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A048768
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Numbers n such that A048767(n) = n.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).
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)
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MATHEMATICA
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wt[n_]:=Times@@Cases[FactorInteger[n], {p_, k_}:>Prime[k]^PrimePi[p]];
Select[Range[1000], wt[#]==#&] (* Gus Wiseman, May 04 2019 *)
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PROG
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(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
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CROSSREFS
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Cf. A008478, A048767, A048769, A056239, A098859, A109297, A109298, A112798, A118914, A217605, A325326, A325368.
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KEYWORD
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nonn,eigen
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AUTHOR
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EXTENSIONS
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a(1) inserted and more terms added by Amiram Eldar, Oct 20 2023
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STATUS
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approved
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