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A146891
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Terminal point of a repeated reduction of usigma starting at 2^n.
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2
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1, 6, 20, 72, 72, 72, 20, 72, 72, 17280, 4800, 17280, 72, 17280, 1152000, 5184, 5184, 5184, 96000, 5184, 345600, 1244160, 320000, 1244160, 82944000, 89579520, 71663616000, 298598400, 1244160, 82944000, 23040000, 82944000, 19906560000
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OFFSET
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0,2
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COMMENTS
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Let PF_p(n) be the highest power of p dividing n. Examples are PF_2(n) = A006519(n), PF_3(n) = A038500(n) and PF_5(n) = 5^A112765(n) for the cases p = 2, 3, and 5.
Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
For each n, we define an auxiliary sequence b(k) starting at b(0) = 2^n by b(k+1) = A034448(b(k))/PF_(2,3,5)(A034448(b(k)), that is, repeated removal of all powers of 2, 3 and 5 from the unitary sigma value. b(k) terminates at some k with b(k)=1. In addition there is an auxiliary parallel sequence c(k) defined by c(0)=2^n and recursively c(k+1) = c(k)*PF_(3,5)(A034448(b(k)))/A006519(A034448(b(k))), reducing 2^n by the powers of 2 which are divided out of the sequence b.
The sequence is defined by a(n) = c(k), the auxiliary sequence c at the point where b terminates.
All values of the sequence a(n) are 5-smooth, i.e., members of A051037.
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LINKS
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EXAMPLE
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n=5
b(n) : 2^5 -> 11 -> 1
c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
So a(5) = c(2) = 2^3*3^2 = 72.
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MAPLE
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PF := proc(n, p) local nshf, a ; a := 1; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; a := a*p ; od: a ; end:
A146891 := proc(n) local b, a, k, t ;
b := [2^n] ;
while op(-1, b) <> 1 do
od:
a := 2^n ;
for k from 2 to nops(b) do
od:
a ;
end:
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MATHEMATICA
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PF[n_, p_] := p^IntegerExponent[n, p];
usigma[n_] := If[n == 1, 1, Times @@ (1+Power @@@ FactorInteger[n])];
A146891[n_] := Module[{b, a, k, t},
b = {2^n};
While[b[[-1]] != 1,
t = usigma[b[[-1]]];
b = Append[b, t/PF[t, 2]/PF[t, 3]/PF[t, 5]]];
a = 2^n;
For[k = 2, k <= Length[b], k++,
t = usigma[b[[k-1]]];
a = a/PF[t, 2]*PF[t, 3]*PF[t, 5]];
a];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Description of relation between a(n) and c(k) corrected by R. J. Mathar, Jul 07 2009
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STATUS
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approved
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