A389160 a(n) = A350388(A228058(n)), where A350388(x) is the largest unitary divisor of x that is a square, and A228058 gives the numbers satisfying Euler's condition for odd perfect numbers.

9, 9, 9, 49, 9, 25, 9, 9, 81, 25, 9, 9, 121, 49, 9, 25, 9, 49, 169, 9, 9, 25, 9, 9, 25, 81, 9, 25, 9, 81, 9, 49, 289, 25, 9, 121, 9, 9, 9, 361, 49, 25, 49, 121, 9, 9, 9, 441, 25, 9, 81, 9, 25, 9, 25, 9, 49, 9, 529, 25, 9, 25, 9, 169, 225, 49, 81, 9, 9, 9, 81, 9, 25, 9, 121, 9, 49, 9, 729, 9, 25, 289, 9, 225, 9, 25

Offset

1,1

Comments

For all odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m (so called "special prime"), r > 1, and gcd(p,r) = 1, a(n) = r^2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = A228058(n) / A389161(n).

Prog

(PARI)
up_to = 20000;
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(k<up_to, n++; if(isA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n];
A350388(n) = { my(m=1, f=factor(n)); for(k=1, #f~, if(0==(f[k, 2]%2), m *= (f[k, 1]^f[k, 2]))); (m); };
A389160(n) = A350388(A228058(n));

Crossrefs

Cf. A228058, A350388, A389161, A389200 [= sigma(a(n))].

Differs from A325320 at n = 9, 26, 30, 48, 51, 65, 67, 71, 79, 84, 91, 105, ...

Differs from A389166 first at n=520, where a(520) = 9, while A389166(520) = 5625.

Sequence in context: A160761 A082049 A282242 * A389166 A325320 A282910

Adjacent sequences: A389157 A389158 A389159 * A389161 A389162 A389163

Keyword

nonn

Author

Antti Karttunen, Sep 28 2025

Status

approved