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URL: https://oeis.org/A389161

⇱ A389161 - OEIS

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A389161
a(n) = A350389(A228058(n)), where A350389(x) is 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.
7
5, 13, 17, 5, 29, 13, 37, 41, 5, 17, 53, 61, 5, 13, 73, 29, 89, 17, 5, 97, 101, 37, 109, 113, 41, 13, 137, 53, 149, 17, 157, 29, 5, 61, 173, 13, 181, 193, 197, 5, 37, 73, 41, 17, 229, 233, 241, 5, 89, 257, 29, 269, 97, 277, 101, 281, 53, 293, 5, 109, 313, 113, 317, 17, 13, 61, 37, 337, 349, 353, 41, 373, 137, 389
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) = p^(1+4k).
Unlike in related A389165 here all terms are not primes. The first composite occurs at a(520) = 3125. See A386428.
LINKS
FORMULA
a(n) = A228058(n) / A389160(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);
xA228058(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); };
CROSSREFS
Cf. A228058, A350389, A386428, A389160, A389201 [= sigma(a(n))].
Differs from A389165 first at n=520, where a(520) = 3125, while A389165(520) = 5.
Sequence in context: A158334 A282747 A088908 * A389165 A386314 A327638
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 28 2025
STATUS
approved