A389166 a(n) is the largest square dividing the numbers satisfying Euler's condition for odd perfect numbers (A228058).

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"), k >= 0, r > 1, and gcd(p,r) = 1, a(n) = p^(4k) * r^2.

Links

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

Formula

a(n) = A008833(A228058(n)).

a(n) = A228058(n) / A389165(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];
A008833(n) = n/core(n)
A389166(n) = A008833(A228058(n));

Crossrefs

Cf. A008833, A228058, A389165.

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

Differs from A389160 first at n=520, where a(520) = 5625, while A389160(520) = 9.

Sequence in context: A082049 A282242 A389160 * A325320 A282910 A282599

Adjacent sequences: A389163 A389164 A389165 * A389167 A389168 A389169

Keyword

nonn

Author

Antti Karttunen, Sep 28 2025

Status

approved