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A324892
Multiplicative with a(p^e) = p^e if sigma(p^e) is prime, and 1 otherwise.
4
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
1,2
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).
EXAMPLE
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.
PROG
(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)); };
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Mar 29 2019
STATUS
approved