A007539 -id:A007539 - cp's OEIS frontend

A227765 First n-fold perfect number divided by 6, or 0 if the quotient is not an integer.

0, 1, 20, 5040, 2363739840, 25724259347628441600, 23551816324573058043308233789747587210723924136427520000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, ..., for n = 1, 2, 3, 4, 5, ...
The first n-fold perfect numbers divided by 120 is A227766.
The first n-fold perfect numbers divided by 30240 is A227767.
The first n-fold perfect numbers divided by 14182439040 is A227768.
For additional comments, references, links, and cross-refs, see A007539.

			The first 3-fold perfect number is A007539(3) = 120, so a(3) = 120/6 = 20.

T. D. Noe, Table of n, a(n) for n = 1..8
E. W. Weisstein's MathWorld, Multiperfect Number
Wikipedia, Multiply perfect number

Cf. A007539, A227766, A227767, A227768, A227769.

a(n) = A007539(n)/6 if 2 <= n <= 8.

A227766 First n-fold perfect number divided by 120, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 0, 1, 252, 118186992, 1286212967381422080, 1177590816228652902165411689487379360536196206821376000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, ..., for n = 1, 2, 3, 4, 5, ...

For additional comments, references, links, and cross-refs, see A007539 and A227765.

			The first 4-fold perfect number is A007539(4) = 30240, so a(4) = 30240/120 = 252.

T. D. Noe, Table of n, a(n) for n = 1..8

Cf. A007539, A227765, A227767, A227768, A227769.

a(n) = A007539(n)/120 if 3 <= n <= 8.

A227767 First n-fold perfect number divided by 30240, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 0, 0, 1, 468996, 5104019711831040, 4672979429478781357799252736061029208476969074688000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, ..., for n = 1, 2, 3, 4, 5, ...

For additional comments, references, links, and cross-refs, see A007539 and A227765.

			The first 5-fold perfect number is A007539(5) = 14182439040, so a(5) = 14182439040/30240 = 468996.

T. D. Noe, Table of n, a(n) for n = 1..8

Cf. A007539, A227765, A227766, A227768, A227769.

a(n) = A007539(n)/30240 if 4 <= n <= 8.

A227768 First n-fold perfect number divided by 14182439040, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 9963793783910270786529635084437882643939328000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, .., for n = 1, 2, 3, 4, 5, ...

For additional comments, references, links, and cross-refs, see A007539 and A227765.

			The first 6-fold perfect number is A007539(6) = 154345556085770649600, and 154345556085770649600/14182439040 = 119711504640/11 is not an integer, so a(6) = 0.

T. D. Noe, Table of n, a(n) for n = 1..8

Cf. A007539, A227765, A227766, A227767, A227769.

a(n) = A007539(n)/14182439040 for n = 5, 7, 8.

A227769 First (n+1)-fold perfect number divided by first n-fold perfect number, or 0 if the quotient is not an integer.

Original entry on oeis.org

6, 20, 252, 468996, 0, 0, 0

Offset: 1

Jonathan Sondow, Jul 30 2013

nonn

The first n-fold perfect numbers are A007539 = 1, 6, 120, 30240, 14182439040, 154345556085770649600, ...

For additional comments, references, links, and crossrefs, see A007539 and A227765.

			The first 4-fold and 5-fold perfect numbers are A007539(4) = 30240 and  A007539(5) = 14182439040, and 14182439040/30240 = 468996 is an integer, so a(4) = 468996.

Cf. A007539, A227765, A227766, A227767, A227768.

a(n) = A007539(n+1)/A007539(n) if n < 5.

A088912 a(n) = smallest m such that sigma(m) = (n+1/2)*m.

Original entry on oeis.org

2, 24, 4320, 8910720, 17116004505600, 170974031122008628879954060917200710847692800, 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000

Offset: 1

Farideh Firoozbakht, Nov 29 2003

2 is the only number m such that sigma(m)=1.5*m.
A direct consequence of Robin's theorem is that a(6)>5E16, a(7)>1.898E29, a(8)>2.144E51, a(9)>9.877E89 and a(10)>6.023E157. - Washington Bomfim, Oct 30 2008
If the Riemann hypothesis (RH) is true then Robin's theorem (Guy Robin, 1984) implies that the n-th term of this sequence is greater than exp(exp((n+1/2)/exp(gamma))) where gamma=0.5772156649... is the Euler-Mascheroni constant (A001620). For the 6th term (which is actually 1.7*10^44) this lower bound is 5.0*10^16. Similarly, if RH is true, the next term (7th term) is at least 1.9*10^29 (and is probably more than 10^90 or so). - Gerard P. Michon, Jun 10 2009
From Gerard P. Michon, Jul 04 2009: (Start)
An upper bound for a(7) is provided by a 97-digit integer of abundancy 15/2 (5.71379...10^96) discovered by Michel Marcus on July 4, 2009. The factorization of that number is: 2^53 3^15 5^6 7^6 11^3 13 17 19^3 23 29 31 37 41 43 61 73 79 97 181 193 199 257 263 4733 11939 19531 21803 87211 262657.
Similarly, an upper bound for a(8) is provided by a 286-digit integer of abundancy 17/2 (3.30181...10^285) equal to x/17, where x is the smallest known number of abundancy 9 (a 287-digit integer discovered by Fred W. Helenius in 1995). This is so because 17 happen to occur with multiplicity 1 in the factorization of x. (End)
A new upper bound for a(7) was found on Aug 15 2009 by Michel Marcus, who broke his own record by finding two "small" multiples of 2^35*3^20*5^5*7^6*11^2*13^2*17 that are of abundancy 15/2. The lower one (1.27494722...10^88) has only 89 digits. - Gerard P. Michon, Aug 15 2009
These are the least hemiperfects of abundancy n + 1/2. - Walter Nissen, Aug 17 2010
On Jul 24 2010, Michel Marcus found a 191-digit integer of abundancy 17/2 (2.7172904...10^190) whose factorization starts with 2^81 3^29 5^9 7^10 11^4 13^3 17^2 19 23^2... This is the best upper bound to a(8) known so far. - Gerard P. Michon, Aug 22 2010

			a(2)=24 because 1+2+3+4+6+8+12+24=2.5*24 and 24 is the earliest m such that sigma(m)=2.5*m.

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon, Multiperfect and hemiperfect integers
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 11/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 13/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 15/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 17/2
Walter Nissen, Abundancy : Some Resources
Wikipedia, Hemiperfect number
Wikipedia, Riemann hypothesis

Cf. A007539, A000396, A005820, A027687, A046060, A046061.

Cf. A159907 (hemiperfect numbers: half-integral abundancy), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A160678 (abundancy = 13/2).

a[n_] := (For[m=1, DivisorSigma[1, m]!=(n+1/2)m, m++ ];m); Do[Print[a[n]], {n, 4}]

a(5)-a(6) from Robert Gerbicz, Apr 19 2009
Cross-references from Gerard P. Michon, Jun 10 2009
Edited by M. F. Hasler, Mar 17 2013
a(7) from Michel Marcus confirmed and added by Max Alekseyev, Jun 05 2025

A317681 a(n) = smallest m such that sigma(m) = n*m/2.

Original entry on oeis.org

1, 2, 6, 24, 120, 4320, 30240, 8910720, 14182439040, 17116004505600, 154345556085770649600, 170974031122008628879954060917200710847692800, 141310897947438348259849402738485523264343544818565120000, 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000

Offset: 2

Jianing Song, Aug 04 2018

Interleaving of A007539 and A088912.
For even n, a(n) is a multiply perfect number; for odd n it is a hemiperfect number.
Note that 1 is the only number with abundancy 1, and 2 is the only number with abundancy 3/2 (in other words, 1 and 2 are solitary numbers; see A014567). For k >= 4 it is not known whether there are finitely many or infinitely many numbers with abundancy k/2. Also it is not known whether a(n) < a(n+1) always holds.
On the Riemann Hypothesis (RH), a(n) > exp(exp(n/(2*exp(gamma)))), where gamma = 0.5772156649... is the Euler-Mascheroni constant (A001620).

			a(7) = 4320 since sigma(4320) = 15120 = 7/2*4320 and 4320 is the smallest m such that sigma(m)/m = 7/2.

Max Alekseyev, Table of n, a(n) for n = 2..16
Achim Flammenkamp, The Multiply Perfect Numbers Page
Fred Helenius, A glossary of MPFN-related terms.
G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy: Some Resources

Cf. A007539, A088912, A246454.

Numbers with abundancy k/2: A000396 (k=4), A141643 (k=5), A005820 (k=6), A055153 (k=7), A027687 (k=8), A141645 (k=9), A046060 (k=10), A159271 (k=11), A046061 (k=12), A160678 (k=13).

Nest[Append[#, Block[{m = #1[[-1]] + 1}, While[DivisorSigma[1, m] != #2 m/2, m++]; m]] & @@ {#, Length@ # + 2} &, {1}, 6] (* Michael De Vlieger, Aug 05 2018 *)

for(n=2, 10, for(m=1, 10^12, if(sigma(m)/m==n/2, print1(m, ", "); break())))

a(n) = my(k=1); while (sigma(k) != k*n/2, k++); k; \\ Michel Marcus, May 15 2025

a(2n) = A007539(n), a(2n+1) = A088912(n), n > 0.

a(15) = A088912(7) added by Max Alekseyev, Jun 05 2025

A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

Original entry on oeis.org

1, 11466, 114660, 411264, 804384, 871416, 4999680, 46332000, 176417280, 378069120, 396168192, 485188704, 709430400, 2004912000, 3921372000, 5600534400, 6128179200, 6956471808, 7556976000, 7746979968, 9904204800, 14092001280, 14182439040, 23423662080, 31998395520

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Sigma preserves both the 2-adic and 3-adic valuation of the terms of this sequence.

All 65 known 5-multiperfect numbers (A046060) are included in this sequence, as well as the smallest 7-multiperfect number, 141310897947438348259849402738485523264343544818565120000 = A007539(7), and probably the majority of other p-multiperfect numbers as well, where p is a prime > 3. However, any term that is in A349747 is not included in this sequence.

Cf. A000203, A003961, A007539, A007814, A007949, A046060, A349747.

Cf. also A349169, A349745.

f[p_, e_] := NextPrime[p, 2]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := n * GCD[(sigma = DivisorSigma[1, n]), (u = s[n])] == sigma * GCD[n, u]; Select[Range[10^6], q] (* Amiram Eldar, Dec 01 2021 *)

A003961twice(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f); };
isA349746(n) = { my(s=sigma(n),u=A003961twice(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)) and A007949(A000203(a(n))) = A007949(a(n)). [See comment]

a(15)-a(25) from Martin Ehrenstein, Dec 17 2021

A349747 Multiply perfect numbers whose 5-adic valuation is larger than their 3-adic valuation.

Original entry on oeis.org

459818240, 51001180160, 41254809330254618094796800000, 417557910137818162642396224651585493401600000, 21908279635853187912360370114977853341696000000, 4602427236053495643738729034543787648483328000000, 1470295051205988580219996701010375287153623040000000, 56532758277786216648694678091722223342645673984000000

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Among the first 1600 multiperfect numbers, there is only one term x for which A007949(x) > A007814(x), and that is x = 6088728021160320, a 4-perfect number, while there are none for which A112765(x) > A007814(x). Only 1, 6, 120, 30240 (incidentally, the first four terms of A007539) seem to be in A025487.

The abundancy index (sigma(n)/n) of the 18 initial terms is: 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 8.

			459818240 = 2^8 * 5 * 7 * 19 * 37 * 73 is included because while it is a multiple of 5, it is not a multiple of 3.
41254809330254618094796800000 = 2^19 * 3^4 * 5^5 * 7^3 * 11^3 * 13 * 19 * 31^3 * 37 * 41 * 61 is included, because the exponent of 5 (which is 5), is larger than the exponent of 3, which is 4.

Index entries for sequences related to sigma(n)

Cf. A000203, A007539, A007691, A007814, A007949, A025487, A112765.

mp = Cases[Import["https://oeis.org/A007691/b007691.txt", "Table"], {, }][[;; , 2]]; Select[mp, IntegerExponent[#, 5] > IntegerExponent[#, 3] &] (* Amiram Eldar, Nov 30 2021 *)

{k in A007691 | A112765(k) > A007949(k)}.

A360642 a(n) is the least number k such that A093653(k)/A000120(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 64, 66, 84, 72, 210, 132, 450, 792, 288, 264, 1044, 672, 5328, 528, 1344, 840, 1026, 1056, 4116, 1800, 4128, 2112, 5124, 3780, 6480, 2184, 3360, 8352, 11088, 8448, 4680, 50700, 4200, 4368, 20880, 8280, 13320, 13440, 12420, 4104, 46200, 8736

Offset: 1

Amiram Eldar, Feb 15 2023

a(n) exists for all n >= 1 since A093653(2^(k-1))/A000120(2^(k-1)) = k for all k >= 1.

Analogous to A007539 as A175522 is analogous to perfect numbers (A000396).

			a(1) = 1 since A093653(1)/A000120(1) = 1/1 = 1.
a(2) = 2 since A093653(2)/A000120(2) = 2/1 = 2, and 2 is the least number with this property.
a(3) = 4 since A093653(4)/A000120(4) = 3/1 = 3, and 4 is the least number with this property.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000120, A000396, A007539, A093653, A093687, A175522, A175526, A360641.

seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n <= nmax, i = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]; If[IntegerQ[i] && i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[50, 10^5]

lista(len, nmax) = {my(s = vector(len), c = 0, n = 1, i); while(c < len && n <= nmax, i = sumdiv(n, d, hammingweight(d))/hammingweight(n); if(denominator(i) == 1 && i <= len && s[i] == 0, c++; s[i] = n); n++); s }

# uses imports and definitions in A093653, A000120
from itertools import count, islice
def f(n): q, r = divmod(A093653(n), A000120(n)); return q if r == 0 else 0
def agen():
    n, adict = 1, dict()
    for k in count(1):
        v = f(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 48))) # Michael S. Branicky, Feb 15 2023

a(n) <= 2^(n-1).

cp's OEIS Frontend

A227765 First n-fold perfect number divided by 6, or 0 if the quotient is not an integer.

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A227766 First n-fold perfect number divided by 120, or 0 if the quotient is not an integer.

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A227767 First n-fold perfect number divided by 30240, or 0 if the quotient is not an integer.

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A227768 First n-fold perfect number divided by 14182439040, or 0 if the quotient is not an integer.

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A227769 First (n+1)-fold perfect number divided by first n-fold perfect number, or 0 if the quotient is not an integer.

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A088912 a(n) = smallest m such that sigma(m) = (n+1/2)*m.

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A317681 a(n) = smallest m such that sigma(m) = n*m/2.

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A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

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