A100696 -id:A100696 - cp's OEIS frontend

A182225 Numbers m such that the sum of their divisors smaller than A033880(m) is greater than A033880(m).

12, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 272, 276, 280, 288, 294, 300, 304, 306, 308, 312

Offset: 1

M. F. Hasler, Apr 19 2012

nonn

Motivated by A100696. Subsequence of A005101: all terms are abundant.

Amiram Eldar, Table of n, a(n) for n = 1..10000

aQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && DivisorSum[n, # &, # < ab &] > ab; Select[Range[312], aQ] (* Amiram Eldar, Sep 08 2019 *)

is_A182225(n)=my(L=sigma(n)-2*n,s=L);fordiv(n,d,d

A138850 Weird numbers (A006037) not divisible by 5.

Original entry on oeis.org

836, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 222952, 243892, 254012, 338572, 343876, 351956, 388076, 407132, 410476, 465652, 470668, 475684, 477356, 482372, 490732, 495748, 500764, 502436, 507452, 512468, 515812

Offset: 1

M. F. Hasler, Apr 02 2008

nonn

Up to 200000, there are only 11 weird numbers not divisible by 5.
Since no A006037(k) < 10^17 is odd, at least up to there, "divisible by 5" is equivalent to "ending in 0" (in base 10).
It appears that 4*11*19*p is an element of this sequence for p=1 and all primes p>547. Moreover, these seem to comprise most of the terms of this sequence.
Up to n=500, the only indices for which a(n) is not of this form are n=2,...,16, 18, 34, 38, 43, 64, 83, 148, 158, 236, 266, 296, 310.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)

Cf. A006037, A100696.

t=0; A138850=vector(50,i,until( (t+=2)%5 && is_A006037(t),);t) \\ [Updated by M. F. Hasler, Jan 07 2014]

A357050 Number of ways A005101(n)+1 can be written as sum of a subset of the proper divisors of A005101(n), the n-th abundant number.

Original entry on oeis.org

2, 1, 1, 4, 4, 7, 2, 2, 10, 2, 2, 32, 2, 1, 26, 1, 6, 24, 1, 19, 20, 2, 1, 1, 20, 4, 1, 270, 11, 14, 1, 14, 116, 12, 9, 12, 3, 195, 1, 2, 719, 1, 42, 1, 8, 9, 8, 2, 148, 142, 6, 1, 8, 6, 6, 2154, 1, 534, 1, 6, 125, 108, 1, 6, 117, 1, 447, 4

Offset: 1

M. F. Hasler, Dec 13 2022

nonn

Obviously, for non abundant numbers (including perfect numbers) N, there is no way to write N+1 as the sum of a subset of N's proper divisors. Therefore we consider only abundant N = A005101(n) here.

The first zero appears for the seventh weird and primitive weird number A006037(7) = A002975(7) = 9272 = A005101(2310) (which surprisingly is w = A100696(1), the first weird number such that the sum of its divisors less than its abundance A033880(w) is larger than that).

Cf. A005101, A006037, A002975, A005835 (abundant, weird, primitive weird and pseudoperfect numbers).

Cf. A033880 (abundance), A100696.

{A357050(n)= sum(b=1, -1+2^#d=divisors(n)[^-1], vecsum(vecextract(d,b))==n+1)} \\ not very efficient, better use code as in is_A005835().

A182227 Sum of divisors of the abundant number m = A005101(n) which are smaller than its abundance A033880(m).

Original entry on oeis.org

6, 3, 1, 24, 27, 55, 20, 19, 76, 21, 14, 108, 23, 3, 123, 12, 66, 140, 3, 144, 156, 22, 12, 1, 172, 52, 12, 240, 123, 204, 12, 126, 259, 147, 236, 138, 66, 312, 12, 42, 546, 12, 316, 7, 171, 165, 198, 44, 366, 384, 174, 12, 112, 218, 117, 744, 12, 476, 12, 198

Offset: 1

M. F. Hasler, Apr 19 2012

nonn

Motivated by A100696. See also A182225 and A182226.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from M. F. Hasler)

Cf. A005101, A033880, A100696, A182225, A182226.

f[n_] := If[(ab = DivisorSigma[1, n] - 2*n) > 0 ,DivisorSum[n, # &, # < ab &],Nothing]; Array[f, 300] (* Amiram Eldar, Apr 06 2024 *)

f(n)=my(A=sigma(n)-2*n,s);fordiv(n,d,(d2*n & print1(f(n)","))

cp's OEIS Frontend

A182225 Numbers m such that the sum of their divisors smaller than A033880(m) is greater than A033880(m).

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A138850 Weird numbers (A006037) not divisible by 5.

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A357050 Number of ways A005101(n)+1 can be written as sum of a subset of the proper divisors of A005101(n), the n-th abundant number.

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PARI

A182227 Sum of divisors of the abundant number m = A005101(n) which are smaller than its abundance A033880(m).

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Author

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Mathematica

PARI