OFFSET
1,1
COMMENTS
Nelson, Penney, & Pomerance call these "Aaron numbers" because 714 is Babe Ruth's lifetime home run record, Hank Aaron's 715th home run broke this record, and 714 and 715 have the same sum of prime divisors. - David W. Wilson
Number of terms < 10^n: 1, 4, 9, 19, 40, 139, 494, 1748, 6650, ..., . - Robert G. Wilson v, Jan 23 2012
REFERENCES
John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
P. Hoffman, The Man Who Loved Only Numbers, pp. 179-181, Hyperion, NY 1998.
J. Roberts, Lure of Integers, pp. 250, MAA 1992.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 159-160, Penguin 1986.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6651 from Robert G. Wilson v)
Joe K. Crump, Ruth-Aaron Pairs-an algorithm
Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017).
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
Dana Mackenzie, Homage to an Itinerant Master, Science, Vol. 275 (1997), p. 759; alternative link.
C. Nelson, D. E. Penney and C. Pomerance, 714 and 715, J. Recreational Math. 7:2 (1994), pp. 87-89.
Ivars Peterson, Playing with Ruth-Aaron pairs
T. Trotter, Jr., Ruth-Aaron Numbers.
Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.
MAPLE
with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n);
for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od:
for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n, s[ n ]); fi; od:
# Alternative:
SumPF := proc(n) option remember; add(NumberTheory:-PrimeFactors(n)) end:
seq(ifelse(SumPF(n) = SumPF(n+1), n, NULL), n = 1..3000); # Peter Luschny, Jun 11 2024
MATHEMATICA
fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ] (* Robert G. Wilson v, Jan 22 2012 *)
PROG
(PARI) sopf(n)=my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1])
is(n)=sopf(n)==sopf(n+1) \\ Charles R Greathouse IV, Jan 27 2012
(Python)
from sympy import factorint
def aupton(terms):
alst, k, sopfk, sopfkp1 = [], 2, 2, 3
while len(alst) < terms:
if sopfkp1 == sopfk: alst.append(k)
k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in factorint(k+2))
return alst
print(aupton(42)) # Michael S. Branicky, May 24 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved