

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.


5



1, 2, 3, 6, 7, 14, 20, 21, 24, 27, 31, 42, 54, 57, 60, 62, 93, 114, 120, 127, 140, 160, 168, 186, 189, 216, 217, 220, 237, 254, 264, 301, 378, 381, 399, 408, 420, 434, 460, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Numbers k such that k is a multiple of A350073(k).
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).
If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
After 2, the only primes present are Mersenne primes, A000668.
(End)


LINKS



MATHEMATICA

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 *)


PROG

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s, 2)))); };


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



