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

1,1

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

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)

Example

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).

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

Prog

(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)));

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).

Sequence in context: A075332 A141105 A047395 * A284794 A187692 A036554

Adjacent sequences: A349163 A349164 A349165 * A349167 A349168 A349169

Keyword

nonn

Author

Antti Karttunen, Nov 09 2021

Status

approved