cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A051177 Perfectly partitioned numbers: numbers k that divide the number of partitions p(k).

Original entry on oeis.org

1, 2, 3, 124, 158, 342, 693, 1896, 3853, 4434, 5273, 8640, 14850, 17928, 110516, 178984, 274534
Offset: 1

Views

Author

M.A. Muller (mam(AT)land.sun.ac.za)

Keywords

Comments

Are there infinitely many perfectly partitioned numbers? Does there exist some k > 3 for which p(k) is a perfectly partitioned number?
No other terms below 10^8. - Max Alekseyev, May 19 2014
A probabilistic analysis suggests that there are infinitely many terms. - Franklin T. Adams-Watters, Oct 07 2018

Examples

			a(4) = 124 because p(124) = 2841940500 is divisible by 124.
a(7) = 693 because partition number of 693 is 43397921522754943172592795 = 693*62623263380598763596815.

References

  • Problem 2464, Journal of Recreational Mathematics 29(4), p. 304.
  • Solution to problem 2464 "Perfect Partitions", Journal of Recreational Mathematics 30(4), pp. 294-295, 1999-2000.

Links

Crossrefs

Cf. A000041.
Cf. A093952 = partition number A000041(n) mod n.
Cf. A056848, A128836, A121015.

Programs

  • Mathematica
    Do[ If[ Mod[ PartitionsP@n, n] == 0, Print@n], {n, 250000}] (* Robert G. Wilson v *)
Select[Range[275000],Divisible[PartitionsP[#],#]&] (* Harvey P. Dale, Aug 21 2013~ *)
  • PARI
    for(n=1,20000,if(numbpart(n)%n==0,print1(n,","))) \\ Klaus Brockhaus, Sep 06 2006

Extensions

More terms from Don Reble, Jul 26 2002

A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

Original entry on oeis.org

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

Examples

			From _Gus Wiseman_, Jan 13 2020: (Start)
Strict partitions of a(1) = 3 through a(4) = 7:
  (3)    (4)    (5)    (7)
  (2,1)  (3,1)  (3,2)  (4,3)
                (4,1)  (5,2)
                       (6,1)
                       (4,2,1)
(End)

Links

Crossrefs

The non-strict version is A046063.
The version for powers of 2 instead of primes is A331022.
The version for factorizations instead of strict partitions is A330991.
The version for strict factorizations instead of strict partitions is A331201.
Cf. A000009, A051005, A056848, A265835.

Programs

Extensions

More terms from Eric W. Weisstein
a(12) from Max Alekseyev, Jul 04 2009
a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015
a(15)-a(17) from Max Alekseyev, Jul 10 2015

A162468 Integers n such that A000009(n) (the number of partitions of n into distinct parts) == 1 (mod n).

Original entry on oeis.org

1, 2, 11, 22, 92, 149, 6919, 25517, 45339, 146635, 167903, 7461583, 14809123, 75788157, 80012043
Offset: 1

Views

Author

Max Alekseyev, Jul 04 2009

Keywords

Comments

Integers n dividing A000009(n)-1.
No other terms below 10^8.

Links

Crossrefs

Cf. A000009, A056848, A128836, A259943

Extensions

a(1)=1 prepended by Max Alekseyev, Dec 28 2011
a(13)-a(15) from Max Alekseyev, Jul 10 2015

A259943 Integers n dividing A000009(n)+1.

Original entry on oeis.org

1, 2, 3, 9, 31, 169, 53281, 10984777, 12245367, 19806045
Offset: 1

Views

Author

Max Alekseyev, Jul 10 2015

Keywords

Comments

No other terms below 10^8.

Crossrefs

Cf. A000009, A056848, A162468, A203023

A299961 Numbers k such that k divides the number of overpartitions of k (A015128).

Original entry on oeis.org

1, 2, 12, 13, 22, 29, 88, 284, 370, 781, 1116, 1472, 1518, 1592, 2431, 2475, 2625, 3286, 5264, 6264, 6444, 7512, 7875, 9900, 22515, 30248, 30946, 31500, 32995, 41580, 69920, 112320, 126000, 140580, 142668, 166084, 166968, 225354, 232000, 272538, 290064, 312000
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

Examples

			284 is in the sequence because A015128(284) = 42480456349401075392 is divisible by 284.

Links

Crossrefs

Cf. A015128, A051177, A056848, A265835, A294086.

Extensions

More terms from Vaclav Kotesovec, Mar 02 2018

A304028 Numbers k such that A033461(k) is divisible by k.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128, 2229, 2929, 3022, 4481, 34542, 34951, 36996, 58091, 292949, 437728, 438237, 2103581, 2237158, 3215950, 3375578
Offset: 1

Views

Author

Vaclav Kotesovec, May 04 2018

Keywords

Comments

A001422 is a finite subsequence.

Examples

			2229 is in the sequence because A033461(2229) = 51267 = 23 * 2229.

Crossrefs

Cf. A001422, A033461, A051177, A056848, A276517.

Programs

  • Mathematica
    max = 100; p = ConstantArray[0, max^2 + 1]; p[[1]] = 1; p[[2]] = 1; Do[Do[p[[j + 1]] += p[[j - k^2 + 1]], {j, max^2, k^2, -1}];, {k, 2, max}]; Select[Range[1, max^2], Divisible[p[[# + 1]], #] &]

A304040 Numbers k such that A026007(k) is divisible by k.

Original entry on oeis.org

1, 2, 4, 7, 11, 22, 61, 101, 217, 4846, 29419
Offset: 1

Views

Author

Vaclav Kotesovec, May 05 2018

Keywords

Comments

No other terms below 225000.

Examples

			217 is in the sequence because A026007(217) = 282948942888849443580867409 = 1303912179211287758437177 * 217.

Crossrefs

Cf. A026007, A051177, A056848, A285223, A304028.

A304043 Numbers k such that A022629(k) is divisible by k.

Original entry on oeis.org

1, 2, 5, 8, 28, 34, 50, 529, 1082, 1888, 42000, 112230, 178219
Offset: 1

Views

Author

Vaclav Kotesovec, May 05 2018

Keywords

Comments

No other terms below 1000000.

Examples

			50 is in the sequence because A022629(50) = 206309050 = 4126181 * 50.

Crossrefs

Cf. A022629, A051177, A056848, A304028, A304040.

A325630 Numbers k such that A000110(k) is divisible by k.

Original entry on oeis.org

1, 2, 35, 16833, 16989, 23684
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 07 2019

Keywords

Comments

No other terms below 50000.
From Amiram Eldar, Jun 20 2024: (Start)
Numbers k such that A166226(k) = 0.
All the terms above 2 are composites since A166226(p) == 2 (mod p) for prime p. (End)
No other terms below 90000. - Michael S. Branicky, Jan 09 2025

Examples

			35 is in the sequence because A000110(35) = 35 * 8045720086273150473238297902.

Crossrefs

Cf. A000110, A051130, A066562, A073877, A113883, A166226, A280727, A325935.
Cf. A014847, A016089, A023172, A051177, A056848.

Programs

  • Mathematica
    Select[Range[1000], Divisible[BellB[#], #] &]

A056872 Numbers k such that k | p(k) + q(k) where p(k) = partition numbers (A000041) and q(k) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 5, 25, 42, 133, 618, 643, 701, 1962, 8150, 147458, 168459, 356038, 415870, 536685, 637757, 1093612, 1207618, 3368325, 3470706, 23400631, 37621653
Offset: 1

Views

Author

Robert G. Wilson v, Sep 02 2000

Keywords

Comments

No other terms below 10^8. - Max Alekseyev, Oct 12 2023

Crossrefs

Cf. A000009, A000041, A051177, A056848, A056873, A121919. A128836, A162468, A259943.

Programs

  • Mathematica
    Do[ If[ Mod[ PartitionsP[ n ] + PartitionsQ[ n ], n ] == 0, Print[ n ] ], {n, 1, 8150} ]

Extensions

a(11)-a(18) from Sean A. Irvine, May 12 2022
a(19)-a(22) from Max Alekseyev, Oct 12 2023
Showing 1-10 of 13 results. Next

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