A175200 -id:A175200 - cp's OEIS frontend

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

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

A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

Original entry on oeis.org

1, 6, 28, 120, 270, 496, 672, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 273000, 293760, 401310, 441936, 446880, 502740, 523776, 614250, 707616, 802620, 819000

Offset: 1

Jud McCranie

nonn

Multiplicities are ignored.
All even perfect numbers are in the sequence. It seems that 1 is the only odd term of the sequence. - Farideh Firoozbakht, Jul 01 2008
sigma() is the multiplicative sum-of-divisors function. - Walter Nissen, Dec 16 2009
Pollack and Pomerance call these "prime-perfect numbers" and show that there are << x^(1/3+e) of these up to x for any e > 0. - Charles R Greathouse IV, May 09 2013
Except for unity for the obvious reason, the primitive terms are the perfect numbers (A000396). - Robert G. Wilson v, Feb 19 2019
If an odd term > 1 exists, it is larger than 5*10^23. - Giovanni Resta, Jun 02 2020

			273000 = 2^3*3*5^3*7*13 and sigma(273000) = 1048320 = 2^8*3^2*5*7*13 so 273000 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B19.

Donovan Johnson, Table of n, a(n) for n = 1..500 (first 100 terms from T. D. Noe)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Intersection of A105402 and A175200. - Amiram Eldar, Jun 02 2020

Cf. A110751, A110819, A055744, A081377, A000203.

Filtered([1..1000000],n->Set(Factors(n))=Set(Factors(Sigma(n)))); # Muniru A Asiru, Feb 21 2019

Select[Range[1000000], Transpose[FactorInteger[#]][[1]] == Transpose[FactorInteger[DivisorSigma[1, #]]][[1]] &] (* T. D. Noe, Dec 08 2012 *)

a(n) = {for (i=1, n, fn = factor(i); fs = factor(sigma(i)); if (fn[,1] == fs[,1], print1(i, ", ")););} \\ Michel Marcus, Nov 18 2012

is(n)=my(f=factor(n),fs=[],t);for(i=1,#f[,1], t=factor((f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))[,1]; fs=vecsort(concat(fs,t~),,8); if(#setminus(fs,f[,1]~), return(0))); fs==f[,1]~ \\ Charles R Greathouse IV, May 09 2013

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 234, 384, 496, 936, 1536, 1638, 6144, 8128, 24576, 42588, 98304, 393216, 1089270, 1572864, 6291456, 25165824, 33550336, 100663296, 115048440, 402653184, 1185125760, 1610612736

Offset: 1

Antti Karttunen, Jul 28 2020

Numbers k such that gcd(sigma(k)-A007947(k), A007947(k)) == A007947(k) are those in A175200. These are equal to k such that gcd(A326143(k), A007947(k)) = gcd(sigma(k)-A007947(k)-k, A007947(k)) are equal to A007947(k).
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

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

Intersection of A175200 and A336552.
Cf. A007947, A326143, A336551.
Cf. A000396, A002023, A326145 (subsequences).
Cf. also A336641 for a similar construction.

A007947(n) = factorback(factorint(n)[, 1]);
isA336550(n) = { my(r=A007947(n), s=sigma(n), u=((n/r)-1)); (!(s%r) && (gcd(u,(s-r-n))==u)); };

A263928 Integers m such that sigma(m)^2 is divisible by m.

Original entry on oeis.org

1, 6, 24, 28, 120, 224, 234, 270, 496, 588, 600, 672, 864, 1080, 1521, 1638, 1782, 2016, 3724, 4320, 4680, 5733, 6048, 6200, 6552, 7128, 8128, 11172, 11466, 15872, 17280, 18144, 18600, 18620, 21600, 22932, 26208, 26460, 27000, 30240, 32640, 32760, 33516, 35640

Offset: 1

Paolo P. Lava, Oct 30 2015

nonn

Previous name was: "Numbers such that the product of the sum of their divisors and the sum of the reciprocals of their divisors is an integer".
The multiply-perfect numbers (A007691) are a subset of this sequence.
This is a subsequence of A175200. - Michel Marcus, Nov 03 2015
Alternative definition: Numbers m such that Sum_{i = 1..k} (sigma(m) - d_i) / d_i is an integer, where d_i are the k divisors of m. - Paolo P. Lava, Mar 23 2017

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their sum is sigma(24) = 60 while the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/24 = 5/2. Finally 60 * 5/2 = 150.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A000203, A007691, A017665, A017666, A175200, A263983.

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

Select[Range[36000],Divisible[DivisorSigma[1,#]^2,#]&] (* Harvey P. Dale, Jul 05 2023 *)

isok(n) = (sigma(n)^2 % n) == 0; \\ Michel Marcus, Nov 03 2015

New name from Michel Marcus, Nov 03 2015

A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

Original entry on oeis.org

1, 6, 28, 30, 42, 66, 84, 102, 120, 138, 186, 210, 270, 282, 318, 330, 364, 420, 426, 462, 496, 510, 546, 570, 642, 672, 690, 714, 762, 840, 868, 870, 924, 930, 966, 1080, 1092, 1122, 1146, 1302, 1320, 1410, 1428, 1488, 1518, 1590, 1638, 1722, 1770, 1782, 1890

Offset: 1

Walter Kehowski, May 01 2005

Also numbers k such that k^k/sigma(k) is integral. - Vicente Izquierdo Gomez, Jan 04 2013.

Pollack and Pomerance call these numbers "prime-deficient numbers". - Amiram Eldar, Jun 02 2020

			102 is a term since 102 = 2*3*17 and sigma(102) = 2^3*3^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Cf. A000203, A027598, A058063, A054027, A014567, A023194, A175200.

A:=select(proc(z) numtheory[factorset](sigma(z)) subset numtheory[factorset](z) end,[$1..100000]); has 716 members.

Select[Range[2000],IntegerQ[#^#/DivisorSigma[1,#]] &] (* Vicente Izquierdo Gomez, Jan 04 2013 *)

Extended by R. J. Mathar, Dec 08 2008

A280581 a(n) = the product of divisors of sum of divisors of n.

Original entry on oeis.org

1, 3, 8, 7, 36, 1728, 64, 225, 13, 5832, 1728, 21952, 196, 331776, 331776, 31, 5832, 1521, 8000, 3111696, 32768, 10077696, 331776, 46656000000, 31, 3111696, 2560000, 9834496, 810000, 139314069504, 32768, 250047, 254803968, 8503056, 254803968, 8281, 1444

Offset: 1

Jaroslav Krizek, Jan 05 2017

nonn

a(n) < A007955(n) for numbers n in A219364.
a(n) | A007955(n) for numbers n in A219363.
A007955(n) | a(n) for numbers n in A219362.
n | a(n) for numbers n in A175200.

			For n = 5; a(n) = product of divisors of sigma(5) = 1*2*3*6 = 36.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to sigma(n).

Cf. A000203, A007955, A219362, A219363, A219364, A280582.

[&*[d: d in Divisors(SumOfDivisors(n))]: n in [1..100]]

Table[Times @@ Divisors@ DivisorSigma[1, n], {n, 37}] (* Michael De Vlieger, Jan 06 2017 *)
a[n_] := (s = DivisorSigma[1, n])^(DivisorSigma[0, s]/2); Array[a, 40] (* Amiram Eldar, Jun 26 2022 *)

a(n) = my(k = 1); fordiv(sigma(n), d, k*=d); k; \\ Michel Marcus, Jan 06 2017

from math import isqrt
from sympy import divisor_count, divisor_sigma
def A280581(n): return (lambda m:isqrt(m)**c if (c:=divisor_count(m)) & 1 else m**(c//2))(divisor_sigma(n)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A000203(n)).

A336554 Odd numbers k such that rad(k) divides sigma(k).

Original entry on oeis.org

1, 135, 891, 1521, 3375, 5733, 10935, 11907, 41067, 43875, 84375, 89667, 92781, 98865, 107811, 108927, 154791, 185625, 200655, 216513, 226233, 273375, 274625, 287469, 307125, 336141, 416745, 448335, 480249, 544635, 557375, 570375, 590625, 627669, 628849, 648675, 692055, 823875, 885735, 931095, 968877, 1071875, 1108809, 1131165

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Only a minority of terms seem to be also in A228058: 5733, 92781, 2069613, 3341637, 4179357, 5929605, 60998353, 67637349, 89162541 among the first 216, the five of them which occur also in A263983.

Question: Is the intersection with A336553 empty?

Amiram Eldar, Table of n, a(n) for n = 1..1200 (terms 1..216 from Antti Karttunen)
Index entries for sequences where odd perfect numbers must occur, if they exist at all.

Odd terms of A175200.
Cf. A000203, A007947, A228058, A263983.
Cf. also A336553.

isA336554(n) = if(!(n%2),0,!(sigma(n) % factorback(factorint(n)[, 1])));

A173615 Numbers k such that rad(k)^2 divides sigma(k).

Original entry on oeis.org

1, 96, 864, 1080, 1782, 6144, 7128, 7776, 17280, 27000, 28512, 54432, 55296, 69984, 87480, 114048, 215622, 276480, 381024, 393216, 432000, 433026, 456192, 497664, 629856, 675000, 862488, 1382400, 1399680, 1677312, 1732104, 1824768, 2187000, 2195424, 2667168

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

rad(n) is the product of the primes dividing n (A007947) and sigma(n) = sum of divisors of n (A000203). Considering the integers k = (2^a)*(3^b), where a+1 = 6*m and b >= 1, we obtain an infinite number of numbers such that rad(k)^2 divides sigma(k).

De Koninck (2000, 2002) asked whether 1 and 1782 are the only numbers k such that rad(k)^2 = sigma(k). - Amiram Eldar, Jan 29 2025

			rad(96)^2 = 6^2 = 36, sigma(96) = 252 and 36 divides 252.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 102.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..300 from Donovan Johnson)
Kevin A. Broughan, Jean-Marie De Koninck, Imre Kátai, and Florian Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), Article 12.7.5, pp. 1-12. See Final remarks pp. 10-11.
Kevin Broughan Daniel Delbourgo, and Qizhi Zhou, A conjecture of De Koninck regarding particular square values of the sum of divisors function, Journal of Number Theory, Vol. 137 (2014), pp. 50-66.
Yong-Gao Chen and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Vol. 154 (2015), pp. 324-364.
Jean-Marie De Koninck, Probelm 000:08, Western Number Theory Problems, 17 & 20 Dec 2000, edited by Gerry Myerson, p. 5.
Jean-Marie De Koninck, Problem 10966, The American Mathematical Monthly, Vol. 109, No. 8 (2002), p. 759; Editorial comment, ibid., Vol. 111, No. 6 (2004), p. 536.
Min Tang and Zhi-Jun Zhou, On a conjecture of De Koninck, INTEGERS, Vol. 18 (2018), Article #A60.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, Vol. 10, No. 3 (2021), pp. 249-260; Correction, ibid., Vol. 10, No. 4 (2021), p. 339; arXiv preprint, arXiv:1906.10001 [math.NT], 2019-2021.

Cf. A000203, A007947, A078615.

Subsequence of A175200.

for n from 1 to 2000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if irem(sigma(n),t2^2) = 0 then print (n): else fi: od :

f[p_, e_] := (p^(e+1) - 1)/(p^2 * (p-1)); q[k_] := IntegerQ[Times @@ f @@@ FactorInteger[k]]; q[1] = True; Select[Range[3*10^6], q] (* Amiram Eldar, Jan 29 2025 *)

isok(n) = my(f=factor(n)); (sigma(f) % factorback(f[, 1])^2) == 0; \\ Michel Marcus, Nov 09 2020

a(30)-a(35) from Donovan Johnson, Jan 14 2012

cp's OEIS Frontend

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

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A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

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A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

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A263928 Integers m such that sigma(m)^2 is divisible by m.

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A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

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A280581 a(n) = the product of divisors of sum of divisors of n.

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A336554 Odd numbers k such that rad(k) divides sigma(k).