A019294 -id:A019294 - cp's OEIS frontend

A019276 Megaperfect numbers: numbers n where A019294(n) = min {m: n divides sigma^(m) (n)} increases to a record; sigma^(m) means apply the sum-of-divisors function m times.

1, 2, 3, 5, 9, 11, 23, 25, 29, 59, 67, 101, 131, 173, 202, 239, 353, 389, 401, 461, 659, 1319, 1579, 1847, 2309, 2797

Offset: 1

Where records occur in A019294. a(n>=23) depend on a few probable primes.

Graeme L. Cohen & Herman J. J. te Riele, Iterating the Sum-of-Divisors Function: Abstract. (Page no longer available; link gives latest snapshot on web.archive.org from Sept. 2006)
Graeme L. Cohen & Herman J. J. te Riele, Iterating the Sum-of-Divisors Function [Broken link to a file "10355A.pdf", maybe the same as NM-R9525.pdf available through the above page of abstract.]
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019277 (the record values), A019294 (min{m: n|sigma^(m)(n)}), A019295 (ratio sigma^(m)(n)/n).

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[Mod[d, m] == 0, 0, d]]; g[n_] := Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1; a = 0; Do[b = g[n]; If[b > a, a = b; Print[ n]], {n, 460}] (* Robert G. Wilson v, Jun 24 2005 *)

m=0;for(n=1,oo,m<(m=max(A019294(n),m))&&print1(n",")) \\ M. F. Hasler, Jan 07 2020

A019277 Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.

Original entry on oeis.org

1, 2, 4, 5, 7, 15, 16, 17, 78, 97, 101, 120, 174, 214, 239, 261, 263, 296, 380, 557, 1287, 1524, 1722, 1911, 2023, 2373

Offset: 1

N. J. A. Sloane

Original name: Let sigma_m(n) be the result of applying the sum-of-divisors function m times to n; let m(n) = min m such that n divides sigma_m (n); let k(n) = sigma_{m(n)}(n)/n; sequence gives k(n) for the megaperfect numbers n, where m(n) increases.
Records in A019294. a(n>=23) depend on a few probable primes.
See also the Cohen-te Riele links under A019276.
The original name mentioned the sequence of ratios k, i.e., A019295(A019276) = (1, 2, 5, 24, 168, 1834560, 6516224, 881280, ...), at present not listed in the OEIS. - M. F. Hasler, Jan 07 2020

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

Cf. A019276 (megaperfect numbers: where A019294 has records), A019294 (min m: n|sigma^m(n)), A019295 (sigma^m(n)/n with m = A019294).

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[Mod[d, m] == 0, 0, d]]; g[n_] := Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1; a = 0; Do[b = g[n]; If[b > a, a = b; Print[ a]], {n, 460}] (* Robert G. Wilson v, Jun 24 2005 *)

{M=0; for(n=1,oo, my(s=n,m=1); while((s=sigma(s))%n,m++); m>M&&print1(M=m,","))} \\ M. F. Hasler, Jan 07 2020

a(n) = A019294(A019276(n)). - M. F. Hasler, Jan 07 2020

Definition corrected by M. F. Hasler, Jan 07 2020

A111227 Numbers n where A019294(n) > n.

Original entry on oeis.org

3, 11, 29, 53, 58, 59, 67, 101, 109, 131, 149, 173, 202, 239, 461, 479, 659, 1031, 1319, 1579, 1847

Offset: 1

Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 26 2005

A number n is called (m,k)-perfect if sigma^m(n) = k*n. A019294(n) gives minimal values of m for n. a(n) are the values where A019294(n) > n.

Corresponding values of m and k are in A111726, A111727.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function Experimental Mathematics, 5 (1996), p 94.

a111227 n = a111227_list !! (n-1)
a111227_list = filter (\x -> a019294 x > x) [1..]
-- Reinhard Zumkeller, Aug 02 2012

A291303 Appears to be an erroneous version of A019294.

Original entry on oeis.org

1, 2, 4, 2, 5, 4, 5

Offset: 1

dead

Included in accordance with OEIS policy of including published but incorrect versions of sequences.

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

A007691 Multiply-perfect numbers: n divides sigma(n).

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Multiply Perfect Numbers [broken link]
Eric Bach, Gary Miller, and Jeffrey Shallit, Sums of divisors perfect numbers and factoring, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.
Robert D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.
Walter Nissen, Abundancy : Some Resources
Kaitlin Rafferty and Judy Holdener, On the form of perfect and multiperfect numbers, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See p. 11.
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Index entries for sequences where any odd perfect numbers must occur

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012

Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)

for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))

from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

More terms from Jud McCranie and then from David W. Wilson.

Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

A019295 a(n) = sigma(sigma(...(sigma(n))...)) / n, where sigma (A000203) is iterated until a multiple of n is reached.

Original entry on oeis.org

1, 2, 5, 2, 24, 2, 24, 3, 168, 12, 1834560, 10, 84480, 12, 4, 2, 92520, 20, 62720, 84, 3, 49920, 6516224, 7, 881280, 28, 3360, 2, 517517500266693633076805172570524811961093324800, 728, 912, 18, 19767296, 46260, 144, 42, 30349648609280, 38644089120, 30, 663, 34042889727216750428160

Offset: 1

N. J. A. Sloane

nonn

The minimal number of iterations of the sigma function until a multiple of n is reached (after the initial n) is given in A019294.

See also the Cohen-te Riele links in A019276.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Experimental Mathematics, Volume 5 (1996) ["Outdated Archival Version" as of 2005.]

Cf. A019276 (megaperfect numbers: where A019294 reaches records), A019276 (record values), A019294 (number of iterations needed to reach a multiple of n).

f:=func; a:=[]; for n in [1..41] do k:=n; while f(k) mod n ne 0 do k:=f(k); end while; Append(~a,f(k) div n); end for; a; // Marius A. Burtea, Jan 11 2020

apply( {A019295(n,s=n)=while((s=sigma(s))%n,);s\n}, [1..50]) \\ M. F. Hasler, Jan 08 2020

More terms from Max Alekseyev, Sep 22 2016

Edited by M. F. Hasler, Jan 08 2020

A173430 Last of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 15, 15, 15, 6, 6, 15, 15, 14, 10, 12, 12, 14, 14, 15, 104, 18, 18, 20, 20, 104, 22, 24, 24, 104, 26, 40, 28, 30, 30, 104, 104, 33, 34, 48, 91, 38, 38, 56, 40, 42, 42, 44, 44, 45, 46, 48, 48, 80, 255, 51, 52, 54, 54, 72, 56, 80, 58, 60, 60, 62, 62, 104, 255, 84, 66, 68, 68

Offset: 1

Walter Nissen, Feb 18 2010

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 4 ) ) ... ) the iterates are 4, 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 4, 7, 8, 15, so a(4) = 15 .
Here sigma ( 4 ) = 7, sigma ( sigma ( 4 ) ) = sigma ( 7 ) = 8, etc.

Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, Vol. 5, No. 2 (1996), pp. 91-100.
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Cf. A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

a[1] = 1; a[n_] := Module[{k = n}, While[CoprimeQ[k, (s = DivisorSigma[1, k])], k = s]; k]; Array[a, 68] (* Amiram Eldar, Sep 02 2019 *)

A073407 Let phi_m(x) denote the Euler totient function applied m times to x. Sequence gives the minimum number of iterations m such that phi_m(n) divides n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 1, 3, 2, 4, 1, 4, 2, 4, 1, 5, 1, 4, 2, 4, 3, 5, 1, 5, 3, 4, 2, 5, 3, 5, 1, 5, 4, 5, 1, 5, 3, 5, 2, 6, 3, 5, 3, 5, 4, 6, 1, 5, 4, 6, 3, 6, 1, 6, 2, 5, 4, 6, 3, 6, 4, 5, 1, 6, 4, 6, 4, 6, 4, 6, 1, 6, 4, 6, 3, 6, 4, 6, 2, 5, 5, 7, 3, 7, 4, 6, 3, 7, 4, 6, 4, 6, 5, 6, 1, 7, 4, 6, 4, 7, 5, 7, 3, 6

Offset: 1

Benoit Cloitre, Aug 23 2002

			phi(22) -> 10, phi(10) -> 4, phi(4) -> 2 and 2 divides 22. Hence 3 iterations are needed and a(22) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A019294.

a[n_] := Module[{c = 0, k = n}, While[c == 0 || !Divisible[n, k], k = EulerPhi[k]; c++]; c]; Array[a, 105] (* Amiram Eldar, Jul 10 2019 *)

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

It seems that sum(k=1, n, a(k)) is asymptotic to C*n*log(n) with C>1.

A073408 Let cophi_m(x) denotes the cototient function applied m times to x (cophi(x)=x-phi(x)). Sequence gives the minimum number of iterations m such that cophi_m(n) divides n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 2, 1, 1, 4, 1, 3, 2, 4, 1, 2, 1, 4, 1, 3, 1, 5, 1, 1, 2, 5, 2, 4, 1, 5, 3, 3, 1, 6, 1, 4, 2, 5, 1, 2, 1, 6, 2, 4, 1, 6, 3, 3, 3, 6, 1, 5, 1, 5, 2, 1, 2, 6, 1, 5, 3, 6, 1, 4, 1, 6, 3, 5, 2, 7, 1, 3, 1, 7, 1, 6, 4, 6, 2, 4, 1, 7, 2, 5, 3, 6, 2, 2, 1, 6, 4, 6, 1, 7, 1, 4, 2, 7

Offset: 2

Benoit Cloitre, Aug 23 2002

			cophi(10) -> 6, cophi(6) -> 4, cophi(4) -> 2 and 2 divides 10. Hence 3 iterations are needed and a(10) = 3.

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A019294, A051953.

Table[Length@ NestWhileList[# - EulerPhi@ # &, n, Or[# == n, ! Divisible[n, #]] &, 1, 12] - 1, {n, 2, 106}] (* Michael De Vlieger, Dec 22 2017 *)

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

It seems that sum(k=1, n, a(k)) is asymptotic to C*n*log(n) with C < 1.

A342718 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> A000203(x)/A000005(x) to reach a number that is not an integer, when starting from x = n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 2, 1, 1, 1, 4, 1, 3, 4, 4, 1, 2, 1, 2, 3, 2, 2, 2, 1, 1, 1, 2, 1, 5, 2, 2, 1, 2, 1, 2, 1, 3, 5, 5, 1, 3, 2, 3, 5, 4, 2, 2, 1, 3, 1, 2, 1, 3, 5, 2, 5, 4, 1, 3, 5, 3, 2, 1, 1, 3, 2, 2, 3, 2, 2, 2, 1, 4, 1, 1, 1, 2, 3, 2, 1, 1, 1

Offset: 1

Ctibor O. Zizka, Mar 19 2021

nonn

a(n) = 1 for n >= 2, n from A049642.

			n = 3; 3 -> s(3)/d(3) = 2 -> s(2)/d(2) = 3/2, a(3) = 2;
n = 11; 11 -> s(11)/d(11) = 6 -> s(6)/d(6) = 3 -> s(3)/d(3) = 2 -> s(2)/d(2) = 3/2, a(11) = 4;
n = 20; 20 -> s(20)/d(20) = 7 -> s(7)/d(7) = 4 -> s(4)/d(4) =  7/3, a(20) = 3;
s(x) = A000203(x), d(x) = A000005(x).

Cf. A000005, A000203, A003601, A049642.

Cf. A019294 for a similar iteration.

f[n_] := Divide @@ DivisorSigma[{1, 0}, n]; a[n_] := Length @ NestWhileList[f, n, IntegerQ[#] && # > 1 &] - 1; Array[a, 100] (* Amiram Eldar, Mar 19 2021 *)

a(n) = if (n==1, 0, my(nb=1, k); while(denominator(k=sigma(n)/numdiv(n)) == 1, n = k; nb++); nb); \\ Michel Marcus, Mar 21 2021

cp's OEIS Frontend

A019276 Megaperfect numbers: numbers n where A019294(n) = min {m: n divides sigma^(m) (n)} increases to a record; sigma^(m) means apply the sum-of-divisors function m times.

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A019277 Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.

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A111227 Numbers n where A019294(n) > n.

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A291303 Appears to be an erroneous version of A019294.

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A007691 Multiply-perfect numbers: n divides sigma(n).

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A019295 a(n) = sigma(sigma(...(sigma(n))...)) / n, where sigma (A000203) is iterated until a multiple of n is reached.

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A173430 Last of consecutive coprime iterations of sum-of-divisors function.

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A073407 Let phi_m(x) denote the Euler totient function applied m times to x. Sequence gives the minimum number of iterations m such that phi_m(n) divides n.

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