A389201 a(n) = A351569(A228058(n)), where A351569(x) is the sum of divisors of the largest unitary divisor of x that is an exponentially odd number, and A228058 gives the numbers satisfying Euler's condition for odd perfect numbers.

6, 14, 18, 6, 30, 14, 38, 42, 6, 18, 54, 62, 6, 14, 74, 30, 90, 18, 6, 98, 102, 38, 110, 114, 42, 14, 138, 54, 150, 18, 158, 30, 6, 62, 174, 14, 182, 194, 198, 6, 38, 74, 42, 18, 230, 234, 242, 6, 90, 258, 30, 270, 98, 278, 102, 282, 54, 294, 6, 110, 314, 114, 318, 18, 14, 62, 38, 338, 350, 354, 42, 374, 138, 390, 30

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) = sigma(p^(1+4k)) = ((p^(2+4k)) - 1) / (p - 1).

Links

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

Index entries for sequences related to sigma(n).

Formula

a(n) = A000203(A389161(n)) = A351569(A228058(n)).

a(n) = A389203(n) / A389200(n).

Example

A228058(520) = 28125 = 3^2 * 5^5, and the largest exponentially odd unitary divisor is 5^5 (its special factor), and sigma(5^5) = ((5^(1+5))-1) / (5-1) = 3906, therefore a(520) = 3906.

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];
A350389(n) = { my(m=1, f=factor(n)); for(k=1, #f~, if(1==(f[k, 2]%2), m *= (f[k, 1]^f[k, 2]))); (m); };
A351569(n) = sigma(A350389(n));
A389201(n) = A351569(A228058(n));

Crossrefs

Cf. A000203, A228058, A351569, A389161, A389200, A389203.

Sequence in context: A088017 A206524 A141082 * A108977 A260114 A340634

Adjacent sequences: A389198 A389199 A389200 * A389202 A389203 A389204

Keyword

nonn

Author

Antti Karttunen, Sep 28 2025

Status

approved