A007593 -id:A007593 - cp's OEIS frontend

A028501 18-hyperperfect numbers: n = 18*(sigma(n)-n-1) + 1.

Original entry on oeis.org

1333, 1909, 2469601, 893748277, 322685352001, 8992165119733, 42052982615431201

Offset: 1

Jud McCranie

1978419655654505210563957 are also terms. - Donovan Johnson, Nov 20 2012

a(8) > 10^17. - Max Alekseyev, Nov 17 2019

Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.

Cf. A000396, A007593, A220290, A028499, A028500, A028502, A034916.

a(5) from Donovan Johnson, Nov 03 2011

a(6)-a(7) from Donovan Johnson confirmed by Max Alekseyev, Nov 21 2019

A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

Original entry on oeis.org

15, 63, 147, 171, 207, 627, 663, 1023, 1647, 1971, 2975, 6399, 18063, 19359, 27639, 40215, 48895, 58563, 78819, 95511, 114231, 133595, 134871, 145915, 147455, 163539, 168507, 172287, 188067, 529983, 680859, 795639, 1207359, 1238571, 1553499, 1588491, 2049219

Offset: 1

Amiram Eldar, Jun 06 2018

nonn

Supersequence of A063906.

A combination of the notions of 2-hyperperfect numbers (A007593) and near-perfect numbers (A181595).

			15 is in the sequence since sigma(15) = 24 and 24 - 3*15/2 - 1/2 = 1 is a proper divisor of 15.

Amiram Eldar, Table of n, a(n) for n = 1..70
Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134.

Cf. A000203, A007593, A063906, A181595, A305617.

aQ[n_] := Module[{d=DivisorSigma[1, n]-3n/2-1/2}, d>0 && d!=n && IntegerQ[d] && Divisible[n,d]]; Select[Range[1000000], aQ]

isok(n) = (n % 2) && (k = sigma(n) - (3*n+1)/2) && (k>0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018

A305617 Deficient 2-hyperperfect numbers: numbers k such that 3*k/2 + 1/2 - sigma(k) is a proper divisor of k.

Original entry on oeis.org

3, 9, 27, 35, 39, 55, 81, 243, 279, 387, 715, 729, 1443, 2187, 2619, 3655, 5635, 6561, 10855, 12635, 19683, 59049, 77283, 177147, 178119, 294759, 443135, 531441, 817167, 1170723, 1594323, 1605987, 1632231, 1710963, 1947159, 2410239, 2624375, 2655747, 3944255

Offset: 1

Amiram Eldar, Jun 06 2018

nonn

Includes all the powers of 3 (A000244).

A combination of the notions 2-hyperperfect numbers (A007593) and deficient-perfect numbers (A271816).

			35 is in the sequence since sigma(35) = 48 and 3*35/2 + 1/2 - 48 = 5 is a proper divisor of 35.

Amiram Eldar, Table of n, a(n) for n = 1..74
Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134.

Cf. A000203, A000244, A007593, A271816, A305616.

aQ[n_] := Module[{d = 3n/2+1/2-DivisorSigma[1,n]}, d>0 && d!=n && IntegerQ[d] && Divisible[n,d]]; Select[Range[2,1000000], aQ]

isok(n) = (n % 2) && (k = (3*n+1)/2 - sigma(n)) && (k > 0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018, corrected by Amiram Eldar, Dec 23 2024

A062967 a(n) = 2*(sigma(n)-n-1)+1, where sigma = A000203, sum of divisors of n.

Original entry on oeis.org

-1, 1, 1, 5, 1, 11, 1, 13, 7, 15, 1, 31, 1, 19, 17, 29, 1, 41, 1, 43, 21, 27, 1, 71, 11, 31, 25, 55, 1, 83, 1, 61, 29, 39, 25, 109, 1, 43, 33, 99, 1, 107, 1, 79, 65, 51, 1, 151, 15, 85, 41, 91, 1, 131, 33, 127, 45, 63, 1, 215, 1, 67, 81, 125, 37, 155, 1, 115, 53, 147, 1, 245, 1, 79, 97, 127, 37, 179, 1, 211, 79, 87, 1, 279, 45, 91, 65

Offset: 1

Jason Earls, Jul 23 2001

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000203, A001065, A007593.

Array[2 (DivisorSigma[1, #] - # - 1) + 1 &, 87] (* Michael De Vlieger, Oct 30 2017 *)

j=[]; for(n=1,200,j=concat(j,2*(sigma(n)-n-1)+1)); j

a(n) = 2*A001065(n) - 1. - Antti Karttunen, Oct 30 2017

cp's OEIS Frontend

A028501 18-hyperperfect numbers: n = 18*(sigma(n)-n-1) + 1.

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A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

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A305617 Deficient 2-hyperperfect numbers: numbers k such that 3*k/2 + 1/2 - sigma(k) is a proper divisor of k.

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A062967 a(n) = 2*(sigma(n)-n-1)+1, where sigma = A000203, sum of divisors of n.

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