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

Author

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

Keywords

• nonn
• nice
• hard
• more

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

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

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