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A048767
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If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.
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12
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1, 2, 4, 3, 8, 8, 16, 5, 9, 16, 32, 12, 64, 32, 32, 7, 128, 18, 256, 24, 64, 64, 512, 20, 27, 128, 25, 48, 1024, 64, 2048, 11, 128, 256, 128, 27, 4096, 512, 256, 40, 8192, 128, 16384, 96, 72, 1024, 32768, 28, 81, 54, 512, 192, 65536, 50, 256, 80, 1024, 2048
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OFFSET
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1,2
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COMMENTS
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If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - Michael De Vlieger, Apr 25 2017
Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - Gus Wiseman, May 04 2019
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LINKS
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EXAMPLE
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For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
1: {} {}
2: {1} {1}
3: {2} {1,1}
4: {1,1} {2}
5: {3} {1,1,1}
6: {1,2} {1,1,1}
7: {4} {1,1,1,1}
8: {1,1,1} {3}
9: {2,2} {2,2}
10: {1,3} {1,1,1,1}
11: {5} {1,1,1,1,1}
12: {1,1,2} {1,1,2}
13: {6} {1,1,1,1,1,1}
14: {1,4} {1,1,1,1,1}
15: {2,3} {1,1,1,1,1}
16: {1,1,1,1} {4}
17: {7} {1,1,1,1,1,1,1}
18: {1,2,2} {1,2,2}
19: {8} {1,1,1,1,1,1,1,1}
20: {1,1,3} {1,1,1,2}
(End)
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MAPLE
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local a, p, e, f;
a := 1 ;
for f in ifactors(n)[2] do
p := op(1, f) ;
e := op(2, f) ;
a := a*ithprime(e)^numtheory[pi](p) ;
end do:
a ;
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MATHEMATICA
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Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* Ivan Neretin, Jun 02 2016 *)
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* Michael De Vlieger, Apr 25 2017 *)
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CROSSREFS
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KEYWORD
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easy,nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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