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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

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

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

URL: https://oeis.org/A051177