A225110 -id:A225110 - cp's OEIS frontend

A226774 Integers a(n) = Sum_{i=1..q} 1/d(i) where d(i) are the divisors of A225110(n) for some q.

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Michel Lagneau, Jun 17 2013

nonn

The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 10, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 24, 4, 4, 4, 4, 4, 4, ...
By convention, a(1)=1. For a majority of n, a(n) = 2.
a(n) = 3 for n = 11, 16, 52, 145, 634, ... where A225110(n) = 120, 180, 672, 1890, 8460, ...
a(n) = 4 for n = 2284, 2476, 6871, ... where A225110(n) = 30240, 32760, 90720, ...

			a(16) = 3 because the divisors of A225110(16) = 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 and 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/30 + 1/36 + 1/45 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A225110.

with(numtheory): for n from 1 to 2000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to
n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 then ii:=1: printf(`%d, `, s):else fi:od:od:

either_A226774_or_0(n) = { if(1==n,return(1)); my(divs=divisors(n),s=0); for(i=1,#divs,s += (1/divs[i]); if((1==denominator(s))&&(i>1),return(s))); return(0); };
up_to = 16384; i=0; n=0; while(iA226774_or_0(n); if(s, i++; write("b226774.txt", i, " ", s)));
\\ Antti Karttunen, Dec 16 2017

A226871 Triangle read by rows: row n gives the first q divisors d(1), d(2), ..., d(q) of A225110(n) such that Sum_{i = 1..q} 1/d(i) is an integer.

Original entry on oeis.org

1, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 4, 7, 14, 28, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 6, 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18

Offset: 1

Michel Lagneau, Jun 20 2013

Rows 2, 3, 5, 6, 7, ... with the divisors {1, 2, 3, 6} are identical;
rows 4, 18, 62, 67, ... with the divisors {1, 2, 4, 7, 14, 28} are identical;
...
The primitive rows are rows 1, 2, 4, 11, 16, 39, 52, 145, ... corresponding to n = 1, 6, 28, 120, 180, 496, 672, 1890, ... (see A226853).
The irregular triangle of divisors is:
[1]
[1, 2, 3, 6]
[1, 2, 3, 6]
[1, 2, 4, 7, 14, 28]
[1, 2, 3, 6]
...

			Row 3 = [1, 2, 3, 6] consists of the first 4 divisors of A225110(3) = 18; 1 + 1/2 + 1/3 + 1/6 = 2 is an integer.

Michel Lagneau, Rows n = 1..2223 of irregular triangle, flattened

Cf. A225110, A226853.

with(numtheory): print({1}):for n from 1 to 5000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 then ii:=1: print({seq(x[i],i=1..q)}) else fi:od:od:

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

A226476 Numbers n with the property that, if tau(n) = k = number of divisors of n, and the d(i) are the divisors [arranged in increasing order], then the sum 1/d(k) + 1/d(k-1) + 1/d(k-2) + ... + 1/d(q) is an integer for some q.

Original entry on oeis.org

1, 6, 24, 28, 120, 496, 672, 2016, 4320, 4680, 8128, 8190, 26208, 30240, 32760, 42336, 45864, 392448, 523776, 714240, 1571328, 2178540, 8910720, 17428320, 20427264, 23569920, 29795040, 33550336, 34369920, 45532800, 61900800

Offset: 1

Michel Lagneau, Jun 11 2013

nonn

By convention, for n = 1, a(1) = 1 with q = 1.
The corresponding pairs (tau(n), q) are (1, 1), (4, 2), (8, 3), (6, 2), (16, 2), (10, 2), (24, 2), (36, 6), (48, 3), (48, 3), (14, 2), (48, 6), (72, 3), (96, 2), (96, 2), (72, 7), (72, 7), (72, 5), (80, 2), (120, 8), (120, 6), (216, 2), (384, 3), (432, 3), (240, 3), (320, 2), (360, 5), (26, 2), (384, 5), (384, 2), (288, 9).
Properties of this sequence:
q = 2 if n = 1, 6, 28, 120, 496, 672, 8128, ... is a multiply-perfect number (see A007691 where it is conjectured that this sequence is infinite), which would imply that this sequence is also infinite because A007691 is a subsequence.

			24 is in the sequence because the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, and the sum 1/24 + 1/12 + 1/8 + 1/6 + 1/4 + 1/3 = 1.
28 is in the sequence because 28 is a multiply-perfect number: the divisors are 1, 2, 4, 7, 14, 28, and the sum of the reciprocals of all the divisors is 1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1 = 2.

Cf. A000005, A000203, A007691, A225110.

with(numtheory): for n from 1 to 10000000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from n1 by -1 to 1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) then ii:=1: printf(`%d, `,n):else fi:od:od:

Edited by Jon E. Schoenfield and N. J. A. Sloane, Sep 09 2017

A226853 Numbers n such that Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of n for some q and n is primitive (the set {d(1), d(2), ..., d(q)} appears only once).

Original entry on oeis.org

1, 6, 28, 120, 180, 496, 672, 1890, 8128, 8415, 20482, 20496, 25410, 30240, 32760, 33345, 34155, 38430, 40128, 47804, 72800, 90720, 103530, 407715, 523776, 806190, 979992, 1084160, 1273725, 1274100, 2178540, 3571050, 7441280, 10782216, 12499150, 23569920, 28360464, 33550336, 45532800

Offset: 1

Michel Lagneau, Jun 19 2013

nonn

Subsequence of A225110 where we find classes of numbers having the same first q divisors; for example, each of the numbers 6, 18, 42, 54, 66, ... has {1, 2, 3, 6} as its first four divisors, and 1/1 + 1/2 + 1/3 + 1/6 = 2; similarly, each of the numbers 28, 196, 812, 868, ... has {1, 2, 4, 7, 14, 28} as its first six divisors, and 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.
This sequence includes only the smallest number having any given set of first divisors {d(1), d(2), ..., d(q)}, i.e., the set of first divisors corresponding to each term occurs only once.
The sets of first divisors (such that Sum_{i = 1..q} 1/d(i) is an integer) corresponding to the first few terms are as follows:
a(1) =   1: [1];
a(2) =   6: [1, 2, 3, 6];
a(3) =  28: [1, 2, 4, 7, 14, 28];
a(4) = 120: [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120];
a(5) = 180: [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45];
a(6) = 496: [1, 2, 4, 8, 16, 31, 62, 124, 248, 496];
a(7) = 672: [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672].

			180 is in the sequence because the divisors are {1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180} and the sum of the reciprocals of the first q = 15 divisors is 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/30 + 1/36 + 1/45 = 3, which is an integer.
Although the first 4 divisors of 18 are {1, 2, 3, 6} and the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/6 = 2, 18 is not in the sequence because 6 has those same first four divisors and is the smallest (i.e., primitive) number having that set of first 4 divisors. Thus, the primitive number 6 is in the sequence, so the non-primitive number 18 is not.

Subsequence of A225110.

with(numtheory): printf ( "%d %d \n",1,6):lst:={6}:for n from 1 to 10000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 and {x[q]} intersect lst <>{x[q]} then lst:=lst union {x[q]}:ii:=1: printf(`%d, `,n):else fi:od:od:

isok(k) = if (k==1, return([1])); my(d=divisors(k), s=1); for (i=2, #d, s += 1/d[i]; if (denominator(s)==1, return(Vec(d, i));));
already(list, v) = for (i=1, #list, if (list[i] == v, return(1)););
lista(nn) = my(listv=List(), listi=List()); for (n=1, nn, my(v=isok(n)); if (v && !already(listv, v), listput(listi, n); listput(listv, v););); Vec(listi); \\ Michel Marcus, Feb 22 2025

Edited by Jon E. Schoenfield, Oct 02 2017

More terms from Michel Marcus, Feb 22 2025

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A226774 Integers a(n) = Sum_{i=1..q} 1/d(i) where d(i) are the divisors of A225110(n) for some q.

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A226871 Triangle read by rows: row n gives the first q divisors d(1), d(2), ..., d(q) of A225110(n) such that Sum_{i = 1..q} 1/d(i) is an integer.

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

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A226476 Numbers n with the property that, if tau(n) = k = number of divisors of n, and the d(i) are the divisors [arranged in increasing order], then the sum 1/d(k) + 1/d(k-1) + 1/d(k-2) + ... + 1/d(q) is an integer for some q.

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A226853 Numbers n such that Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of n for some q and n is primitive (the set {d(1), d(2), ..., d(q)} appears only once).

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