A007691 -id:A007691 - cp's OEIS frontend

A098203 Euler totient function phi values of multiperfect numbers.

1, 2, 12, 32, 240, 192, 4032, 6912, 6912, 153600, 435456, 5529600, 16773120, 11059200, 33177600, 143327232, 286654464, 433520640, 4294901760, 2627665920, 6019743744, 10705305600, 15925248000, 15606743040, 68719214592, 31850496000, 93640458240, 100329062400, 172009474560

Offset: 1

Labos Elemer, Sep 23 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..730

Cf. A000010, A007691, A098204.

a(n) = A000010(A007691(n)).

More terms from Amiram Eldar, May 10 2024

A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

Original entry on oeis.org

120, 672, 32760, 2178540, 1379454720, 14182439040, 518666803200, 30823866178560, 71065075104190073088, 154345556085770649600, 9186050031556349952000, 680489641226538823680000

Offset: 1

Sven Simon, Feb 19 2006

nonn

From a list of about 5000 multiperfect numbers, 38 numbers were found with the property, all having k <= 9, the largest was the only one having k=9. A091443 uses sopfr with repetition.

Conjecture: the sequence is finite.

			a(0) = 120 = 2^3*3*5, sopf(120) = 2+3+5 = 10.

Sven Simon, Table of n, a(n) for n = 1..38 [Conjectured to be complete]
Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A091443.

Intersection of A007691 and A089352. - Michel Marcus, Oct 08 2017

A153801 Index of Mersenne number A000225 that is also Mersenne prime A000668, minus n-th prime: a(n) = A000043(n) - A000040(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 2, 12, 38, 60, 76, 90, 480, 564, 1232, 2150, 2222, 3156, 4186, 4352, 9616, 9862, 11130, 19848, 21604, 23108, 44394, 86136, 110394, 131936, 215964, 756708, 859296, 1257648, 1398120, 2976070, 3021220, 6972430, 13466750, 20995838, 24036404, 25964770

Offset: 1

Omar E. Pol, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A000040, A000043, A000668, A000225, A000396, A007691, A153800, A153802, A153803.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Prime[Range[max]]] (* Amiram Eldar, Oct 21 2024 *)

More terms from R. J. Mathar, Feb 19 2009

More terms from Jinyuan Wang, Mar 02 2020

A165701 Numbers n such that 5^n-6 is prime.

Original entry on oeis.org

2, 4, 5, 6, 10, 53, 76, 82, 88, 242, 247, 473, 586, 966, 1015, 1297, 1825, 2413, 2599, 2833, 5850, 5965, 6052, 27199, 49704, 79000

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Oct 30 2009

Numbers corresponding to the a(n) for n>11 are probable prime.
If Q is a 4-perfect number and gcd(Q, 5*(5^a(n)-6))=1 then m=5^(a(n)-1)
(5^a(n)-6)*Q is a solution of the equation sigma(x)=5(x+Q)(see comment lines of the sequence A058959). 142990848 is the smallest 4-perfect number m such that 5 doesn't divide m.
a(27) > 10^5. - Robert Price, Feb 03 2014

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1
H. Lifchitz, R. Lifchitz: PRP Top Records Search for 5^n-6.

Cf. A007691, A054030, A058959.

Mathematica
```
Do[If[PrimeQ[5^n-6],Print[n]],{n,8888}]
```

is(n)=ispseudoprime(5^n-6) \\ Charles R Greathouse IV, Jun 13 2017

a(24)-a(26) from Robert Price, Feb 03 2014

A166070 Sorted sequence of primes and multiply perfect numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Jaroslav Krizek, Oct 06 2009, Oct 16 2009

nonn

Numbers k such that sigma(k) is divisible by all proper divisors of k.

			a(4) = 6: all proper divisors of 6 (1, 2, 3) divide sigma(6) = 12.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A053813.

Select[Range[300],And@@Divisible[DivisorSigma[1,#],Most[Divisors[#]]]&] (* Harvey P. Dale, Jan 18 2015 *)

A000040 Union A007691.
{1} Union A053813 Union A166069.
{1} Union A000040 Union A000396 Union A166069.

A183013 Largest members of k-sociable cycles of order r.

Original entry on oeis.org

6, 28, 120, 284, 496, 672, 1210, 2924, 5564, 6368, 8128, 10856, 14595, 15472, 18416, 30240, 32760, 66992, 71145, 76084, 87663, 88730, 123152, 124155, 139815, 153176, 168730, 176336, 180848, 202444, 203432, 365084, 389924, 399592, 430402

Offset: 1

William Rex Marshall, Jan 08 2011

nonn

A k-sociable (or multisociable) cycle of order r consists of r distinct positive integers such that the sum of the aliquot divisors (or proper divisors) of each is equal to k times the next term in the cycle, where k (the multiplicity) is a fixed positive integer.
A183014(n) gives the multiplicity of the cycle with largest term a(n).
A183015(n) gives the order of the cycle with largest term a(n).
If examples of two or more multisociable cycles with the same largest term exist, the largest term is repeated in this sequence, and corresponding multiplicities listed in order of increasing size in A183014. (No such examples are known. Do any exist?)

Cf. A000396, A001065, A002046, A007691, A183014 (multiplicities), A183015 (orders), A183019, A183020.

A183016 Conjectured list of smallest terms of k-sociable cycles of order r.

Original entry on oeis.org

6, 28, 120, 220, 496, 672, 1184, 2620, 5020, 6232, 8128, 10744, 12285, 12496, 14316, 17296, 30240, 32760, 63020, 66928, 67095, 69615, 79750, 100485, 122265, 122368, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319550

Offset: 1

William Rex Marshall, Jan 08 2011

nonn

A k-sociable (or multisociable) cycle of order r consists of r distinct positive integers such that the sum of the aliquot divisors (or proper divisors) of each is equal to k times the next term in the cycle, where k (the multiplicity) is a fixed positive integer.
A183017(n) gives the multiplicity of the cycle with smallest term a(n).
A183018(n) gives the order of the cycle with smallest term a(n).
If examples of two or more multisociable cycles with the same smallest term exist, the smallest term is repeated in this sequence, and corresponding multiplicities listed in order of increasing size in A183017. (No such examples are known. Do any exist?)

Cf. A000396, A001065, A002025, A003416, A007691, A183017 (multiplicities), A183018 (orders), A183019, A183021.

A183019 Conjectured list of multisociable numbers.

Original entry on oeis.org

6, 28, 120, 220, 284, 496, 672, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14536, 14595, 15472, 17296, 18416, 30240, 32760, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633

Offset: 1

William Rex Marshall, Jan 08 2011

nonn

Cf. A000396, A001065, A007691, A063990, A122726, A183013, A183016, A183022, A183029.

A227303 Numbers k such that k divides sigma(3*k).

Original entry on oeis.org

1, 2, 4, 28, 40, 78, 90, 224, 360, 496, 546, 2016, 2184, 8128, 10080, 10920, 11880, 66528, 145236, 174592, 714240, 726180, 1571328, 4333056, 6168960, 7856640, 12065760, 15177600, 33550336, 47663616, 69521760, 80196480, 91963648, 99993600, 156854880, 459818240, 492101632

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

If k belongs to the sequence, then sigma(3*k)/k is an integer, so sigma(3*k)/(3*k) is either an integer or a third of an integer, so 3*k is either multiperfect or belongs to A160320 or A160321. - Michel Marcus, Jul 09 2013

Jud McCranie, Table of n, a(n) for n = 1..65

Cf. A000203, A007691, A227302.

Cf. A160320, A160321.

k = 0; lst = {}; While[k < 10^11, If[ Mod[ DivisorSigma[1, 3 k], k] == 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 07 2021 *)

isok(k) = !(sigma(3*k) % k); \\ Michel Marcus, Mar 07 2021

A227306 Numbers k that divide sigma(k) + sigma(k-1).

Original entry on oeis.org

2, 6, 34, 50, 216, 236, 262, 386, 898, 924, 945, 1456, 2380, 5356, 6468, 6624, 8362, 14100, 23496, 26938, 46594, 80876, 196344, 212796, 1661136, 4070200, 4160920, 4626700, 5244548, 5462384, 17062316, 60464628, 217408416, 248621604, 262792908, 265371336, 323987588

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

Is 945 the only odd term? - Zak Seidov, Jul 06 2013
945 and 19910536425 are the only odd terms below 2^36. - Alex Ratushnyak, Jul 08 2013
The third odd term is a(58) = 841488503841. - Giovanni Resta, Apr 04 2014

Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 10^13)

Cf. A000203, A092403, A007691, A068078, A186103, A072188.

With[{nn=324*10^6},Select[Thread[{Total/@Partition[DivisorSigma[ 1,Range[ nn]],2,1],Range[ 2,nn]}],Divisible[#[[1]],#[[2]]]&][[All,2]]] (* Harvey P. Dale, May 29 2020 *)

cp's OEIS Frontend

A098203 Euler totient function phi values of multiperfect numbers.

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A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

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A153801 Index of Mersenne number A000225 that is also Mersenne prime A000668, minus n-th prime: a(n) = A000043(n) - A000040(n).

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A165701 Numbers n such that 5^n-6 is prime.

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Mathematica

PARI

Extensions

A166070 Sorted sequence of primes and multiply perfect numbers.

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Examples

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Programs

Mathematica

Formula

A183013 Largest members of k-sociable cycles of order r.

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A183016 Conjectured list of smallest terms of k-sociable cycles of order r.

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A183019 Conjectured list of multisociable numbers.

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A227303 Numbers k such that k divides sigma(3*k).

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Links

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Programs

Mathematica

PARI

A227306 Numbers k that divide sigma(k) + sigma(k-1).

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Keywords

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