A072868 -id:A072868 - cp's OEIS frontend

A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

4, 8, 32, 60, 128, 8192, 43200, 69360, 120960, 131072, 524288, 4146912, 6549984, 12927600, 13335840, 16329600, 34715520, 51603840, 57879360, 59633280, 107775360, 160797000, 169155840, 252067200, 371226240, 391789440, 436230144, 439883136, 489888000, 657296640

Offset: 1

Paolo P. Lava, Oct 04 2013

nonn

A072868 is a subsequence of this sequence. Any term x of A072868 can be expressed as x = 2*sigma(sigma(x/2)).
Note the analogy with amicable pair sums (A180164) which satisfy a similar condition: k = sigma(x) = sigma(y) where k = x + y. - Michel Marcus, Oct 07 2013
When terms do not belong to A072868, then they belong to A159886, and the (x,y) couples are (23,37), (14999,28201), (34673,34687), (55373,65587), (2056961,2089951), (2458187,4091797), (4586987,8340613), (5174363,8161477), (6204767,10124833), (15788453,18927067), (25748273,25855567), (20699927,37179433), (22239647,37393633), ... - Michel Marcus, Oct 08 2013

			4 = 2 + 2 = 2*sigma(sigma(2)).
8 = 4 + 4 = 2*sigma(sigma(4)).
32 = 16 + 16 = 2*sigma(sigma(16)).
60 = 23 + 37 = sigma(sigma(23)) = sigma(sigma(37)).
128 = 64 + 64 = 2*sigma(sigma(64)).
8192 = 4096 + 4096 = 2*sigma(sigma(4096)).

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experiment. Math. (1996) vol. 5, no. 2, pp. 91-100 (see merge at n=60 in tree of section 4 page 97).

Cf. A000203, A072868.

with(numtheory); P:=proc(q) local j,n;
for n from 1 to q do for j from 1 to trunc(n/2) do
if sigma(sigma(j))=sigma(sigma(n-j)) and sigma(sigma(j))=n then print(n);
fi; od; od; end: P(10^6);

a(7)-a(20) from Giovanni Resta, Oct 06 2013

a(21)-a(30) from Donovan Johnson, Oct 08 2013

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

A387218 Numbers k such that k = sigma(s(s(s(k)))) where s(k) = sigma(k)-k and sigma = A000203.

Original entry on oeis.org

44, 248, 11904, 565838, 583730, 16588800

Offset: 1

Hugo Cuéllar, Aug 22 2025

			k = 44:
s(44) = sigma(44) - 44 = 84 - 44 = 40,
s(40) = sigma(40) - 40 = 90 - 40 = 50,
s(50) = sigma(50) - 50 = 93 - 50 = 43,
sigma(43) = 44.
k = 248:
s(248) = sigma(248) - 248 = 480 - 248 = 232,
s(232) = sigma(232) - 232 = 450 - 232 = 218,
s(218) = sigma(218) - 218 = 330 - 218 = 112,
sigma(112) = 248.
k = 11904:
s(11904) = sigma(11904) - 11904 = 32640 - 11904 = 20736,
s(20736) = sigma(20736) - 20736 = 61831 - 20736 = 41095,
s(41095) = sigma(41095) - 41095 = 49320 - 41095 = 8225,
sigma(8225) = 11904.

Cf. A000203 (sigma(n)), A001065 (s(n)).

Cf. A072868 (sigma(sigma(k)-k) = k).

s[k_]:=DivisorSigma[1,k]-k;Select[Range[10^6],DivisorSigma[1,s[s[s[#]]]]==#&] (* James C. McMahon, Aug 30 2025 *)

isok(k) = my(x=sigma(k)-k); if (x>0, x=sigma(x)-x; if (x>0, x=sigma(x)-x; if (x>0, sigma(x) == k))); \\ Michel Marcus, Aug 24 2025

a(6) from Michel Marcus, Aug 24 2025

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A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

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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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A387218 Numbers k such that k = sigma(s(s(s(k)))) where s(k) = sigma(k)-k and sigma = A000203.

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