A234286 -id:A234286 - cp's OEIS frontend

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

1, 5, 9, 11, 13, 15, 21, 23, 25, 27, 29, 33, 35, 37, 43, 45

Offset: 1

N. J. A. Sloane, Dec 28 2013

The terms shown here are conjectural, based on a search up to 10^20 made by Davis et al. (2013).
Comments from Farideh Firoozbakht, Jan 12 2014: (Start)
1. Mersenne primes are not in this sequence. Because if M=2^p-1 is prime then M=sigma(m)-2m, where m=2^(p-1)*(2^p-1)^2=(1/2)*(M+1)*M^2 (please see Proposition 2.1 of Firoozbakht-Hasler, 2010).
2. If M = 2^p - 1 is a Mersenne prime then M^2 + 3M + 1 = 4^p + 2^p - 1 is not in the sequence. Because M^2 + 3M + 1 = sigma(m) - 2m where m = M^3 + M^2 = 2^p(2^p-1)^2 (please see Proposition 2.5, op. cit.).
Examples:
p = 2, M = 3, 4^p + 2^p - 1 = 19, m = M^3 + M^2 = 2^p(2^p-1)^2 = 36; sigma(m) - 2m = 19
p = 3, M = 7, 4^p + 2^p - 1 = 71, m = M^3 + M^2 = 2^p(2^p-1)^2 = 392; sigma(m) - 2m = 71
3. Note that if r is an even number and if for a number k p = 2^k - r - 1 is an odd prime then r = sigma(m) - 2m where m = 2^(k-1)*p. Namely r is not in the sequence (see Theorem 1.1, op. cit.).
It seems that for each even number r, there exists at least one odd prime of the form 2^k - r - 1. This means there is no even term in the sequence.
Moreover I conjecture that, for each even number r, there exist infinitely many primes p of the form 2^k - r - 1, or equivalently, I conjecture that: For each odd number s, there exists infinitely many primes p of the form 2^k - s.
Special cases:
(i): s = 1, there exist infinitely many Mersenne primes.
(ii): s = -1, there exist infinitely many Fermat primes.
(iii): s = 3, sequence A050414 is infinite.
(iv): s = -3, sequence A057732 is infinite.
(v): s = -5, sequence A059242 is infinite.
and so on. (End)
Cohen (1983) showed that 203^2 is not a term since sigma(m) - 2*m = 203^2 has a solution m = 742^2. - Max Alekseyev, Aug 29 2025

Graeme L. Cohen, Generalised quasiperfect numbers, Ph.D. Dissertation, University of New South Wales, Sydney, 1982. Abstract in Bull. Australian Math. Soc., 27 (1983), 153-155.
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
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. #10.3.1.

Cf. A000203, A033879 (2n - sigma(n)).

For negative values of n see A234286.

Edited by Max Alekseyev, Aug 29 2025

A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

Original entry on oeis.org

6, 1, 3, 18, 5, 9, 7, 50, 22

Offset: 0

Omar E. Pol, Oct 29 2017

If nonzero, a(9) > 10^9. - Michel Marcus, Oct 29 2017
From Robert Israel, Oct 29 2017: (Start)
If n is odd, then a(n) must be a square or twice a square (A028982).
If nonzero, a(9) > 10^13.
Some other values: a(11)=244036, a(17)=100, a(19)=25, a(25)=98, a(31)=15376, a(37)=484, a(39)=162, a(41)=49, a(47)=225, a(51)=72. (End)
a(n) > 10^20 for n in (9, 13, 15, 21, 23, 27, 29, 33, 35, 43, 45); see the intersection of A234285 and A234286. - Michel Marcus, Oct 30 2017
For the intersection mentioned above see A294406. - Omar E. Pol, Nov 01 2017

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.
Raven Dean, Rick Erdman, Dominic Klyve, Emily Lycette, Melissa Pidde, and Derek Wheel, Families of Values of the Excedent Function sigma(n)-2n, Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 37-46.

Cf. A000203, A000396, A005100, A005101, A028982, A033879, A033880, A234285, A234286, A294386, A294393, A294406.

Table[k = 1; While[Abs[2 k - DivisorSigma[1, k]] != n, k++]; k, {n, 0, 8}] (* Michael De Vlieger, Oct 30 2017 *)

a(n) = {my(k=1); while (abs(2*k-sigma(k)) != n, k++); k;} \\ Michel Marcus, Oct 29 2017

A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

Original entry on oeis.org

5, 0, 0, 0, 13, 0, 0, 12, 0, 0, 25, 0, 87, 31, 0, 0, 217, 22, 0, 16, 0, 0, 49, 0, 11, 55, 0, 32, 27, 0, 0, 22, 43, 0, 73, 0, 0, 637, 81, 0, 75, 0, 320, 28, 0, 50, 313, 24, 0, 103, 0, 0, 109, 0, 0, 34, 0, 62, 301, 24, -1, 120, 67, 0, 133, 0, 128, 139, 0, 0, 433, 42, 23, 151, 0, 0, 219, 82, 0, 28, 119

Offset: 0

Omar E. Pol, Oct 30 2017

sign

Note that a(60) = -1.

			--------------------------------------
n    A294386(n) - A005408(n)  =  a(n)
--------------------------------------
0        6            1           5
1        3            3           0
2        5            5           0
3        7            7           0
4       22            9          13
...

Cf. A000203, A000396, A005100, A005101, A033879, A033880, A234285, A234286, A294347, A294386.

f(n) = abs(2*n-sigma(n));
a(n) = my(k=1); while(f(k) != 2*n, k++); k - (2*n+1); \\ Michel Marcus, Oct 31 2017

a(n) = A294386(n) - A005408(n).

A294406 Positive odd numbers k such that both (sigma(m) - 2m) and (2m - sigma(m)) are never equal to k, where sigma(.) is the sum of divisors function A000203 (conjectured).

Original entry on oeis.org

9, 13, 15, 21, 23, 27, 29, 33, 35, 43, 45

Offset: 1

Omar E. Pol, Oct 30 2017

Intersection of A234285 and A234286.
Inspired by Michel Marcus's comment in A294347.
Cf. A000203, A033879, A033880, A005100, A005101.

cp's OEIS Frontend

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

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A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

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A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

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A294406 Positive odd numbers k such that both (sigma(m) - 2m) and (2m - sigma(m)) are never equal to k, where sigma(.) is the sum of divisors function A000203 (conjectured).

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cp's OEIS Frontend

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A294406 Positive odd numbers k such that both (sigma(m) - 2*m) and (2*m - sigma(m)) are never equal to k, where sigma(.) is the sum of divisors function A000203 (conjectured).

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Author

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Crossrefs

A294406 Positive odd numbers k such that both (sigma(m) - 2m) and (2m - sigma(m)) are never equal to k, where sigma(.) is the sum of divisors function A000203 (conjectured).