A387166 -id:A387166 - cp's OEIS frontend

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

117, 153, 333, 369, 425, 477, 549, 637, 657, 845, 873, 909, 925, 1017, 1053, 1233, 1325, 1377, 1413, 1421, 1445, 1525, 1557, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2097, 2169, 2225, 2313, 2493, 2525, 2529, 2597, 2637, 2725, 2817, 2825, 2853, 2989, 2997, 3033, 3177, 3321, 3357, 3425, 3509, 3573, 3577, 3609, 3725

Offset: 1

Antti Karttunen, Aug 28 2025

Terms k of A228058 for which A322361(k) = A342671(k), or equally, such that A319626(k) = A349164(k).

Intersection of A228058 and A349174.
Union of A387166 and A387167.
Differs from its subsequence A387167 for the first time at n=201, where a(201) = 14157, while A387167(201) = 14225.
Cf. A000203, A003961, A319626, A322361, A342671, A349164.
Cf. also A371082.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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));
isA349174(n) = if(!(n%2), 0, my(u=A003961(n)); gcd(u, sigma(n))==gcd(u, n));
isA387164(n) = (isA228058(n) && isA349174(n));

A387167 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) = 1, and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

Offset: 1

Antti Karttunen, Aug 28 2025

Terms k of A228058 for which k and A003961(k) are relatively prime, and also sigma(k) and A003961(k) are coprime.

Intersection of A228058 and A349177.
Intersection of A387164 and A319630, or equally, intersection of A387164 and A349165.
Setwise difference A387164 \ A387166.
Cf. A000203, A003961.
Subsequence of A387164 from which this differs for the first time at n=201, where a(201) = 14225, while A387164(201) = 14157.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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));
isA349177(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (1==t)&&(gcd(u,sigma(n))==t));
isA387167(n) = (isA228058(n) && isA349177(n));

cp's OEIS Frontend

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

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