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

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

Offset

1,2

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

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

Table of n, a(n) for n=1..17.

Carlos Rivera, Puzzle 1029. p that divides the number of partitions of p, The Prime Puzzles and Problems Connection.

Example

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

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~ *)

Prog

(PARI) for(n=1, 20000, if(numbpart(n)%n==0, print1(n, ", "))) \\ Klaus Brockhaus, Sep 06 2006)

Crossrefs

Cf. A000041.

Cf. A093952 = partition number A000041(n) mod n.

Cf. A056848, A128836, A121015.

Sequence in context: A041813 A065842 A065841 * A371271 A334661 A258968

Adjacent sequences: A051174 A051175 A051176 * A051178 A051179 A051180

Keyword

nonn,nice,hard,more

Author

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

Extensions

More terms from Don Reble, Jul 26 2002

Status

approved