cp's OEIS Frontend

A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

%I A349756 #19 Jul 28 2022 21:15:44
%S A349756 1,2,3,6,7,14,20,21,24,27,31,42,54,57,60,62,93,114,120,127,140,160,
%T A349756 168,186,189,216,217,220,237,254,264,301,378,381,399,408,420,434,460,
%U A349756 474,480,513,540,552,602,620,651,660,744,762,792,798,837,840,889,903,940,1026,1080,1120,1128,1140,1302,1320,1380,1392,1512
%N A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.
%C A349756 Numbers k for which A161942(k) = A342671(k).
%C A349756 From _Antti Karttunen_, Jul 23 2022: (Start)
%C A349756 Numbers k such that k is a multiple of A350073(k).
%C A349756 For any square s in this sequence, A349162(s) = 1, i.e. sigma(s) divides A003961(s), and also A286385(s). Question: Is 1 the only square in this sequence? (see the conjecture in A350072).
%C A349756 If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
%C A349756 After 2, the only primes present are Mersenne primes, A000668.
%C A349756 (End)
%H A349756 Antti Karttunen, <a href="/A349756/b349756.txt">Table of n, a(n) for n = 1..23008; terms < 2^31</a>
%H A349756 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349756 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%t A349756 f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; oddpart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := oddpart[(sigma = DivisorSigma[1, n])] == GCD[sigma, s[n]]; Select[Range[1500], q] (* _Amiram Eldar_, Dec 04 2021 *)
%o A349756 (PARI)
%o A349756 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349756 A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };
%o A349756 isA349756(n) = A355946(n);
%Y A349756 Positions of 1's in A348992.
%Y A349756 Positions where the powers of 2 (A000079) occur in A349162.
%Y A349756 Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).
%Y A349756 Cf. A000668, A046528 (subsequences).
%Y A349756 Cf. also A348943.
%K A349756 nonn
%O A349756 1,2
%A A349756 _Antti Karttunen_, Dec 03 2021