A007691 -id:A007691 - cp's OEIS frontend

A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

6, 28, 120, 496, 672, 963, 1036, 5871, 8128, 10479, 164284, 264768, 523776, 2308203, 6511664, 33550336, 41240261, 75384301, 400902412, 459818240, 581013140, 1253768516, 1476304896, 2114464203, 8589869056

Offset: 1

Antti Karttunen, Apr 08 2021

Composite numbers k for which A342926(k) = p*A003415(k), for some prime p.

Corresponding prime p for the first 25 terms is: 2, 2, 3, 2, 3, 3, 3, 11, 2, 11, 2, 3, 3, 5, 2, 2, 101, 397, 2, 3, 5, 7, 3, 5, 2. - Antti Karttunen, Feb 25 2022

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Odd terms in this sequence form a subsequence of A347884.
Cf. A342925, A342926.
Cf. A000396, A005820, A046060, A065997 (subsequences).
Cf. also A342922, A342923, A007691.

Block[{f}, f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Select[Range[4, 10^6], And[CompositeQ[#], PrimeQ[(f[DivisorSigma[1, #]] - #)/f[#] ]] &]] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
isA342924(n) = if((n<2)||isprime(n),0,my(q=(A342925(n)-n)/A003415(n)); ((1==denominator(q))&&isprime(q)));

Terms a(21) - a(25) from Antti Karttunen, Feb 25 2022

A353759 Numbers k for which A353749(k) <= A353749(sigma(k)), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 64, 68, 72, 74, 75, 76, 80, 81, 84, 88, 90, 96, 98, 100, 104, 108, 112, 116, 117, 120, 121, 126, 128, 132, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 192, 196, 198, 200, 204, 208, 216, 220

Offset: 1

Antti Karttunen, May 10 2022

nonn

If an odd number is in the intersection of A348749 and A353684, then it is certainly included in this sequence also.

Positions of nonnegative terms in A353757. Positions of 0's in A353760.
Cf. A353749, A353750, A353758 (complement).
Cf. A007691, A353764 (subsequences).
Cf. also A348749, A353684, A353685.

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
A353760(n) = { my(s=sigma(n)); (A353749(s)<A353749(n)); };
isA353759(n) = !A353760(n);

A046999 Numbers k whose average divisor is nonintegral and divides k.

Original entry on oeis.org

28, 496, 8128, 950976, 2178540, 33550336, 142990848, 301953024, 459818240, 675347400, 714954240, 995248800, 1379454720, 2701389600, 3288789504, 6720569856, 8589869056, 10200236032, 14254365440, 30600708096, 42763096320, 43861478400, 66433720320, 71271827200

Offset: 1

Labos Elemer

nonn

The sequence contains perfect numbers (A000396) and others. Most of them have only small prime factors.
The first three terms are in A007691 (multiply perfect numbers) but 950976 is not since sigma_1/k is not an integer.
sigma_0(k) is the number of divisors of k (A000005).
sigma_1(k) is the sum of the divisors of k [same as sigma(k)] (A000203).
Harmonic numbers that are not arithmetic numbers. Of the 937 harmonic numbers below 10^14 there are just 90 such terms, of them 13 are multiply perfect numbers. - Amiram Eldar, Jun 08 2020

			k=28, sigma_0=6, sigma_1=56, m=sigma_1/sigma_0=9.333... is not an integer, but k/m=3 is;
k=950976, m=2958592/84=3521.333... but k/m=27 is integral.

Amiram Eldar, Table of n, a(n) for n = 1..90 (terms below 10^14)

In A001599 but not in A003601.

Cf. A007691, A046985, A046986, A046987, A000396.

Average divisor = m = sigma_1(k)/sigma_0(k) is not an integer but k/m is.

More terms from Jud McCranie, Dec 25 2000

a(16)-a(24) from Donovan Johnson, Apr 22 2008

A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

Original entry on oeis.org

1, 56, 60, 96, 132, 184, 204, 248, 276, 348, 376, 480, 492, 504, 564, 568, 612, 632, 636, 672, 708, 824, 852, 864, 996, 1016, 1056, 1068, 1208, 1212, 1248, 1284, 1336, 1356, 1520, 1528, 1572, 1592, 1632, 1644, 1656, 1784, 1788, 1824, 1908, 1912, 1980

Offset: 1

Labos Elemer

nonn

			x = 56, sigma(x) = 120, number of divisors of x = 8. 120/8 and 56/8 are integers.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A007691, A001599, A046985.

adiQ[n_]:=Module[{ds1=DivisorSigma[1,n],ds0=DivisorSigma[0,n]} ,Divisible[ ds1,ds0]&&Divisible[n,ds0]]; Select[Range[2000],adiQ] (* Harvey P. Dale, Apr 27 2012 *)

isok(n) = my(d = numdiv(n)); !(n % d) && !(sigma(n) % d); \\ Michel Marcus, Oct 15 2016

Both sigma_1(x)/sigma_0(x) and x/sigma_0(x) are integers. - clarified by Harvey P. Dale, Apr 27 2012

A071707 Numbers k that divide tau(k)*sigma(k).

Original entry on oeis.org

1, 2, 6, 12, 18, 24, 28, 40, 84, 120, 224, 234, 240, 252, 360, 468, 496, 672, 864, 936, 1638, 1920, 2016, 2480, 3276, 4320, 4680, 6048, 6528, 6552, 7440, 8128, 9360, 10880, 22320, 22932, 26208, 30240, 32640, 32760, 47616, 56896, 58752, 65520, 74880, 79360, 84480

Offset: 1

Reinhard Zumkeller, Jun 03 2002

nonn

			The divisors of 18 are {1,2,3,6,9,18}, so tau(18) = 6 and sigma(18) = 1+2+3+6+9+18 = 39, 18 is a term as 18*13 = 6*39 = tau(18)*sigma(18).

Donovan Johnson, Table of n, a(n) for n = 1..300

A007691 is a subsequence.

Cf. A064840 (A000005(n)*A000203(n)).

Select[Range[10^5], Divisible[Times @@ DivisorSigma[{0, 1}, #], #] &] (* Amiram Eldar, Apr 16 2025 *)

isok(k) = {my(f = factor(k)); !((numdiv(f) * sigma(f)) % k);} \\ Amiram Eldar, Apr 16 2025

A076231 Numbers k such that sigma(k)/k, sigma_3(k)/k and sigma_5(k)/k are all integers.

Original entry on oeis.org

1, 6, 120, 672, 8128, 30240, 32760, 33550336, 459818240, 1379454720, 1476304896, 8589869056, 31998395520, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 30823866178560, 796928461056000, 6088728021160320, 14942123276641920, 212517062615531520

Offset: 1

Labos Elemer, Oct 03 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..86

Intersection of A007691, A046763 and A055709.

Cf. A000203, A001158, A001160, A007691, A046763, A055709, A076233, A076234.

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n) && !(sigma(n, 5) % n); \\ Michel Marcus, Dec 26 2013

a(16)-a(20) from Donovan Johnson, May 08 2010

a(21)-a(23) from Amiram Eldar, May 09 2024

A076233 Numbers k such that sigma(k)/k and sigma_3(k)/k are both integers.

Original entry on oeis.org

1, 6, 120, 496, 672, 8128, 30240, 32760, 523776, 23569920, 33550336, 459818240, 1379454720, 1476304896, 8589869056, 31998395520, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 30823866178560, 796928461056000, 6088728021160320, 14942123276641920

Offset: 1

Labos Elemer, Oct 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..156

Cf. A000203, A001158, A007691, A046763, A055709, A076231, A076234.

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n); \\ Michel Marcus, Dec 26 2013

a(19)-a(25) from Amiram Eldar, May 09 2024

A076234 Numbers k such that sigma(k)/k, sigma_3(k)/k, sigma_5(k)/k and sigma_7(k)/k are all integers.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 33550336, 459818240, 1379454720, 8589869056, 31998395520, 51001180160, 137438691328, 153003540480, 30823866178560, 796928461056000, 6088728021160320, 212517062615531520, 2305843008139952128, 69357059049509038080, 143573364313605309726720

Offset: 1

Labos Elemer, Oct 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000203, A001158, A001160, A013955.
Cf. A007691, A046763, A055709, A076231, A076233.
Cf. A066289 (k divides sigma_m(k) for all odd m).

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n) && !(sigma(n, 5) % n) && !(sigma(n, 7) % n); \\ Michel Marcus, Dec 26 2013

a(13)-a(18) from Donovan Johnson, May 08 2010

a(19)-a(21) from Amiram Eldar, May 09 2024

A088844 Multiply perfect numbers k for which the quotient sigma_3(k)/k = A001158(k)/k is nonintegral.

Original entry on oeis.org

28, 2178540, 45532800, 142990848, 14182439040, 43861478400, 518666803200, 704575228896, 13661860101120, 181742883469056, 740344994887680, 20158185857531904, 275502900594021408, 71065075104190073088, 87934476737668055040, 154345556085770649600, 1161492388333469337600

Offset: 1

Labos Elemer, Nov 05 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..574

Cf. A007691, A001158, A066269.

More terms from Amiram Eldar, May 09 2024

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

cp's OEIS Frontend

A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A353759 Numbers k for which A353749(k) <= A353749(sigma(k)), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

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A046999 Numbers k whose average divisor is nonintegral and divides k.

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A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

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A071707 Numbers k that divide tau(k)*sigma(k).

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A076231 Numbers k such that sigma(k)/k, sigma_3(k)/k and sigma_5(k)/k are all integers.

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A076233 Numbers k such that sigma(k)/k and sigma_3(k)/k are both integers.

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A076234 Numbers k such that sigma(k)/k, sigma_3(k)/k, sigma_5(k)/k and sigma_7(k)/k are all integers.

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A088844 Multiply perfect numbers k for which the quotient sigma_3(k)/k = A001158(k)/k is nonintegral.