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Numbers n such that A048767(n) = n.
8

%I #23 Oct 21 2023 01:49:56

%S 1,2,9,12,18,40,112,125,250,352,360,675,832,1008,1125,1350,1500,2176,

%T 2250,2401,3168,3969,4802,4864,7488,7938,11776,14000,19584,21609,

%U 28812,29403,29696,43218,43776,44000,58806,63488,75600,96040,104000,105984,123201,126000

%N Numbers n such that A048767(n) = n.

%C 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

%H Amiram Eldar, <a href="/A048768/b048768.txt">Table of n, a(n) for n = 1..128</a>

%e 12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).

%e From _Gus Wiseman_, May 04 2019: (Start)

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 9: {2,2}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 40: {1,1,1,3}

%e 112: {1,1,1,1,4}

%e 125: {3,3,3}

%e 250: {1,3,3,3}

%e 352: {1,1,1,1,1,5}

%e 360: {1,1,1,2,2,3}

%e 675: {2,2,2,3,3}

%e 832: {1,1,1,1,1,1,6}

%e 1008: {1,1,1,1,2,2,4}

%e 1125: {2,2,3,3,3}

%e 1350: {1,2,2,2,3,3}

%e 1500: {1,1,2,3,3,3}

%e 2176: {1,1,1,1,1,1,1,7}

%e 2250: {1,2,2,3,3,3}

%e 2401: {4,4,4,4}

%e (End)

%t wt[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];

%t Select[Range[1000],wt[#]==#&] (* _Gus Wiseman_, May 04 2019 *)

%o (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

%Y A subsequence of A130091.

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

%K nonn,eigen

%O 1,2

%A _Naohiro Nomoto_

%E a(1) inserted and more terms added by _Amiram Eldar_, Oct 20 2023