A048768 Numbers n such that A048767(n) = n.

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 11776, 14000, 19584, 21609, 28812, 29403, 29696, 43218, 43776, 44000, 58806, 63488, 75600, 96040, 104000, 105984, 123201, 126000

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions that are fixed points under the map described in A217605 (which interchanges the parts with their multiplicities). The enumeration of these partitions by sum is given by A217605. - Gus Wiseman, May 04 2019

Links

Amiram Eldar, Table of n, a(n) for n = 1..128

Example

12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).
From Gus Wiseman, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   125: {3,3,3}
   250: {1,3,3,3}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
  1008: {1,1,1,1,2,2,4}
  1125: {2,2,3,3,3}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  2176: {1,1,1,1,1,1,1,7}
  2250: {1,2,2,3,3,3}
  2401: {4,4,4,4}
(End)

Mathematica

wt[n_]:=Times@@Cases[FactorInteger[n], {p_, k_}:>Prime[k]^PrimePi[p]];
Select[Range[1000], wt[#]==#&] (* Gus Wiseman, May 04 2019 *)

Prog

(PARI) is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); #Set(e) == #e && prod(i = 1, #e, prime(e[i])^primepi(p[i])) == n; } \\ Amiram Eldar, Oct 20 2023

Crossrefs

A subsequence of A130091.

Cf. A008478, A048767, A048769, A056239, A098859, A109297, A109298, A112798, A118914, A217605, A325326, A325368.

Sequence in context: A360453 A324570 A109297 * A070226 A273669 A129829

Adjacent sequences: A048765 A048766 A048767 * A048769 A048770 A048771

Keyword

nonn,eigen

Author

Naohiro Nomoto

Extensions

a(1) inserted and more terms added by Amiram Eldar, Oct 20 2023

Status

approved