A046061 -id:A046061 - cp's OEIS frontend

A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

1, 2, 3, 6, 7, 10, 14, 15, 20, 21, 22, 24, 27, 28, 30, 31, 33, 34, 40, 42, 46, 54, 57, 60, 62, 66, 69, 70, 84, 87, 91, 93, 94, 102, 105, 106, 110, 114, 120, 127, 130, 138, 140, 141, 142, 154, 160, 168, 170, 174, 177, 182, 186, 189, 190, 195, 198, 210, 214, 216, 217, 220, 224, 230, 231, 237, 238, 254, 260, 264, 270, 273

Offset: 1

Antti Karttunen, Feb 16 2022

Numbers k for which A351555(k) = 0. This is a necessary condition for the terms of A349169 and of A349745, therefore they are subsequences of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Moreover, all multiperfect numbers (A007691) seem to be in this sequence.
From Antti Karttunen, Aug 27 2025: (Start)
Multiperfect number m is included in this sequence only if its abundancy sigma(m)/m has only such odd prime factors p that prevprime(p) [A151799] divides m for each p.  E.g., all 65 known 5-multiperfects are multiples of 3, and all known terms of A005820 and A046061 are even.
This sequence contains natural numbers k such that the odd primes in the prime factorization of sigma(k) have the same valuation there as in k, except that the primes in A003961(k) [or equally in A003961(A007947(k))] stand for "don't care primes", that are "masked off" from the comparison.
(End)

Cf. A000203, A003961, A151799, A351546.
Positions of zeros in A351555.
Subsequences: A000396, A351553 (even terms), A386430 (odd terms), A351551, A349169, A349745, A387160 (terms of the form prime * m^2), also these, at least all the currently (Feb 2022) known terms: A005820, A007691, A046060.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351554(n) = (0==A351555(n));

isA351554(n) = { my(sh=A351546(n),f=factor(sh)); for(i=1,#f~, if((f[i,1]%2)&&valuation(n,f[i,1])!=f[i,2],return(0))); (1); }; \\ Uses also program given in A351546.

Definition corrected by Antti Karttunen, Aug 22 2025

A216782 Numbers such that numerator(sigma(n)/n) is even and denominator(sigma(n)/n) is odd.

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102

Offset: 1

Michel Marcus, Sep 16 2012

a(n) contains odd primes (A065091), odd squarefree semiprimes (A046388), perfect numbers (A000396), and 2n-multiperfect (A027687, A046061).

			sigma(3)/3 = 4/3 (even/odd).

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A216780, A216781, A324903 (characteristic function).

Subsequences: A000396, A027687, A043305 (without its initial 1), A046061, A046388, A065091, A336702 (without its initial 1).

Select[Range[1000], EvenQ[Numerator[DivisorSigma[1, #] / # ]] && OddQ[Denominator[DivisorSigma[1, #]/#]]&] (* Vincenzo Librandi, Jun 24 2014 *)
nedoQ[n_]:=Module[{ds=DivisorSigma[1,n]/n},EvenQ[Numerator[ds]]&&OddQ[ Denominator[ ds]]]; Select[Range[200],nedoQ] (* Harvey P. Dale, Feb 28 2015 *)

eoab(n) = {for (i=1, n, ab = sigma(i)/i; if ((numerator(ab) % 2 == 0) && (denominator(ab) % 2 == 1), print1(i, ", ")););}

A249670 a(n) = A017665(n)*A017666(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 2, 56, 120, 117, 45, 132, 21, 182, 84, 40, 496, 306, 78, 380, 210, 672, 198, 552, 10, 775, 273, 1080, 2, 870, 60, 992, 2016, 176, 459, 1680, 3276, 1406, 570, 2184, 36, 1722, 112, 1892, 231, 390, 828, 2256, 372, 2793, 4650, 408, 1274, 2862

Offset: 1

Michel Marcus, Nov 03 2014

nonn

If n is a k-multiperfect, then a(n) = k.

Allan C. Wechsler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)).
Cf. A017665/A017666 (abundancy of n).
Cf. A009194 (gcd(n, sigma(n))), A064987 (n*sigma(n)).
Cf. A000396, A005820, A027687, A046060, A046061.

a249670 n = div (n * s) (gcd n s ^ 2)
 where s = sum (filter (\k -> mod n k == 0) [1..n])
-- Allan C. Wechsler, Mar 31 2023

a249670[n_Integer] := Numerator[DivisorSigma[-1, n]]*Denominator[DivisorSigma[-1, n]]; a249670 /@ Range[80] (* Michael De Vlieger, Nov 10 2014 *)

a(n) = my(ab = sigma(n)/n); numerator(ab)*denominator(ab);

a(n) = A064987(n)/A009194(n)^2.
a(A000396(n)) = 2 (perfect).
a(A005820(n)) = 3 (tri-perfect).
For p prime, a(p) = p*(p+1).

A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 31998395520, 796928461056000, 212517062615531520, 680489641226538823680000, 13297004660164711617331200000, 1534736870451951230417633280000, 6070066569710805693016339910206758877366156437562171488352958895095808000000000

Offset: 1

Labos Elemer, Dec 12 2001

nonn

Tested for each k and j < 200. Otherwise the proof for all j seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.
Tested for each k and j <= 1000. - Thomas Baruchel, Oct 10 2003
The given terms have been tested for all j. - Don Reble, Nov 03 2003
This is a proper subset of the multiply perfect numbers A007691. E.g., 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.

Cf. A066135, A066284, A007691, A066290.

DivisorSigma(2*j-1, k)/k is an integer for all j = 1, 2, 3, ..., 200, ...

The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061). - Thomas Baruchel, Oct 10 2003

Extended to 13 confirmed terms by Don Reble, Nov 04 2003. There is a question whether there are other members below a(13). However, there are none in Achim's list of multiperfect numbers (see A007691); Richard C. Schroeppel has suggested that that list is complete to 10^70 - if so, a(1..12) are correct; as for a(13), Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble

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

A091443 Multiperfect numbers n which are divisible by sopfr(n) (multiperfect number: sigma(n) = k*n with k integer, sopfr: Sum of prime factors with repetition).

Original entry on oeis.org

1379454720, 14182439040, 212517062615531520, 27099073228001299660800, 680489641226538823680000, 15229814702070563916152832000, 34111227434420791224041472000, 59023729003862626557345792000

Offset: 1

Sven Simon, Jan 10 2004

nonn

The sequence contains multiperfect numbers with multiplicity k from 3..8. They are extracted from a list with about 5000 multiperfect numbers with multiplicity from 2..11. Because of the size of these numbers, no numbers with multiplicity k > 8 were found, even though there were about 3000 of them in the list. 95% of the multiperfect numbers with multiplicity from 3..8 are known.
Conjecture: the sequence is finite.
There are 5255 multiperfect numbers known with multiplicity 3 to 11. No more findings for A091443 so we still have 33 multiperfect numbers divisible by their sopfr (without the trivial case 1). With multiplicity 3..8 quite surely all are found (only very few - if any - missing). It is estimated that there are about 2200 with multiplicity 9 and 2091 of them are already found. With multiplicity 10 of estimated 4500 1161 are known. So far no multiperfect number with multiplicity 9 or 10 is divisible by its sopfr (with repetition). Using sopfr without repetition (A114887), there is one number with multiplicity 9 (or more). - Sven Simon, Feb 12 2012

			a(1): 1379454720 = 2^8*3*5*7*19*37*73, sopfr(n)= 2^5*5.

Sven Simon, Table of n, a(n) for n = 1..33 [Conjectured to be complete]
Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Achim Flammenkamp, he Multiply Perfect Numbers Page [Local copy in pdf format of the page as of Jun 04 2020, made with permission of the author]
Achim Flammenkamp, Datafile_2013_12_31 (with 5311 multiperfect numbers). [_Sven Simon_, Feb 15 2014]
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A000203, A001414, A005820, A027687, A046060, A046061.

Intersection of A007691 and A036844. - Michel Marcus, Oct 08 2017

A171265 Primes of the form 1 + 6-multiperfect numbers.

Original entry on oeis.org

9186050031556349952001, 837294585717161066223413745156096001, 102833200965458113792221682653890347008001, 8564360335859979576063023115658227351552001

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

Primes of the form 1+A046061(n). A subsequence of A093034. This sequence has only 8 terms. a(5)=1+A046061(76),a(6)=1+A046061(79),
a(7)=1+A046061(204) and a(8)=1+A046061(209).
If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=6(x-1). See comment lines of the sequence A093034.

			p=9186050031556349952001 is prime and sigma(p-1)/(p-1)=6, so p is in the sequence.

Cf. A046061, A093034, A171263, A171264.

A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which xphi(n) + ysigma(n) is a multiple of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 3, 2, 4, 5, 2, 2, 3, 2, 4, 7, 4, 2, 2, 5, 4, 3, 1, 2, 5, 2, 2, 9, 4, 18, 3, 2, 4, 9, 4, 2, 7, 2, 6, 3, 4, 2, 3, 7, 5, 9, 5, 2, 3, 7, 3, 9, 4, 2, 5, 2, 4, 7, 2, 11, 7, 2, 9, 9, 10, 2, 3, 2, 4, 12, 4, 10, 7, 2, 5, 3, 4, 2, 3, 13, 4, 9, 4, 2, 5, 9, 9, 9, 4, 19, 3, 2, 7, 5, 5, 2, 7

Offset: 1

Walter Nissen, May 20 2004

nonn

a(n) is a generalization of the multiperfect numbers in A007691.

			a(6) = 1 as 1*sigma(6) is a multiple of 6.
a(4) = 2 as 2*phi(4) + 0*sigma(4) = 4. - Example added by _Antti Karttunen_, Feb 24 2020
a(14) = 4 as 3*phi(14) + 1*sigma(14) = 3*6 + 24 = 3*14, where 3+1 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..12857
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A000203, A000396, A005820, A007691 (positions of ones), A027687, A046060, A046061.

A094701(n) = { my(x=eulerphi(n),y=sigma(n)); for(s=1,oo,for(t=0,s,if(!(((t*x)+((s-t)*y))%n),return(s)))); }; \\ Antti Karttunen, Feb 24 2020

a(multiperfect) = 1.
a(prime) = 2 as 1*phi(prime) + 1*sigma(prime) and 1+1 = 2.
For primes > 5, a(2*prime) = 4.

Name clarified by Antti Karttunen, Feb 24 2020

A134639 Conjectured number of numbers k such that sigma(k)/k = n.

Original entry on oeis.org

6, 36, 65, 245, 516

Offset: 3

T. D. Noe, Nov 05 2007

These numbers come from Guy and Flammenkamp. Sequences A000396, A005820, A027687, A046060 and A046061 give the k for which the abundancy sigma(k)/k is 2, 3, 4, 5 and 6, respectively. Sequence A054030 gives the abundancy of each multiperfect number A007691.

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B2.

Achim Flammenkamp, The Multiply Perfect Numbers Page

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

cp's OEIS Frontend

A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A216782 Numbers such that numerator(sigma(n)/n) is even and denominator(sigma(n)/n) is odd.

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A249670 a(n) = A017665(n)*A017666(n).

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A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

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

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A091443 Multiperfect numbers n which are divisible by sopfr(n) (multiperfect number: sigma(n) = k*n with k integer, sopfr: Sum of prime factors with repetition).

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A171265 Primes of the form 1 + 6-multiperfect numbers.

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A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which x*phi(n) + y*sigma(n) is a multiple of n.

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A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which xphi(n) + ysigma(n) is a multiple of n.