A387167 -id:A387167 - 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));

A387166 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

14157, 33525, 101025, 118825, 129605, 281025, 300713, 301725, 335405, 348525, 358925, 438525, 573525, 618525, 686025, 688205, 696725, 742577, 776025, 838125, 909225, 911025, 978525, 1046025, 1079225, 1099805, 1226025, 1293525, 1316025, 1322893, 1428889, 1451025, 1529045, 1563525, 1698525, 1721025, 1788525, 1991025, 2036025

Offset: 1

Antti Karttunen, Aug 28 2025

Intersection of A228058 and A349176.
Intersection of A387164 and A104210, or equally, intersection of A387164 and A349166.
Setwise difference A387164 \ A387167.
Cf. A000203, A003961.

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));
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));
isA387166(n) = (isA228058(n) && isA349176(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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