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Perfectly partitioned numbers: numbers k that divide the number of partitions p(k).
31

%I #43 Aug 12 2024 01:27:20

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

%T 178984,274534

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

%C Are there infinitely many perfectly partitioned numbers? Does there exist some k > 3 for which p(k) is a perfectly partitioned number?

%C No other terms below 10^8. - _Max Alekseyev_, May 19 2014

%C A probabilistic analysis suggests that there are infinitely many terms. - _Franklin T. Adams-Watters_, Oct 07 2018

%D Problem 2464, Journal of Recreational Mathematics 29(4), p. 304.

%D Solution to problem 2464 "Perfect Partitions", Journal of Recreational Mathematics 30(4), pp. 294-295, 1999-2000.

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1029.htm">Puzzle 1029. p that divides the number of partitions of p</a>, The Prime Puzzles and Problems Connection.

%e a(4) = 124 because p(124) = 2841940500 is divisible by 124.

%e a(7) = 693 because partition number of 693 is 43397921522754943172592795 = 693*62623263380598763596815.

%t Do[ If[ Mod[ PartitionsP@n, n] == 0, Print@n], {n, 250000}] (* _Robert G. Wilson v_ *)

%t Select[Range[275000],Divisible[PartitionsP[#],#]&] (* _Harvey P. Dale_, Aug 21 2013~ *)

%o (PARI) for(n=1,20000,if(numbpart(n)%n==0,print1(n,","))) \\ _Klaus Brockhaus_, Sep 06 2006

%Y Cf. A000041.

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

%Y Cf. A056848, A128836, A121015.

%K nonn,nice,hard,more

%O 1,2

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

%E More terms from _Don Reble_, Jul 26 2002