A351541 Even numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k), but p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

364, 760, 1092, 1148, 1160, 1358, 1490, 1782, 1990, 2324, 2360, 2716, 2912, 2980, 3160, 3276, 3388, 3430, 3444, 3490, 3560, 3564, 3892, 3980, 4004, 4074, 4102, 4360, 4490, 4676, 4990, 5068, 5302, 5320, 5432, 5510, 5560, 5960, 5990, 6188, 6244, 6804, 6860, 6916, 6972, 6980, 7028, 7128, 7160, 7462, 7960, 8120, 8148

Offset

1,1

Comments

Even numbers k that have an odd prime factor prime(i) such that prime(i-1) is not a factor of k, and prime(i)^(1+A286561(k,prime(i))) divides sigma(k).

Links

Table of n, a(n) for n=1..53.

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Example

364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Mathematica

Select[Range[2, 8200, 2], Function[{k, s, facs, t}, AnyTrue[DeleteCases[facs[[All, 1]], 2], And[Mod[s, #^(1 + IntegerExponent[k, #])] == 0, Mod[t, #] != 0] &]] @@ {#1, #2, #3, Apply[Times, (NextPrime[#1])^#2 & @@@ #3]} & @@ {#, DivisorSigma[1, #], FactorInteger[#]} &] (* Michael De Vlieger, Feb 16 2022 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Aux351541(n) = { my(f=factor(n), s=sigma(n), u=A003961(n)); sum(k=1, #f~, (f[k, 1]%2) && 0!=(u%f[k, 1]) && (0==(s%(f[k, 1]^(1+f[k, 2]))))); };
isA351541(n) = (!(n%2) && Aux351541(n)>0);

Crossrefs

Cf. A000203, A003961.

Subsequence of A351540, and of A351542 and of A351543.

Sequence in context: A256087 A260837 A272359 * A023697 A038468 A131348

Adjacent sequences: A351538 A351539 A351540 * A351542 A351543 A351544

Keyword

nonn

Author

Antti Karttunen, Feb 16 2022

Status

approved