A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.

30, 84, 264, 273, 286, 325, 351, 364, 390, 441, 490, 525, 624, 756, 784, 810, 840, 874, 900, 988, 1000, 1173, 1197, 1254, 1330, 1425, 1495, 1632, 1771, 2079, 2156, 2178, 2204, 2294, 2310, 2420, 2475, 2750, 2958, 3219, 3393, 3648, 3726, 3770, 3864, 3944, 4042

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A326852.

Links

Table of n, a(n) for n=1..47.

Example

The sequence of terms together with their prime indices begins:
    30: {1,2,3}
    84: {1,1,2,4}
   264: {1,1,1,2,5}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   351: {2,2,2,6}
   364: {1,1,4,6}
   390: {1,2,3,6}
   441: {2,2,4,4}
   490: {1,3,4,4}
   525: {2,3,3,4}
   624: {1,1,1,1,2,6}
   756: {1,1,2,2,2,4}
   784: {1,1,1,1,4,4}
   810: {1,2,2,2,2,3}
   840: {1,1,1,2,3,4}
   874: {1,8,9}
   900: {1,1,2,2,3,3}
   988: {1,1,6,8}

Mathematica

Select[Range[1000], With[{y=Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]}, !SameQ@@y&&Divisible[Total[y], Max[y]]&&Divisible[Total[y], Length[y]]]&]

Crossrefs

The possibly constant case is A326837.

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848, A326851.

Sequence in context: A277980 A241025 A326309 * A098996 A130862 A070756

Adjacent sequences: A326835 A326836 A326837 * A326839 A326840 A326841

Keyword

nonn

Author

Gus Wiseman, Jul 26 2019

Status

approved