A347383 - OEIS

A347383

Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

189, 1271, 2125, 9261, 63767, 133907, 142859, 161257, 189209, 226967, 368063, 426373, 777923, 801727, 925101, 961193, 1003043, 4566661, 5244091, 5588327, 6031163, 6064439, 8135263, 8639879, 10074227, 10150571, 11234875, 12489107

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OFFSET

1,1

COMMENTS

Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.

Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. Proof: such an odd number cannot be a prime power (although it must be a square), thus it must have at least two nontrivial unitary divisors (with A034444(x) >= 4). Because sigma(x) = 3*x, it must be a term of A347391. From the illustration given there, we see that any odd square y in that sequence (i.e. with A347381(y)=1) would have an abundancy index of at least three (sigma(y)/y >= 3). But because abundancy index is multiplicative and always > 1 for n > 1, any nontrivial unitary divisor d of an odd triperfect number x must have sigma(d)/d < 3, thus for all such d, A347381(d) <> 1. And neither such divisor d can be a term of A336702, because 3*x is odd, therefore we must have A347381(d) > 1 for all nontrivial unitary divisors d of such a hypothetical x.

Any odd term of A000396, i.e., an odd perfect number, if such a hypothetical number exists, must also be a term of this sequence, by reasoning similar to above. See also illustration in A347392.

LINKS

Table of n, a(n) for n=1..28.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

EXAMPLE

189 is a term, because A347381(189) = 1, and the only way to factor 189 into nontrivial unitary divisors is 7*27, and A347381(7) = A347381(27) = 3 > 1.

63767 = 11^2 * 17 * 31 is a term, as its nontrivial unitary divisors are [17, 31, 121, 527, 2057, 3751], at which points A347381 obtains values [6, 10, 5, 11, 6, 8], every one which is larger than A347381(63767) = 3.

PROG

(PARI) isA347383(n) = if((1==n)||!(n%2)||isprimepower(n), 0, my(w=A347381(n)); fordiv(n, d, if((d>1)&&(d<n)&&(1==gcd(d, n/d)) && (A347381(d)<=w), return(0))); (1));

CROSSREFS

Cf. A000396, A005820, A005940, A034444, A336702, A347381, A347391, A347392.

Subsequence of A347390, which is a subsequence of A347384.

Sequence in context: A347390 A292173 A231394 * A297229 A211819 A266135

Adjacent sequences: A347380 A347381 A347382 * A347384 A347385 A347386

KEYWORD

nonn,hard,more

AUTHOR

Antti Karttunen, Sep 10 2021

STATUS

approved