07 2000

STATUS

approved

A352171

a(n) is the start of a sequence of exactly n members of A023200 under the iteration p -> 3*p+4.

+20
0

7, 13, 3, 1547803

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Let s(1) = a(n) and s(k+1) = 3*s(k)+4. Then s(1), ..., s(n) are in A023200 but s(n+1) is not in A023200, and a(n) is the least value of s(n) for which this is the case.

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

a(3) = 3 because with s(1) = 3 we have s(2) = 3*3+4 = 13, s(3) = 3*13+4 = 43, s(4) = 3*43+4 = 133; 3, 13, and 43 are in A023200 because 3, 7, 13, 17, 42, 47 are prime, but 133 is not in A023200 because 133 is composite.

MAPLE

f:= proc(p) option remember;

if isprime(p) and isprime(p+4) then 1 + procname(3*p+4) else 0 fi

end proc:

V:= Vector(5): V[1]:= 7: V[3]:= 3: count:= 2:

for p from 13 by 30 while count < 5 do

v:= f(p);

if v > 0 and V[v] = 0 then count:= count+1; V[v]:= p; fi

od:

convert(V, list);

CROSSREFS

Cf. A023200.

KEYWORD

nonn,more

AUTHOR

J. M. Bergot and Robert Israel, Mar 07 2022

STATUS

approved

A046132

Larger member p+4 of cousin primes (p, p+4).

+10
73

7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A pair of cousin primes are primes of the form p and p+4 (where p+2 may or may not be a prime). - N. J. A. Sloane, Mar 18 2021

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Cousin Primes

FORMULA

a(n) = A023200(n) + 4 = A087679(n) + 2.

a(n) = 3*A157834(n-1) + 2 = A029710(n-1) + 4 = 6*A056956(n-1) + 5 (thus a(n) mod 6 == 5), for all n>1. - M. F. Hasler, Jan 15 2013

MATHEMATICA

lst={}; Do[p=Prime[n]; If[PrimeQ[p4=p+4], (*Print[p4]; *)AppendTo[lst, p4]], {n, 10^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

Select[Prime[Range[300]], PrimeQ[#+4]&]+4 (* Harvey P. Dale, Dec 15 2017 *)

PROG

(PARI) forprime(p=2, 1e5, if(isprime(p-4), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

(Haskell)

a046132 n = a046132_list !! (n-1)

a046132_list = filter ((== 1) . a010051') $ map (+ 4) a000040_list

-- Reinhard Zumkeller, Aug 01 2014

CROSSREFS

Essentially the same as A031505. Cf. A023200, A029710, A098429.

Cf. A000040, A010051.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

A007694

Numbers k such that phi(k) divides k.
(Formerly M0992)

+10
43

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 256, 288, 324, 384, 432, 486, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8192, 8748, 9216

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) divides p^a(n) - 1 for all primes p >= 5. - Benoit Cloitre, Mar 22 2002

Also k such that Sum_{d divides k} mu(d)/d has numerator 1. - Benoit Cloitre, Apr 15 2002

k is here if and only if phi(k) also divides cototient(k). On the other hand, cototient(k) divides phi(k) if and only if k is a prime or power of a prime. - Labos Elemer, May 03 2002

It follows that k/phi(k) = 2 if k is a power of 2 and equal to 3 if k is of the form 6*A003586. - Gary Detlefs, Jun 28 2011

1 and even 3-smooth numbers, cf. A003586. - Reinhard Zumkeller, Jan 06 2014

Numbers k such that k = (1+omega(k))*phi(k). - Farideh Firoozbakht, Oct 02 2014

These are the integers whose largest squarefree divisor is 1, 2 or 6. As such, this sequence is equal to the set V_infinite, defined as the intersection of the V_k for k >= 1, where V_k(x) = {phi_k(n) <= x} and phi_k is the k-th iterate of phi, the Euler function; for instance, V_1 is given by A002202 (see Theorem 7 in Pomerance and Luca). - Michel Marcus, Nov 09 2015

This sequence is contained in A068997. The terms of A068997 not in this sequence have largest squarefree divisor other than 1, 2, or 6, beginning with 10. - Torlach Rush, Dec 07 2017

REFERENCES

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 526 pp. 71; 256, Ellipses Paris 2004.

Sárközy A. and Suranyi J., Number Theory Problem Book (in Hungarian), Tankonyvkiado, Budapest, 1972.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Roohollah Ebrahimian, When Does phi(n) divide n? A Number Theory Math Competition Problem., YouTube video, 2023.

Michael W. Ecker and Scott J. Beslin, Problem E3037, Amer. Math. Monthly, Vol. 93, No. 8 (1986), pp. 656-657.

Florian Luca and Carl Pomerance, On the range of the iterated Euler function, Article 8, Integers: Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference 2007, Volume 9 Supplement (2009).

Mathematics Stack Exchange, Is there phi(n)/n = 6

Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

FORMULA

k/phi(k) is an integer if and only if k = 1 or k = 2^w * 3^u for w > 0 and u >= 0.

k/phi(k) = 3 if and only if phi(k)|k and 3|k. - Thomas Ordowski, Nov 03 2014

a(n) is approximately exp(sqrt(2*log(2)*log(3)*n))/sqrt(3/2). - Charles R Greathouse IV, Nov 10 2015

From Amiram Eldar, Oct 29 2020: (Start)

a(n) = 2 * A003586(n) for n > 1.

Sum_{n>=1} 1/a(n) = 5/2. (End)

EXAMPLE

12 is in the sequence because 12/phi(12) = 12/4 = 3, which is an integer.

16 is in the sequence because 16/phi(16) = 16/8 = 2, which is an integer.

20 is not in the sequence because 20/phi(20) = 20/8 = 5/2 = 2.5, which is not an integer.

MAPLE

select(n -> n mod numtheory:-phi(n) = 0, [$1..5000]); # Robert Israel, Nov 03 2014

MATHEMATICA

Select[ Range[5000], IntegerQ[ #/EulerPhi[ # ]] &]

m = 5000; Join[{1}, Sort @ Flatten @ Table[2^i*3^j, {i, 1, Log2[m]}, {j, 0, Log[3, m/2^i]}]] (* Amiram Eldar, Oct 29 2020 *)

PROG

(R) library(numbers); j=N=1

while(j<200) if(isNatural((N=N+1)/eulersPhi(N))) dtot[(j=j+1)]=N # Christian N. K. Anderson, Apr 04 2013

(PARI) for(n=1, 10^6, if (n%eulerphi(n)==0, print1(n, ", "))); \\ Joerg Arndt, Apr 04 2013

(PARI) list(lim)=my(v=List([1]), t); for(i=1, logint(lim\1, 2), listput(v, t=2^i); for(j=1, logint(lim\t, 3), listput(v, t*=3))); Set(v) \\ Charles R Greathouse IV, Nov 10 2015

(Haskell)

a007694 n = a007694_list !! (n-1)

a007694_list = 1 : filter even a003586_list

-- Reinhard Zumkeller, Jan 06 2014

(Sage)

is_A007694 = lambda n: euler_phi(n).divides(n)

A007694_list = lambda len: filter(is_A007694, (1..len))

A007694_list(4100) # Peter Luschny, Oct 03 2014

CROSSREFS

Cf. A000010, A049237, A007694, A007947, A003557, A023200, A003586, A001221, A033950, A235353 (subsequence), A068997 (subsequence).

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

A029710

Primes such that next prime is 4 greater.

+10
38

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Union with A124588 gives A124589. - Reinhard Zumkeller, Dec 23 2006

For any prime p > 3, if p + 4 is prime then necessarily it is the next prime. But there cannot be three consecutive primes with mutual distance 4: If p and p + 4 are prime, then p+8 is an odd multiple of 3 (cf. formula). - M. F. Hasler, Jan 15 2013

The smaller members p of cousin prime pairs (p,p+4) excluding p=3. - Marc Morgenegg, Apr 19 2016

LINKS

Marius A. Burtea, Table of n, a(n) for n = 1..14741 ( first 1000 terms from R. Zumkeller )

Index entries for primes, gaps between

FORMULA

a(n) = A031505(n + 1) - 4 = A029708(n) - 2.

a(n) = 1 (mod 6) for all n; (a(n) + 2)/3 = A157834(n), i.e., a(n) = 3*A157834(n) - 2. - M. F. Hasler, Jan 15 2013

EXAMPLE

79 is a term as the next prime is 79 + 4 = 83. 3 is not a term even though 3 + 4 = 7 is prime, since it is not the next one.

MAPLE

for i from 1 to 226 do if ithprime(i+1) = ithprime(i) + 4 then print({ithprime(i)}); fi; od; # Zerinvary Lajos, Mar 19 2007

MATHEMATICA

Select[Prime[Range[225]], NextPrime[#] == # + 4 &] (* Alonso del Arte, Jan 17 2013 *)

Transpose[Select[Partition[Prime[Range[300]], 2, 1], #[[2]]-#[[1]]==4&]] [[1]] (* Harvey P. Dale, Mar 28 2016 *)

PROG

(PARI) forprime(p=1, 1e4, if(nextprime(p+1)-p==4, print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

(Magma) [p:p in PrimesUpTo(1700)| IsPrime(p+4) and NextPrime(p) eq p+4] // Marius A. Burtea, Jan 24 2019

(MATLAB)

p=primes(1700); m=1;

for u=1:length(p)-4

if and(isprime(p(u)+4)==1, p(u+1)==p(u)+4); sol(m)=p(u); m=m+1; end

end

sol % Marius A. Burtea, Jan 24 2019

CROSSREFS

Essentially the same as A023200.

Cf. A001359, A029708, A031505, A124588, A124589, A157834.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A029709

Numbers k such that k-th and (k+1)st primes differ by 4.

+10
31

4, 6, 8, 12, 14, 19, 22, 25, 27, 29, 31, 38, 44, 48, 50, 59, 63, 65, 70, 75, 78, 85, 88, 90, 93, 95, 112, 117, 122, 131, 134, 136, 143, 147, 149, 151, 153, 155, 159, 163, 169, 181, 183, 198, 207, 211, 213, 224, 226, 229, 235, 237, 244, 247, 249, 251

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Positions of 4 in A001223. - Zak Seidov, Apr 28 2015

LINKS

N. J. A. Sloane and K. D. Bajpai, Table of n, a(n) for n = 1..10213 (first 56 terms from N. J. A. Sloane)

FORMULA

A029710(n) = prime(a(n)). - R. J. Mathar, Apr 30 2024

MATHEMATICA

Select[Range[2, 300], 4 == (Prime[# + 1] - Prime[#]) &] (* Vincenzo Librandi, Apr 28 2015 *)

PROG

(Magma) [n: n in [0..300] | NthPrime(n+1) - NthPrime(n) eq 4]; // Vincenzo Librandi, Apr 28 2015

CROSSREFS

Cf. A023200, A029710, A001223.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A060308

Largest prime <= 2n.

+10
30

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) is the smallest k such that C(2n,n) divides k!. - Benoit Cloitre, May 30 2002

a(n) is largest prime factor of C(2n,n) = (2n)!/(n!)^2. - Alexander Adamchuk, Jul 11 2006

a(n) is also the largest prime in the interval [n,2n]. - Peter Luschny, Mar 04 2011

Odd prime p repeats (q-p)/2 times, where q > p is the next prime. In particular, every lesser of twin primes (A001359) occurs 1 time, every lesser more than 3 of cousin primes (A023200) occurs 2 times, etc. - Vladimir Shevelev, Mar 12 2012

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Max[FactorInteger[(2n)!/(n!)^2]]. - Alexander Adamchuk, Jul 11 2006

a(n) = A006530(A000142(2*n)) and a(n) = A006530(A056040(2*n)). - Peter Luschny, Mar 04 2011

a(n) ~ 2*n as n tends to infinity. - Vladimir Shevelev, Mar 12 2012

a(n) = A007917(A005843(n)) = A226078(n, A067434(n)). - Reinhard Zumkeller, May 25 2013

EXAMPLE

n=1, 2n=2, p(1) = 2 = a(1) is the largest prime not exceeding 2.

MAPLE

seq (prevprime(2*i+1), i=1..256);

seq(max(op(select(isprime, [$n..2*n]))), n=1..66); # Peter Luschny, Mar 04 2011

MATHEMATICA

Table[Max[FactorInteger[(2n)!/(n!)^2]], {n, 1, 100}] (* Alexander Adamchuk, Jul 11 2006 *)

NextPrime[2*Range[80]+1, -1] (* Harvey P. Dale, Apr 23 2017 *)

PROG

(PARI) a(n)=precprime(2*n) \\ Charles R Greathouse IV, May 24 2013

(Haskell)

a060308 = a007917 . a005843 -- Reinhard Zumkeller, May 25 2013

(Magma) [NthPrime(#PrimesUpTo(2*n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

CROSSREFS

Apart from initial term, same as A060265.

Cf. A007917 (largest prime <= n), A005843 (2n).

Cf. A020482, A049653, A060266, A060267, A060268, A060264.

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Mar 27 2001

EXTENSIONS

More terms from Alexander Adamchuk, Jul 11 2006

STATUS

approved

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