A349166 Numbers k such that sigma(k) and A003961(k) share a prime factor, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

2, 6, 8, 10, 14, 18, 20, 22, 24, 26, 27, 30, 32, 34, 38, 40, 42, 44, 46, 50, 54, 56, 57, 58, 60, 62, 65, 66, 68, 70, 72, 74, 78, 80, 82, 86, 87, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 130, 132, 134, 135, 136, 138, 140, 142, 146, 150, 152, 154, 158, 160, 162, 164

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn

Comments

The only prime term is 2. A prime power prime(j)^k with k > 1 is a term if and only if k+1 is divisible by the multiplicative order of prime(j) mod prime(j+1). - Robert Israel, May 22 2025

Examples

			For n = 2, A000203(2) = A003961(2) = 3, therefore they share a prime factor 3, and 2 is included in this sequence.
For n = 10 = 2*5, sigma(10) = 18 = 2 * 3^2, while A003961(10) = 21 = 3*7, therefore 10 is included, as there is a shared prime factor (3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961.

Positions of terms larger than ones in A342671, and also in A349163.

Positions of zeros in A349167.

Cf. A349165 (complement), A349168 (subsequence).

Programs

Maple

filter:= proc(n) local F,a,b,t;
   F:= ifactors(n)[2];
   b:= convert(map(nextprime,F[..,1]),`*`);
   a:= mul((t[1]^(t[2]+1)-1)/(t[1]-1),t=F);
   igcd(a,b) <> 1
end proc;
select(filter, [$1..1000]); # Robert Israel, May 21 2025

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349166(n) = (1!=gcd(sigma(n), A003961(n)));