A007691 -id:A007691 - cp's OEIS frontend

A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 14182439040, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Numbers m such that sigma(m)/m is an integer f and simultaneously m*tau(m)/sigma(m) is an integer g. Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ...
Complement of A325024 with respect to A007691.
Even perfect numbers from A000396 are terms.
Intersection of A325020 and A007691.
Conjecture: Numbers m such that all values of sigma(m)/m, m*tau(m)/sigma(m) and m*(m-tau(m))/sigma(m) are any integers (f, g, and h respectively). Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, ...

			Multi-perfect number 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..512

Cf. A000005, A000203, A000396, A007691, A049820, A325020, A325021, A325022, A325024.

[n: n in [1..1000000] | IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[#3, #1] == 0, IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(m) = my(s=sigma(m)); (frac(m*(m-numdiv(m))/s) == 0) && (frac(s/m) == 0); \\ Michel Marcus, Mar 25 2019

A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

120, 523776, 459818240, 1476304896, 31998395520, 51001180160, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Jaroslav Krizek, May 12 2019

nonn

Numbers m such that m divides sigma(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325023 with respect to A007691.

			120 is a term because 120*(120-tau(120))/sigma(120) = 120*(120-16)/360 = 104/3.

Amiram Eldar, Table of n, a(n) for n = 1..218

Cf. A000005, A000203, A007691, A049820, A325020, A325021, A325022, A325023.

[n: n in [1..1000000] | not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[#3, #1] == 0, !IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Amiram Eldar, Jul 10 2019 after Michael De Vlieger at A325023 *)

isA325024(m) = { my(s=sigma(m)); ((1==denominator(s/m)) && (1!=denominator(m*(m-numdiv(m))/s))); }; \\ Antti Karttunen, May 25 2019

A134740 Number of distinct prime factors in the n-th multiply perfect number A007691(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 5, 4, 6, 5, 2, 5, 6, 6, 7, 5, 2, 7, 7, 6, 6, 6, 2, 7, 7, 7, 6, 8, 8, 8, 8, 6, 7, 8, 8, 9, 7, 8, 2, 9, 9, 9, 11, 10, 9, 10, 10, 8, 9, 11, 10, 10, 8, 11, 11, 10, 10, 11, 11, 9, 11, 9, 12, 11, 13, 13, 12, 11, 12, 12, 11, 13, 12, 12, 12, 11, 12, 13, 12, 11, 10, 13, 12, 11, 10

Offset: 1

T. D. Noe, Nov 07 2007

nonn

If a multiply perfect number k has abundancy m = sigma(k)/k, then k must have at least A005579(m) distinct prime factors.

T. D. Noe, Table of n, a(n) for n=1..5000 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page

Cf. A007691.

A153800 Indices of perfect numbers (A000396) in the sequence of multiply-perfect numbers (A007691).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 25, 41, 110, 192, 229, 294

Offset: 1

Omar E. Pol, Jan 13 2009

The sequence contains no further entries between 294 and 1600. - R. J. Mathar, May 26 2011

			a(7) is 25 because the 25th multiply-perfect number A007691(25)=137438691328 is also the 7th perfect number A000396(7).

Cf. A000396, A007691.

A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 8589869056, 14182439040, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Multi-perfect numbers from A007691 that are harmonic numbers (A001599). Complement of A325026 with respect to A001599.
Harmonic numbers from A001599 that are multi-perfect numbers (A007691). Complement of A140798 with respect to A007691.
Numbers m such that sigma(m)/m is an integer g and simultaneously m*tau(m)/sigma(m) is an integer h, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203). Corresponding values of integers g: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 3, 4, 2, 5, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, 120, ...
Even perfect numbers from A000396 are terms.

			28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..528

Cf. A000005, A000203, A000396, A001599, A007691, A140798, A325026.

A325021 and A325023 are closely related sequences. - N. J. A. Sloane, May 03 2019

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[DivisorSigma[1, #], #] == 0, IntegerQ@ HarmonicMean@ Divisors@ #] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(n) = my(s=sigma(n)); !frac(s/n) && !frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019

A342659 The least prime that does not divide the n-th multiply perfect number: a(n) = A053669(A007691(n)).

Original entry on oeis.org

2, 5, 3, 7, 3, 5, 3, 11, 11, 5, 11, 7, 3, 7, 5, 3, 11, 5, 3, 13, 11, 7, 3, 7, 3, 11, 5, 11, 5, 13, 11, 5, 11, 11, 7, 7, 5, 13, 5, 5, 3, 11, 5, 11, 23, 11, 11, 11, 11, 11, 11, 23, 11, 13, 7, 13, 23, 11, 5, 5, 13, 5, 11, 5, 17, 17, 17, 17, 17, 17, 17, 17, 13, 17, 17, 11, 19, 11, 23, 19, 11, 5, 7, 11, 5, 7, 17, 17, 7, 7, 5

Offset: 1

Antti Karttunen, Mar 19 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1600

Cf. A007691, A053669.

Cf. also A134740, A342658, A342660.

Map[Block[{k = 1}, While[Mod[#, Prime[k]] == 0, k++]; Prime[k]] &, Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] ] (* or, using the b-file at A007691: *)
Map[Block[{k = 1}, While[Mod[#, Prime[k]] == 0, k++]; Prime[k]] &, DeleteCases[Import["https://oeis.org/A007691/b007691.txt", "Data"], ?(Length@ # == 0 &)][[1 ;; 91, -1]] ] (* _Michael De Vlieger, Mar 19 2021 *)

a(n) = A053669(A007691(n)).

A134665 Exponent of highest power of 2 dividing the n-th multiply perfect number A007691.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 3, 9, 2, 9, 12, 7, 9, 8, 8, 13, 16, 7, 7, 10, 14, 13, 18, 14, 13, 11, 5, 10, 10, 8, 11, 11, 7, 7, 14, 15, 5, 9, 30, 11, 8, 17, 15, 11, 11, 14, 20, 20, 20, 15, 21, 19, 25, 19, 15, 17, 17, 17, 22, 25, 20, 25, 19, 19, 19, 27, 19, 19, 27, 17, 19, 27, 19, 21, 23, 20

Offset: 1

T. D. Noe, Nov 05 2007

nonn

The large terms, for example at n=41, correspond to the perfect numbers A000396 in A007691. See A134666 for the number of multiply perfect numbers having a given highest exponent of 2.

T. D. Noe, Table of n, a(n) for n=1..5000 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page

Cf. A007814, A007691.

a(n) = A007814(A007691(n)). - Michel Marcus, Oct 10 2017

A342658 Number of prime factors in the n-th multiply perfect number, counted with multiplicity: a(n) = bigomega(A007691(n)).

Original entry on oeis.org

0, 2, 3, 5, 5, 7, 7, 10, 8, 12, 9, 15, 13, 14, 15, 13, 14, 17, 17, 17, 18, 18, 19, 20, 19, 20, 20, 21, 16, 21, 22, 18, 24, 26, 22, 19, 24, 29, 19, 22, 31, 29, 21, 32, 31, 26, 28, 28, 33, 35, 34, 34, 37, 37, 35, 34, 35, 35, 36, 33, 38, 38, 37, 39, 42, 42, 39, 46, 40, 42, 48, 39, 38, 46, 40, 40, 47, 41, 44, 46, 43, 39

Offset: 1

Antti Karttunen, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..1600

Cf. A001222, A007691.

Cf. also A134740, A342659, A342660.

Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* or, using the b-file at A007691: *)
Map[PrimeOmega, DeleteCases[Import["https://oeis.org/A007691/b007691.txt", "Data"], ?(Length@ # == 0 &)][[1 ;; 82, -1]] ] (* _Michael De Vlieger, Mar 19 2021 *)

a(n) = A001222(A007691(n)).

a(n) >= A134740(n).

A342660 The largest prime that divides the n-th multiply perfect number: a(n) = A006530(A007691(n)); a(1) = 1.

Original entry on oeis.org

1, 3, 7, 5, 31, 7, 127, 7, 13, 31, 19, 31, 8191, 31, 31, 73, 73, 127, 131071, 19, 19, 89, 151, 127, 524287, 151, 127, 31, 127, 89, 89, 127, 61, 17, 3851, 1093, 151, 257, 911, 331, 2147483647, 3851, 911, 127, 257, 1093, 331, 331, 337, 337, 337, 257, 683, 3851, 8191, 1093, 257, 911, 3851, 1093, 178481, 8191, 337, 8191

Offset: 1

Antti Karttunen, Mar 19 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1600

Cf. A000668, A072002 (subsequences), A006530, A007691.

Cf. also A134740, A342658, A342659.

Map[FactorInteger[#][[-1, 1]] &, Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] ] (* or, using the b-file at A007691: *)
Map[FactorInteger[#][[-1, 1]] &, DeleteCases[Import["https://oeis.org/A007691/b007691.txt", "Data"], ?(Length@ # == 0 &)][[1 ;; 64, -1]] ] (* _Michael De Vlieger, Mar 19 2021 *)

a(n) = A006530(A007691(n)).

A320024 The odd part of those numbers which divide the sum of their divisors (A007691).

Original entry on oeis.org

1, 3, 7, 15, 31, 21, 127, 945, 4095, 1023, 544635, 46035, 8191, 355725, 279279, 1796165, 5388495, 180213, 131071, 110800305, 249987465, 42833475, 3112865, 8109585, 524287, 9338595, 49198149, 253255275, 22017975903, 13341660255, 30101431815, 709933138551

Offset: 1

Peter Luschny, Oct 03 2018

nonn

The data was derived from the data in A007691.
The Mersenne primes A000668 are a subsequence; in fact a prime is in this sequence if and only if it is a Mersenne prime.
Note that John Voight's 'spoof odd perfect number' 22017975903 is included. - Peter Luschny, Oct 30 2020

Alois P. Heinz, Table of n, a(n) for n = 1..1600
BYU Computational Number Theory Group, Odd, spoof perfect factorizations, arXiv:2006.10697 [math.NT], 2020.
J. Voight, On the nonexistence of odd perfect numbers.

Cf. A000265, A000668, A007691.

a_list := proc(len) local L, n; L := NULL; for n from 1 to len do
if numtheory:-sigma(n) mod n = 0 then L := L, n/2^padic[ordp](n, 2) fi od; L end:

A007691 = Select[Range[1000000], Divisible[DivisorSigma[1, #], #] &]; Table[A007691[[n]]/2^IntegerExponent[A007691[[n]], 2], {n, 1, Length[A007691]}] (* Vaclav Kotesovec, Oct 14 2018 *)

a(n) = A000265(A007691(n)).

cp's OEIS Frontend

A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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A134740 Number of distinct prime factors in the n-th multiply perfect number A007691(n).

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A153800 Indices of perfect numbers (A000396) in the sequence of multiply-perfect numbers (A007691).

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A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599).

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A342659 The least prime that does not divide the n-th multiply perfect number: a(n) = A053669(A007691(n)).

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A134665 Exponent of highest power of 2 dividing the n-th multiply perfect number A007691.

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A342658 Number of prime factors in the n-th multiply perfect number, counted with multiplicity: a(n) = bigomega(A007691(n)).

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A342660 The largest prime that divides the n-th multiply perfect number: a(n) = A006530(A007691(n)); a(1) = 1.

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