A019276 -id:A019276 - cp's OEIS frontend

A331035 a(n) = sigma^m(N)/N for N = A019276(n) (megaperfect numbers), where m(N) = min {m: N | sigma^m(N)} reaches record values; sigma^m is m-fold iteration of A000203.

1, 2, 5, 24, 168, 1834560, 6516224, 881280, 517517500266693633076805172570524811961093324800, 12291248474277267848395134563637563923813851715476607482451722240, 9432427640268436606451425375305719340471711381221905226790680395776

Offset: 1

M. F. Hasler, Jan 08 2020

nonn

See A019294 for m as function of the starting value.

See the main sequence A019276 for further information and references.

Cf. A019276 (megaperfect numbers: where A019294 reaches records), A019277 (the record values), A019294 (min{m>0: n|sigma^m(n)}), A019295 (ratio sigma^m(n)/n).

a(n) = A019295(A019276(n)).

A019278 Numbers j such that sigma(sigma(j)) = k*j for some k.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 21, 24, 42, 60, 64, 84, 160, 168, 240, 336, 480, 504, 512, 960, 1023, 1344, 1536, 4092, 4096, 10752, 13824, 16368, 29127, 32256, 32736, 47360, 57120, 58254, 61440, 65472, 65536, 86016, 116508, 217728, 262144, 331520, 343976, 466032, 550095

Offset: 1

N. J. A. Sloane

nonn

Let sigma^m (j) be the result of applying the sum-of-divisors function (A000203) m times to j; call j (m,k)-perfect if sigma^m (j) = k*j; then this is the sequence of (2,k)-perfect numbers.
From Michel Marcus, May 14 2016: (Start)
For these numbers, the quotient k = sigma(sigma(j))/j is an integer (see A098223). Then also k = (sigma(s)/s)*(sigma(j)/j) with s = sigma(j). That is, k = abundancy(s)*abundancy(j).
So looking at the abundancy of these terms may be interesting. Indeed we see that 459818240 and 51001180160 are actually 3-perfect numbers (A005820), and the reason they are here is that they are coprime to 3. So their sums of divisors are 4-perfect numbers (A027687), yielding q=12.
In a similar way, we can see that the 5-perfect numbers (A046060) that are coprime to 5 will be terms of this sequence with q=30. There are 20 such numbers, the smallest being 13188979363639752997731839211623940096. (End)
From Michel Marcus, May 15 2016: (Start)
It is also interesting to note that for a(2)=8, s=sigma(8)=15 is also a term. This happens to be the case for chains of several terms in a row:
8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;
512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;
29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;
1556480, 3932040, 14008320 with k = 9,13,14;
106151936, 251650560, 955367424 with k = 9,13,14;
312792480, 1505806848 with k = 19,20;
6604416000, 30834059256 with k = 19,20;
9623577600, 46566269568 with k = 19,20.
When j is a term, we can test if s=sigma(j) is also a term; this way we get 6 more terms: 572941926400, 845734196736, 1422976331052, 4010593484800, 11383810648416, 36095341363200.
And the corresponding chains are:
173238912000, 845734196736 with k = 19,20;
355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)
From Altug Alkan, May 17 2016: (Start)
Here are additional chains for the above list:
57120, 217728 with k = 13,14;
343976, 710400 with k = 7,8;
1980342, 5621760 with k = 10,14;
4404480, 14913024 with k = 11,12;
238608384, 775898880 with k = 11,12. (End)
Currently, the coefficient pairs are [1, 1], [3, 4], [4, 7], [7, 8], [8, 9], [9, 10], [9, 13], [10, 14], [11, 12], [13, 14], [16, 17], [16, 21], [17, 18], [19, 20], [23, 24], [25, 26], [25, 31], [27, 28], [29, 30], [31, 32], [32, 33], [37, 38]. It is interesting to note that for some of them, the pair (s,t) also satisfies t=sigma(s). - Michel Marcus, Jul 03 2016; Sep 06 2016
Using these empirical pairs of coefficients in conjunction with the first comment allows us to determine whether some term is the sum of divisors of another yet unknown smaller term. - Michel Marcus, Jul 04 2016
For m in A090748 = A000043 - 1 and c in A205597 (= odd a(n)), c*2^m is in the sequence, unless 2^(m+1)-1 | sigma(c). Indeed, from sigma(x*y) = sigma(x)*sigma(y) for gcd(x,y) = 1, we get sigma(sigma(c*2^m)) = sigma(sigma(c))*2^(m+1), so c*2^m is in the sequence if sigma(sigma(c))/c = k/2 (where k can't be odd: A330598 has no odd c). - M. F. Hasler, Jan 06 2020

Giovanni Resta, Table of n, a(n) for n = 1..145 (first 130 terms from Jud McCranie)
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, List of terms, grouped by quotient, not exhaustive.

Cf. A098219, A098220, A098221, A098222, A098223, A008333, A051027, A019276.
For sigma see A000203 and A007691.
Cf. A205597 (odd terms), A323653 (those terms that are in A007691, i.e., for which sigma(n)/n is also an integer), A330598 (half-integer ratio).

[m: m in [1..560000]| IsIntegral(DivisorSigma(1,DivisorSigma(1,m))/m)]; // Marius A. Burtea, Nov 16 2019

Select[Range[100000], Mod[DivisorSigma[1, DivisorSigma[1, #]], #] == 0 &] (* Carl Najafi, Aug 22 2011 *)

is_A019278(n)=sigma(sigma(n))%n==0 \\ M. F. Hasler, Jul 02 2016

from sympy.ntheory import divisor_sigma as D
print([i for i in range(1, 10000) if D(D(i, 1), 1)%i==0]) # Indranil Ghosh, Mar 17 2017

Simpler definition from M. F. Hasler, Jul 02 2016

A019283 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,6)-perfect numbers.

Original entry on oeis.org

42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304

Offset: 1

N. J. A. Sloane

If 2^p-1 is a Mersenne prime then m = 21*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(sigma(21*2^(p-1))) = sigma(32*(2^p-1)) = 63*2^p = 6*(21*2^(p-1)) = 6*m. So 21*(A000668+1)/2 is a subsequence of this sequence. This is the subsequence 42, 84, 336, 1344, 86016, 1376256, 5505024, 22548578304, 24211351596743786496, ... - Farideh Firoozbakht, Dec 05 2005
See also the Cohen-te Riele links under A019276.
No other terms < 5 * 10^11. - Jud McCranie, Feb 08 2012
Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n, thus in combination they must satisfy sigma(sigma(n)) = 6n. Note that any odd perfect number should occur also in A326181. - Antti Karttunen, Jun 16 2019
a(11) > 4*10^12. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Index entries for sequences where any odd perfect numbers must occur

Cf. A000668, A019278, A019279, A019282.

Cf. A000203, A000396, A005820, A051027, A326051, A326181.

Do[If[DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 6000000}] (* Farideh Firoozbakht, Dec 05 2005 *)

isok(n) = sigma(sigma(n))/n  == 6; \\ Michel Marcus, May 12 2016

a(10) by Jud McCranie, Feb 08 2012

A019284 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,7)-perfect numbers.

Original entry on oeis.org

24, 1536, 47360, 343976, 572941926400

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
572941926400 is also a term. See comment in A019278. - Michel Marcus, May 15 2016
a(6) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000668, A019278, A019279, A019281, A019282, A019283, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

Select[Range[50000], DivisorSigma[1, DivisorSigma[1, #]]/# == 7 &] (* Robert Price, Apr 07 2019 *)

isok(n) = sigma(sigma(n))/n  == 7; \\ Michel Marcus, May 12 2016

a(5) from Giovanni Resta, Feb 26 2020

A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

Original entry on oeis.org

1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15

Offset: 1

N. J. A. Sloane

Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Riele links under A019276.
a(A111227(n)) > A111227(n). - Reinhard Zumkeller, Aug 02 2012
a(659) > 870. - Michel Marcus, Jan 04 2017

			If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - _Don Reble_, Jun 23 2005

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

Don Reble, Table of n, a(n) for n = 1..1578 (Terms a(1..400) from T. D. Noe, Nov 2007; a(401..659) from Michel Marcus, Jan 02 2017), Feb 20 2022.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100. See Table 2, p. 95.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Carl Pomerance, On the composition of the functions sigma and phi, Colloq. Math., 59 (1989), 11-15.
Wikipedia, Iterated function, as of Jan 02 2020.
Zeraoulia Rafik, On congruence of the iterated form sigma^k(m) = 0 mod m, arXiv:2102.09941 [math.NT], 2021.

Cf. A019295 (ratio sigma^m(n)/n), A019276 (indices of records), A019277 (records), A000396.

a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
                        (\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
-- Reinhard Zumkeller, Aug 02 2012

a:=[]; f:=func; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 11 2020

A019294 := proc(n)
    local a,nitr ;
    a := 1 ;
    nitr := numtheory[sigma](n);
    while modp(nitr,n) <> 0 do
        nitr := numtheory[sigma](nitr) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 22 2016

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1,#]&,DivisorSigma[1,n], !Divisible[ #,n]&]],{n,90}] (* Harvey P. Dale, Mar 04 2015 *)

a(n)=if(n<0,0,c=1; s=n; while(sigma(s)%n>0,s=sigma(s); c++); c)

apply( A019294(n,s=n)=for(k=1,oo,(s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020

Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002
From Michel Marcus, Jan 02 2017: (Start)
a(n) = 1 for n in A007691.
a(n) = 2 for n in A019278 unless it belongs to A007691.
a(n) = 3 for n in A019292 unless it belongs to A007691 or A019278. (End)

Additional comments from Labos Elemer, Jun 20 2001

Edited by M. F. Hasler, Jan 07 2020

A019281 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,3)-perfect numbers.

Original entry on oeis.org

8, 21, 512

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No further term < 10^9 [see Table 1].
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(4) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Experimental Mathematics, Home Page.

Cf. A019278, A019279, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

A019282 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,4)-perfect numbers.

Original entry on oeis.org

15, 1023, 29127, 355744082763

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(5) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

Select[Range[100000], DivisorSigma[1, DivisorSigma[1, #]]/# == 4 &] (* Robert Price, Apr 07 2019 *)

isok(n) = sigma(sigma(n))/n  == 4; \\ Michel Marcus, May 12 2016

a(4) from Jud McCranie, Feb 08 2012

A019285 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.

Original entry on oeis.org

60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360, 711488165526, 1098437885952, 1422976331052

Offset: 1

N. J. A. Sloane

If 2^p-1 is a Mersenne prime greater than 3 then m = 15*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(15*2^(p-1)) = sigma(24*(2^p-1)) = 60*2^p = 8*(15*2^(p-1)) = 8*m. So for n>1 15/2*(A000668(n)+1) is in the sequence. 60, 240, 960, 61440, 983040, 3932160, 16106127360 and 1729382256910270464042 are such terms. - Farideh Firoozbakht, Dec 05 2005
See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
1422976331052 is also a term. See comment in A019278. - Michel Marcus, May 15 2016
a(25) > 4*10^12. - Giovanni Resta, Feb 26 2020

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000668, A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

isok(n) = sigma(sigma(n))/n == 8; \\ Michel Marcus, May 15 2016

a(19) from Jud McCranie, Nov 13 2001
a(20)-a(21) from Jud McCranie, Jan 29 2012
a(22)-a(24) from Giovanni Resta, Feb 26 2020

A019286 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,9)-perfect numbers.

Original entry on oeis.org

168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
4010593484800 is also a term. See comment in A019278. - Michel Marcus, May 15 2016

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000668, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

isok(n) = sigma(sigma(n))/n  == 9; \\ Michel Marcus, May 12 2016

a(8) by Jud McCranie, Jan 28 2012

a(9) from Giovanni Resta, Feb 26 2020

A019287 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,10)-perfect numbers.

Original entry on oeis.org

480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(11) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019288, A019289, A019290, A019291.

More terms from Jud McCranie, Nov 13 2001; a(9) Jan 29 2012, a(10) Feb 08 2012

cp's OEIS Frontend

A331035 a(n) = sigma^m(N)/N for N = A019276(n) (megaperfect numbers), where m(N) = min {m: N | sigma^m(N)} reaches record values; sigma^m is m-fold iteration of A000203.

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A019278 Numbers j such that sigma(sigma(j)) = k*j for some k.

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A019283 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,6)-perfect numbers.

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A019284 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,7)-perfect numbers.

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A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

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A019281 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,3)-perfect numbers.

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A019282 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,4)-perfect numbers.

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A019285 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.