A351544 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor also divides A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 9, 1, 15, 1, 3, 5, 1, 1, 9, 1, 9, 1, 27, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 9, 1, 21, 1, 3, 1, 1, 7, 9, 1, 9, 1, 9, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 5, 9, 1, 9, 1, 3, 1, 9, 1, 9, 1, 9, 13, 7, 1, 27

Offset: 1

Author

Antti Karttunen, Feb 16 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A351545, A351546.

Cf. A342671, A349161, A349162.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351544(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); prod(k=1,#f~,if(!(u%f[k,1]), f[k,1]^f[k,2], 1)); };

Formula

a(n) = Product_{p^e || A000203(n)} p^(e*[p divides A003961(n)]), where [ ] is the Iverson bracket, returning 1 if p is a divisor of A003961(n), and 0 otherwise. Here p^e is the largest power of prime p dividing sigma(n).

a(n) = A000203(n) / A351546(n).

For all n >= 1, a(n) is a multiple of A351545(n).