A072869 -id:A072869 - cp's OEIS frontend

A178601 a(n) = s(s(n)), where s(n) = sigma(n)-n = A001065(n).

0, 0, 1, 0, 6, 0, 1, 3, 7, 0, 15, 0, 8, 4, 9, 0, 11, 0, 14, 1, 10, 0, 55, 6, 15, 1, 28, 0, 54, 0, 1, 9, 22, 1, 17, 0, 14, 1, 43, 0, 66, 0, 50, 15, 16, 0, 64, 7, 1, 11, 26, 0, 78, 1, 63, 1, 31, 0, 172, 0, 20, 1, 41, 1, 90, 0, 32, 13, 40, 0, 45, 0, 50, 8, 63, 1, 144, 0, 56, 50, 40, 0, 196, 1, 26, 15, 76, 0, 259, 11, 64, 13, 43, 6, 236, 0

Offset: 2

Roger L. Bagula, May 30 2010

nonn

Antti Karttunen, Table of n, a(n) for n = 2..16384
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
Index entries for sequences related to sums of divisors

Cf. A001065, A000203, A051027, A072869.

Cf. A206708 (fixed points, union of A000396 and A063990). - Antti Karttunen, Nov 01 2017

Table[Apply[ Plus, Divisors[Apply[ Plus, Divisors[n]] - n]] - (Apply[Plus, Divisors[n]] - n), {n, 0, 100}]

A001065(n) = (sigma(n) - n);
A178601(n) = A001065(A001065(n)); \\ Antti Karttunen, Nov 01 2017

Edited by N. J. A. Sloane, May 30 2010

Terms a(0) and a(1) removed and more terms added by Antti Karttunen, Nov 01 2017

A247111 Integers k such that sigma(sigma(k) - k) = 2*k, where sigma is the sum of divisors, A000203.

Original entry on oeis.org

6, 28, 36, 496, 8128, 33550336, 8589869056

Offset: 1

Michel Marcus, Nov 19 2014

That is, integers k such that A072869(k) = 2*k.
All perfect numbers (A000396) belong to this sequence.
Is there another term like 36 that is not perfect?
a(8) > 10^11. - Hiroaki Yamanouchi, Sep 11 2015
a(8) <= 137438691328. - David A. Corneth, Jun 04 2021

			For k=36, sigma(sigma(36)-36) = sigma(91-36) = sigma(55) = 72, hence 36 is in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203 (sigma(n)), A000396 (perfect numbers), A001065 (sigma(n)-n), A072869 (sigma(sigma(n)-n)).

Cf. also A019283, A326181, A342922.

Select[Range[1,10000],DivisorSigma[1,DivisorSigma[1,#]-#]==2*#&] (* Julien Kluge, Sep 20 2016 *)

PARI
```
isok(n) = (sigma(sigma(n) - n) == 2*n);
```

a(7) from Michel Marcus, Nov 22 2014

A291881 Numbers n such that sigma(sigma(n)) = sigma(sigma(n)-n) + sigma(n); that is, f(g(n)) = g(f(n)) where f = A000203 and g = A001065.

Original entry on oeis.org

2, 38040, 51888, 236644, 260880, 3097024, 5283852, 5667312, 11777472, 46120848, 52981252, 69128640, 121352208, 330364848, 485906400, 662736600, 769422720, 1111869360, 1267978320, 1272335760, 1426817904, 1807128528, 2107406448, 2381691312, 2452404544, 2691587568

Offset: 1

Altug Alkan, Sep 05 2017

nonn

Initial motivation for this sequence was that question: Can be an odd number k such that f(g(k)) = g(f(k)) where f = A000203 and g = A001065?
Non-abundant terms are 2, 236644, 52981252,...
If an odd term exists, it is larger than 2*10^11. - Giovanni Resta, Sep 15 2017

			38040 is a term because sigma(38040) = 114480 and sigma(114480) = sigma(76440) + 114480.

Cf. A000203, A001065, A051027, A072869.

inQ[n_] :=  DivisorSigma[1, DivisorSigma[1, n]] == DivisorSigma[1, DivisorSigma[1, n] - n] + DivisorSigma[1, n]; (* Robert G. Wilson v, Sep 10 2017 *)

a001065(n) = sigma(n)-n;
isok(n) = sigma(a001065(n))==a001065(sigma(n));

A300658 Numbers m that divide sigma(sigma(m) - m) where sigma is the sum of divisors function (A000203).

Original entry on oeis.org

4, 6, 8, 28, 32, 36, 78, 84, 128, 168, 252, 496, 504, 532, 756, 1488, 2808, 3720, 4464, 5928, 8128, 8192, 13392, 24384, 61236, 73152, 78120, 131072, 183708, 217728, 219456, 425880, 458640, 524288, 1084752, 1834560, 2204280, 3254256, 6120432, 6386688, 11007360

Offset: 1

Jaroslav Krizek, Mar 24 2018

nonn

Numbers m that divide A072869(m).
Numbers m such that sigma(sigma(m)-m) = k*m for k = 1 - 5:
k = 1:    4, 8, 32, 128, 8192, 131072, 524288, 2147483648, ... (A072868),
k = 2:    6, 28, 36, 496, 8128, 33550336, 8589869056, ... (A247111),
k = 3:    78, 532, ...,
k = 4:    84, 252, 756, 1488, 4464, 13392, 24384, 61236, 73152, ...,
k = 5:    168, 2808, 5928, 6120432, ...
Perfect numbers (A000396) are terms.
Corresponding values of (sigma(sigma(m) - m)) / m for numbers m from this sequence: 1, 2, 1, 2, 1, 2, 3, 4, 1, 5, 4, 2, 6, 3, 4, 4, 5, 7, 4, 5, 2, 1, 4, 4, 4, 4, 10, 1, 4, 8, 4, 12, 10, 1, 4, 11, 9, ...
Sequence of the smallest numbers k such that sigma(sigma(k) - k) = n*k for n >= 1: 4, 6, 78, 84, 168, 504, 3720, 217728, 2204280, 78120, 1834560, 425880, ...

			6 is a term because sigma(sigma(6) - 6) / 6 = 12 / 6 = 2 (integer).

Cf. A000203, A000396, A072868, A072869, A247111.

[n: n in[2..1000000] | SumOfDivisors(SumOfDivisors(n)- n) mod n eq 0];

isok(n) = (n!=1) && !(sigma(sigma(n)-n) % n); \\ Michel Marcus, Mar 25 2018

cp's OEIS Frontend

A178601 a(n) = s(s(n)), where s(n) = sigma(n)-n = A001065(n).

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A247111 Integers k such that sigma(sigma(k) - k) = 2*k, where sigma is the sum of divisors, A000203.

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A291881 Numbers n such that sigma(sigma(n)) = sigma(sigma(n)-n) + sigma(n); that is, f(g(n)) = g(f(n)) where f = A000203 and g = A001065.

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A300658 Numbers m that divide sigma(sigma(m) - m) where sigma is the sum of divisors function (A000203).

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