A082730 -id:A082730 - cp's OEIS frontend

A082731 a(n) is the smallest number k such that A033880(k)= n, or 0 if no such number exists, where A033880 is the abundance of k.

6, 0, 20, 18, 12, 0, 8925, 196, 56, 0, 40, 0, 24, 0, 272, 0, 550, 100, 208, 36, 176, 0, 1312, 0, 112, 0, 80, 0, 48, 0, 945, 15376, 572, 0, 928, 0, 2205, 0, 5696, 162, 736, 1352, 9555, 0, 350, 0, 490, 0, 60, 0, 416, 72, 352, 0, 90, 0, 84, 0, 160, 968, 96, 0, 24704, 0, 108, 200

Offset: 0

Amarnath Murthy, Apr 14 2003

nonn

Caution: so far a(n)=0 only indicates no k < 3*10^6 exists; nonexistence is not proved. - R. J. Mathar, Jul 26 2007

For each term listed as 0 in the Data section, there is no such k < 10^14. - Jon E. Schoenfield, Jan 12 2021

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A082730.

A082731 := proc(n) local k; k := 1 ; while numtheory[sigma](k)-2*k <> n do k := k+1 ; if k = 3000000 then RETURN(0) ; fi ; od ; RETURN(k) ; end: seq(A082731(n),n=0..200) ; # R. J. Mathar, Nov 07 2016

More terms from R. J. Mathar, Jul 26 2007

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A082731 a(n) is the smallest number k such that A033880(k)= n, or 0 if no such number exists, where A033880 is the abundance of k.

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