This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351554 #33 Aug 27 2025 18:21:56
%S A351554 1,2,3,6,7,10,14,15,20,21,22,24,27,28,30,31,33,34,40,42,46,54,57,60,
%T A351554 62,66,69,70,84,87,91,93,94,102,105,106,110,114,120,127,130,138,140,
%U A351554 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
%N 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.
%C A351554 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.
%C A351554 All six known 3-perfect numbers (A005820) are included in this sequence.
%C A351554 All 65 known 5-multiperfects (A046060) are included in this sequence.
%C A351554 Moreover, all multiperfect numbers (A007691) seem to be in this sequence.
%C A351554 From _Antti Karttunen_, Aug 27 2025: (Start)
%C A351554 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.
%C A351554 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.
%C A351554 (End)
%H A351554 Antti Karttunen, <a href="/A351554/b351554.txt">Table of n, a(n) for n = 1..20000</a>
%H A351554 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A351554 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H A351554 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%o A351554 (PARI)
%o A351554 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A351554 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)); };
%o A351554 isA351554(n) = (0==A351555(n));
%o A351554 (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.
%Y A351554 Cf. A000203, A003961, A151799, A351546.
%Y A351554 Positions of zeros in A351555.
%Y A351554 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.
%K A351554 nonn,changed
%O A351554 1,2
%A A351554 _Antti Karttunen_, Feb 16 2022
%E A351554 Definition corrected by _Antti Karttunen_, Aug 22 2025