A349165 Numbers k such that sigma(k) and A003961(k) are relatively prime, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 21, 23, 25, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 51, 52, 53, 55, 59, 61, 63, 64, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 84, 85, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 112, 113, 115, 117, 119, 121, 123, 124, 125, 127, 129, 131, 133, 137, 139

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn

Comments

Includes all odd primes. A prime power prime(j)^k with k > 1 is a term if and only if k+1 is not 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)

Crossrefs

Cf. A000203, A003961.

Positions of ones in A342671, and also in A349163.

Cf. A349166 (complement), A349167 (characteristic function).

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); };
isA349165(n) = (1==gcd(sigma(n), A003961(n)));