A141548 -id:A141548 - cp's OEIS frontend

A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

6, 3, 5, 7, 22, 11, 13, 27, 17, 19, 46, 23, 124, 58, 29, 31, 250, 57, 37, 55, 41, 43, 94, 47, 1264, 106, 53, 87, 118, 59, 61, 85, 134, 67, 142, 71, 73, 712, 158, 79, 166, 83, 405, 115, 89, 141, 406, 119, 97, 202, 101, 103, 214, 107, 109, 145, 113, 177, 418, 143

Offset: 0

Jayanta Basu, Mar 28 2013

nonn

a(0) = 6 corresponds to the smallest perfect number.
Is n = 144 the first number for which a(n) = 0? - T. D. Noe, Mar 28 2013
No, a(144) = 95501968. - Giovanni Resta, Mar 28 2013
We can instead compute k - sigma(k)/2 for increasing k, which is computationally much faster. In this case, we stop computing when all n have been found for a range of numbers. - T. D. Noe, Mar 28 2013
Also, the first number whose deficiency is 2n. This is the even bisection of A082730. Hence, the first number in the following sequences: A000396, A191363, A125246, A141548, A125247, A101223, A141549, A141550, A125248, A223608, A223607, A223606. - T. D. Noe, Mar 29 2013
10^12 < a(654) <= 618970019665683124609613824. - Donovan Johnson, Jan 04 2014

			a(4)=22, since 22 is the least number such that sigma(22)=36=2*(22-4).

Donovan Johnson, Table of n, a(n) for n = 0..653 (first 241 terms from T. D. Noe)
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.

Cf. A000203, A000396.

Cf. A087998 (negative n).

Table[Min[Select[Range[2000], DivisorSigma[1, #] == 2*(# - i) &]], {i, 0, 60}]
nn = 144; t = Table[0, {nn}]; k = 0; While[k++; Times @@ t == 0, s = (2*k - DivisorSigma[1, k])/2; If[s >= 0 && s < nn && IntegerQ[s] && t[[s + 1]] == 0, t[[s + 1]] = k]]; t (* T. D. Noe, Mar 28 2013 *)

A162302 Numbers n such that (A000203(n)+28)/n is an integer.

Original entry on oeis.org

1, 28, 29, 62, 84, 182, 230, 252, 344, 756, 944, 2268, 6710, 6804, 20264, 20412, 36224, 61236, 183708, 538112, 551124, 1653372, 2085710, 4960116, 14503550, 14880348, 33665024, 44641044, 55328384, 133923132, 134438912, 401769396, 615206030, 1082574464

Offset: 1

Ctibor O. Zizka, Jun 30 2009

nonn

Contains the subset of all n of the form 28*3^k.
Generalized sequences are defined by A*A000203(n)+ B = C*n with A,B,C integers.
Then we get for different settings of A, B, C hyperperfect numbers:
A=1, C=2, B=0 gives A000396. A=1, C=2, B=1 gives A000079.
A=1, C=2, B=2 gives A056006. A=1, C=2, B=4 gives A125246. A=1, C=2, B=6 gives A141548.
A=1, C=2, B=8 gives A125247. A=1, C=2, B=10 gives A101223. A=1, C=2, B=12 gives A141549.
A=1, C=2, B=14 gives A141550. A=1, C=2, B=16 gives A125248. A=1, C=2, B=0 gives A000396.
A=1, C=2, B=0 gives A000396. A=1, C=3, B=0 gives A005820.
Not in the OEIS: A=1, C=3, B=12,18,28,... A=2, C=3, B=21,27,33,45,... A=3, C=4, B=20,...
Terms not of the form 28*3^n: 1, 29, 62, 182, 230, 344, 944, 6710, 20264, 36224, 538112, 2085710, 14503550, 33665024, 55328384, ..., . [Robert G. Wilson v, Sep 05 2010]

Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol.5 (1996), No. 2, pp.91-100.

Cf. A000203, A000396, A000079, A005820, A056006, A125246, A141548, A125247, A101223, A141549, A141550, A125248,

A000203 := proc(n) numtheory[sigma](n) ; end proc:
isA152302 := proc(n) (A000203(n)+28) mod n = 0 ; end proc:
for n from 1 to 1000000 do if isA152302(n) then printf("%d,",n) ; end if ; end do: # R. J. Mathar, Aug 25 2010

fQ[n_] := Divisible[ DivisorSigma[1, n] + 28, n]; lst = {}; k = 1; While[k < 10^9/4, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Sep 05 2010 *)

Missing terms (1, 29, 182,..) inserted, 7 terms added, comment corrected - R. J. Mathar, Aug 25 2010
a(22)-a(30) from Robert G. Wilson v, Sep 05 2010
a(31)-a(34) from Donovan Johnson, Nov 03 2011

A302125 Numbers whose deficiency is a perfect number.

Original entry on oeis.org

7, 15, 29, 52, 62, 182, 230, 315, 344, 592, 944, 998, 1155, 2012, 2570, 4028, 6710, 15128, 19688, 20264, 30248, 36224, 38252, 40730, 43964, 52088, 90332, 96128, 168116, 195224, 258512, 262112, 451952, 538112, 991904, 1209632, 1237856, 1659128, 2080544, 2085710, 2102272, 2186132

Offset: 1

Robert G. Wilson v and Waldemar Puszkarz, Apr 01 2018

nonn

Subsequence of deficient numbers (A005100) whose deficiency (A033879) is a member of perfect numbers (A000396).

			52 is in the sequence since the divisors of 52 are {1, 2, 4, 13, 26 & 52} so d(52) = 98 and 2*52 - 98 = 6, a perfect number.

Cf. A000396 (perfect numbers), A033879 (deficiency), A005100 (deficient numbers), A141548 (subsequence), A301859 (related sequence).

fQ[n_] := PerfectNumberQ[2n - DivisorSigma[1, n]]; Select[ Range@ 2500000, fQ]

for(n=1,25*10^5, d=2*n-sigma(n); d>0&&sigma(d)==2*d&&print1(n ","))

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A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

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A162302 Numbers n such that (A000203(n)+28)/n is an integer.

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A302125 Numbers whose deficiency is a perfect number.

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