A007691 -id:A007691 - cp's OEIS frontend

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 228, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6

Offset: 1

Labos Elemer, Dec 06 2001

nonn

a(n) <= 2p, where p = A002586(n) is the smallest prime factor of (1 + 2^n). (Proof. Since sigma_n(2p) = (1 + 2^n)(1 + p^n) and p is odd, 2p divides sigma_n(2p).) - Jonathan Sondow, Nov 23 2012

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A001158, A001159, A002586, A007691, A046762-A046764, A055709-A055717.

Cf. A218860, A218861 (unique values and where they first occur).

Table[m = 2; While[Mod[DivisorSigma[n, m], m] > 0, m++]; m, {n, 100}] (* T. D. Noe, Nov 23 2012 *)

Sum{d^n} = ka(n), d runs over the divisors of a(n), where k is an integer and a(n) is the smallest suitable number.

Definition and formulas corrected by Jonathan Sondow, Nov 23 2012

A046985 Multiply perfect numbers whose average divisor is an integer and divides the number itself.

Original entry on oeis.org

1, 6, 672, 30240, 32760, 23569920, 45532800, 14182439040, 51001180160, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 9186050031556349952000, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920

Offset: 1

Labos Elemer

nonn

			k = 45532800 is a term since, s0 = 384, s1 = 182131200, and the three quotients s1/k = 182131200/45532800 = 4, (k * s0)/s1 = (45532800 * 384)/182131200 = 96, and s1/s0 = 182131200/384 = 474300 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..321

Intersection of A003601, A007691 and A001599.

Cf. A000005, A000203, A046986, A046987.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d]]; Select[Range[33000], q] (* Amiram Eldar, May 09 2024 *)

isok(n) = s1 = sigma(n); s0 = numdiv(n); !(s1 % n) && !(s1 % s0) && !((n*s0) % s1); \\ Michel Marcus, Dec 10 2013

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d);} \\ Amiram Eldar, May 09 2024

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/k, (k * s0)/s1, and s1/s0 are all integers.

a(10)-a(15) from Donovan Johnson, Nov 30 2008

Edited and a(16)-a(18) added by Amiram Eldar, May 09 2024

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

Original entry on oeis.org

1, 2, 10, 34, 106, 120, 216, 260, 340, 408, 440, 580, 672, 696, 820, 1060, 1272, 1666, 1780, 1940, 2136, 2340, 2464, 3320, 3576, 3960, 4280, 4536, 5280, 5380, 5860, 6456, 6960, 7520, 8746, 8840, 9120, 9632, 10040, 10776, 12528, 12640, 13464, 14560, 16180, 16660, 17400, 17620, 19040, 19416, 19992, 21320, 22176, 22968

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Numbers k for which A351546(k) is a unitary divisor of k.
The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.
The condition is also a necessary condition for A349745, therefore it is a subsequence 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.
Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).
Subsequence of A351552 and of A351554.
Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
isA351551(n) =  { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

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.

Original entry on oeis.org

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

A066961 Numbers k such that sigma(k) divides sigma(sigma(k)).

Original entry on oeis.org

1, 5, 12, 54, 56, 87, 95, 276, 308, 427, 429, 446, 455, 501, 581, 611, 9120, 9180, 9504, 9720, 9960, 10296, 10620, 10740, 10824, 11070, 11310, 11480, 11484, 11556, 11628, 11748, 11934, 11960, 12024, 12036, 12072, 12084, 12376, 12460, 12510, 12570

Offset: 1

Benoit Cloitre, Jan 26 2002

nonn

Is this sequence finite?

These are numbers k such that sigma(k) is a multiply-perfect number (A007691). - Ivan N. Ianakiev, Sep 13 2016

			12 is in the sequence since sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28 divides sigma(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56. - _Michael B. Porter_, Sep 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..3718 (first 1000 terms from Harry J. Smith)
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A051027.

Subsequences: A323653 (intersection with A007691, or equally, with A019278), A353365 (where the quotient is a power of 2).

[n: n in [1..13000] | (SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0)]; // Vincenzo Librandi, Sep 13 2016

with(numtheory): A066961:=n->`if`(sigma(sigma(n)) mod sigma(n) = 0, n, NULL): seq(A066961(n), n=1..2*10^4); # Wesley Ivan Hurt, Sep 22 2016

Select[Range[30000], Divisible[DivisorSigma[1, DivisorSigma[1, #]], DivisorSigma[1, #]] &] (* Ivan N. Ianakiev, Sep 13 2016 *)

isok(n) = my(s=sigma(n)); s && ((sigma(s) % s) == 0); \\ Michel Marcus, Sep 17 2016

More terms from Lior Manor, Feb 06 2002

A166069 Multiply perfect numbers k such that sigma(k)/k > 2.

Original entry on oeis.org

120, 672, 30240, 32760, 523776, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720, 1476304896, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, Oct 06 2009

nonn

Subsequence of multiply perfect numbers (A007691). Numbers k = A007691(n) such that sigma(A007691(n))/A007691(n) > 2. Numbers k = A007691(n) such that A054030(n) > 2.

			For n = 1 the a(1) = 120, sigma(120) / 120 = 360 / 120 = 3, i.e. > 2.

Charles R Greathouse IV, Table of n, a(n) for n=1..1587
Achim Flammenkamp, The Multiply Perfect Numbers Page

isok(n) = sn = sigma(n)/n ; (type(sn) == "t_INT") && (sn > 2); \\ Michel Marcus, Oct 24 2013

Extended by Charles R Greathouse IV, Oct 12 2009

A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 6, 60, 90, 87360

Offset: 1

Antti Karttunen, Aug 28 2019

10^13 < a(6) <= 146361946186458562560000. - Giovanni Resta, Aug 29 2019

Peter Hagis, Jr., Lower bounds for unitary multiperfect numbers, Fibonacci Quarterly, Vol. 22, No. 2 (1984), pp. 140-143.

Fixed points of A323166, positions of zeros in A327164.
Cf. A002827 (a subsequence), A034448, A327163.
Cf. also A007691.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA327158(n) = (gcd(n,A034448(n))==n);

A348942 a(n) = A326042(n) / gcd(n, A326042(n)).

Original entry on oeis.org

1, 1, 2, 11, 1, 1, 2, 3, 29, 1, 5, 11, 4, 1, 2, 49, 3, 29, 2, 11, 4, 5, 6, 1, 34, 2, 22, 11, 1, 1, 17, 55, 10, 3, 2, 319, 10, 1, 8, 3, 7, 2, 2, 5, 29, 3, 8, 49, 85, 17, 2, 11, 6, 11, 1, 3, 4, 1, 29, 11, 13, 17, 58, 1091, 4, 5, 4, 33, 4, 1, 31, 29, 3, 5, 68, 11, 10, 4, 10, 49, 469, 7, 12, 11, 3, 1, 2, 15, 25, 29, 8

Offset: 1

Antti Karttunen, Nov 04 2021

Numerator of ratio A326042(n)/n. Ratio A326042(n)/n is multiplicative because both A326042 and A000027 are.

Cf. A000203, A003961, A005820, A007691, A046060, A064989, A326042, A348739, A348940, A348941 (denominators), A348943.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (sn = s2[n])/GCD[n, sn]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };

a(n) = A326042(n) / A348940(n) = A326042(n) / gcd(n, A326042(n)).

For all n >= 1, A348943(A064989(n)) = 1.

A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 30823866178560, 69357059049509038080, 4010059765937523916800, 27099073228001299660800, 686498980761986918441287680, 2827987212986831882236723200, 115131961034430181728489308160, 13361233986454282110797768294400, 32789312424503984621373515366400

Offset: 1

Labos Elemer

nonn

			k = 523776 is a term since s0 = d(k) = 80, s1 = sigma(k) = 1571328, s1/k = 1571328/523776 = 3 is an integer, but (k * s0)/s1 = 80/3 and s1/s0 = 98208/5 are not integers.

Amiram Eldar, Table of n, a(n) for n = 1..100

In A007691 but neither in A003601 nor in A001599.

Cf. A000005, A000203, A046985, A046986.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && !Divisible[n * d, s] && !Divisible[s, d]]; Select[Range[6*10^5], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && ((k * d) % s) && (s % d);} \\ Amiram Eldar, May 09 2024

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, but (k * s0) is not divisible by s1, and s1 is not divisible by s0.

a(5)-a(10) from Donovan Johnson, Nov 30 2008

Edited and a(11)-a(13) added by Amiram Eldar, May 09 2024

A160678 Numbers n whose abundancy is equal to 13/2; sigma(n)/n = 13/2.

Original entry on oeis.org

170974031122008628879954060917200710847692800, 1893010442758976546037991125738431754692198400, 54361481238923605327597493185154939181072384000

Offset: 1

Gerard P. Michon, Jun 06 2009

nonn

This sequence includes many terms but it is conjectured to be finite.

			a(1) = 2^23 3^9 5^2 7^5 11^5 13^2 17 19^3 31 37 43 61^2 97 181 241.
As the "sum of divisors" function (sigma) is a multiplicative function, sigma(a(1)) is the product of the values of sigma at the above prime powers, respectively given as follows, in factorized form:
sigma(a(1)) = (3^2 5 7 13 17 241) (2^2 11^2 61) (31) (2^3 3 19 43) (2^2 3^2 7 19 37) (3 61) (2 3^2) (2^3 5 181) (2^5) (2 19) (2^2 11) (3 13 97) (2 7 13) (2 7^2) (2 11^2).
a(1) belongs to the sequence because the latter product boils down to 13/2 times the former.

Michel Marcus, Table of n, a(n) for n = 1..307
G. P. Michon, Multiperfect and hemiperfect integers
G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 13/2
Walter Nissen, Abundancy: Some Resources

Cf. A000203 (sigma function, sum of divisors), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A159907 (half-integral abundancy, "hemiperfect numbers"), A088912 (least numbers of given half-integer abundancy). A007691 (multiperfect numbers, abundancy is an integer), A000396 (perfect numbers, abundancy = 2), A005101 (abundant numbers, abundancy is greater than 2), A005100 (deficient numbers, abundancy is less than 2).

is(n)=sigma(n,-1)==13/2 \\ Charles R Greathouse IV, Feb 21 2017

cp's OEIS Frontend

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

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A046985 Multiply perfect numbers whose average divisor is an integer and divides the number itself.

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A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

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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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A066961 Numbers k such that sigma(k) divides sigma(sigma(k)).

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A166069 Multiply perfect numbers k such that sigma(k)/k > 2.

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A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

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A348942 a(n) = A326042(n) / gcd(n, A326042(n)).