A351545 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor p also divides A003961(n), and valuation(sigma(n),p) >= valuation(A003961(n),p).

1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 9, 1, 5, 1, 3, 1, 1, 1, 9, 1, 1, 1, 27, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 5, 9, 1, 7, 1, 3, 1, 1, 7, 9, 1, 9, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 3, 5, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 9, 13, 1, 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, A351544, A351547.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351545(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); prod(k=1,#f~,if(!(u%f[k,1]) && (f[k,2]>=valuation(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) but p^(1+e) does not divide A003961(n)]), where [ ] is the Iverson bracket, returning 1 if the condition holds, and 0 otherwise. Here p^e is the largest power of prime p dividing sigma(n).

a(n) = A000203(n) / A351547(n).

For all n >= 1, a(n) is a divisor of A351544(n), which is a divisor of A000203(n).