A324892 Multiplicative with a(p^e) = p^e if sigma(p^e) is prime, and 1 otherwise.

1, 2, 1, 4, 1, 2, 1, 1, 9, 2, 1, 4, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 1, 25, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 36, 1, 2, 1, 1, 1, 2, 1, 4, 9, 2, 1, 16, 1, 50, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 25, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 1, 1, 18, 1, 4, 1, 2, 1, 1, 1, 2, 9, 100, 1, 2, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Mar 29 2019

Keywords

• nonn
• mult

Examples

			For n = 150 = 2 * 3 * 5^2, sigma(2) = 3 is a prime, sigma(3) = 4 is not prime, and sigma(25) = 31 is a prime, thus a(150) = 2 * 25 = 50.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A010051, A324894.

Programs

PARI

A324892(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i,1]^isprime(sigma(f[i,1]^f[i,2])))^f[i,2]); };

PARI

A324892(n) = { my(f=factor(n)); prod(i=1, #f~, if(isprime(((f[i,1]^(1+f[i,2]))-1)/(f[i,1]-1)), f[i,1]^f[i,2], 1)); };

Formula

Multiplicative with a(p^e) = p^e if (p^(1+e) - 1)/(p-1) = 1 + p + p^2 + ... + p^e is prime, and 1 otherwise.

a(n) = n / A324894(n).