A349756 - OEIS

%I #19 Jul 28 2022 21:15:44

%S 1,2,3,6,7,14,20,21,24,27,31,42,54,57,60,62,93,114,120,127,140,160,

%T 168,186,189,216,217,220,237,254,264,301,378,381,399,408,420,434,460,

%U 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 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 Numbers k for which A161942(k) = A342671(k).

%C From _Antti Karttunen_, Jul 23 2022: (Start)

%C Numbers k such that k is a multiple of A350073(k).

%C 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 If both x and y are terms and gcd(x, y) = 1, then x*y is also present.

%C After 2, the only primes present are Mersenne primes, A000668.

%C (End)

%H Antti Karttunen, <a href="/A349756/b349756.txt">Table of n, a(n) for n = 1..23008; terms < 2^31</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>

%t 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 (PARI)

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

%o A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };

%o isA349756(n) = A355946(n);

%Y Positions of 1's in A348992.

%Y Positions where the powers of 2 (A000079) occur in A349162.

%Y Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).

%Y Cf. A000668, A046528 (subsequences).

%Y Cf. also A348943.

%K nonn

%O 1,2

%A _Antti Karttunen_, Dec 03 2021