A351554 - OEIS

%I #16 Feb 25 2022 17:02:59

%S 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 62,66,69,70,84,87,91,93,94,102,105,106,110,114,120,127,130,138,140,

%U 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 Numbers k such that there are no odd prime factors p of k such that p would not divide A003961(k) and the valuation(k, p) would be 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 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 All six known 3-perfect numbers (A005820) are included in this sequence.

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

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

%H Antti Karttunen, <a href="/A351554/b351554.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o 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 isA351554(n) = (0==A351555(n));

%Y Cf. A000203, A003961.

%Y Positions of zeros in A351555.

%Y Subsequences: A351553 (even terms), A351551, A349169, A349745, also these, at least all the currently (Feb 2022) known terms: A005820, A007691, A046060.

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 16 2022