A325814 -id:A325814 - cp's OEIS frontend

A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

1, 3465, 72981, 78651, 80937, 152703, 199341, 201771, 241605, 253287, 492507, 631881, 880821, 933147, 985473, 1063755, 1209285, 1244133, 1292445, 1313235, 1327095, 1347885, 1360881, 1451835, 1521135, 1597365, 1620375, 1814373, 2015475, 2664585, 6058233, 6676371, 8186751, 11119761, 17496243, 18379935, 28695627

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977(k) and A325978(k) are never zero for the same k, these are odd numbers k such that A325978(k) is not zero and divides A325977(k).

Of the first 281 terms, only a(5) = 80937, a(51) = 86086881, a(175) = 43024468437, and a(262) = 564858541521 are in A228058. - Updated Jul 20 2025

Giovanni Resta, Table of n, a(n) for n = 1..281 (terms < 10^12; first 65 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur.

Cf. A228058, A325975, A325977, A325978, A325981.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325979(n) = ((n%2)&&(A325975(n)==abs(A325978(n))));
\\ Or alternatively as:
isA325979(n) = if(!(n%2),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(y&&!(x%y)));

A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

Original entry on oeis.org

1, 6, 12, 28, 56, 60, 108, 120, 132, 168, 264, 280, 312, 408, 420, 440, 456, 496, 528, 540, 552, 696, 700, 728, 744, 756, 760, 888, 984, 992, 1032, 1128, 1140, 1188, 1272, 1404, 1416, 1456, 1464, 1608, 1704, 1710, 1752, 1836, 1896, 1992, 2052, 2136, 2328, 2424, 2472, 2484, 2568, 2616, 2646, 2712, 3048, 3132, 3144, 3288, 3336, 3344

Offset: 1

Antti Karttunen, May 23 2019

nonn

Numbers k for which A325813(k) is equal to abs(A325814(k)).
Numbers k such that A325814(k) is not zero (not in A064591) and divides A034460(k).
Conjecture: after the initial one all other terms are even. If this holds then there are no odd perfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences where any odd perfect numbers must occur

Cf. A034448, A034460, A048146, A064591, A325813, A325814, A325822.

Cf. A000396 (a subsequence).

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));
isA325812(n) = (A325813(n)==abs(A325814(n)));
\\ Alternatively:
isA325812(n) = (A325814(n) && !(A034460(n)%A325814(n)));

cp's OEIS Frontend

A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

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