A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

1, 2, 3, 6, 7, 10, 14, 15, 20, 21, 22, 24, 27, 28, 30, 31, 33, 34, 40, 42, 46, 54, 57, 60, 62, 66, 69, 70, 84, 87, 91, 93, 94, 102, 105, 106, 110, 114, 120, 127, 130, 138, 140, 141, 142, 154, 160, 168, 170, 174, 177, 182, 186, 189, 190, 195, 198, 210, 214, 216, 217, 220, 224, 230, 231, 237, 238, 254, 260, 264, 270, 273

Offset: 1

Author

Antti Karttunen, Feb 16 2022

Keywords

• nonn
• changed

Comments

Numbers k for which A351555(k) = 0. This is a necessary condition for the terms of A349169 and of A349745, therefore they are subsequences of this sequence.

All six known 3-perfect numbers (A005820) are included in this sequence.

All 65 known 5-multiperfects (A046060) are included in this sequence.

Moreover, all multiperfect numbers (A007691) seem to be in this sequence.

From Antti Karttunen, Aug 27 2025: (Start)

Multiperfect number m is included in this sequence only if its abundancy sigma(m)/m has only such odd prime factors p that prevprime(p) [A151799] divides m for each p. E.g., all 65 known 5-multiperfects are multiples of 3, and all known terms of A005820 and A046061 are even.

This sequence contains natural numbers k such that the odd primes in the prime factorization of sigma(k) have the same valuation there as in k, except that the primes in A003961(k) [or equally in A003961(A007947(k))] stand for "don't care primes", that are "masked off" from the comparison.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)
• Index entries for sequences where odd perfect numbers must occur, if they exist at all

Crossrefs

Cf. A000203, A003961, A151799, A351546.

Positions of zeros in A351555.

Subsequences: A000396, A351553 (even terms), A386430 (odd terms), A351551, A349169, A349745, A387160 (terms of the form prime * m^2), also these, at least all the currently (Feb 2022) known terms: A005820, A007691, A046060.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351554(n) = (0==A351555(n));

PARI

isA351554(n) = { my(sh=A351546(n),f=factor(sh)); for(i=1,#f~, if((f[i,1]%2)&&valuation(n,f[i,1])!=f[i,2],return(0))); (1); }; \\ Uses also program given in A351546.

Extensions

Definition corrected by Antti Karttunen, Aug 22 2025