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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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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EXAMPLE
| 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
| A034448 := proc(n) local ans, i: ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: ans ; end:
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:
A006519 := proc(n) PF(n, 2) ; end:
A038500 := proc(n) PF(n, 3) ; end:
A146891 := proc(n) local b, a, k, t ; b := [2^n] ; while op(-1, b) <> 1 do t := A034448(op(-1, b)) ; b := [op(b), t/A006519(t)/ A038500(t)/PF(t, 5) ] ; od: a := 2^n ; for k from 2 to nops(b) do t := A034448(op(k-1, b)) ; a := a/ A006519(t) *A038500(t)*PF(t, 5) ; od: a ; end:
seq(A146891(n), n=0..60) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 24 2009]
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CROSSREFS
| Cf. A146892, A151659.
Sequence in context: A050930 A074353 A075055 * A189604 A153372 A028402
Adjacent sequences: A146888 A146889 A146890 * A146892 A146893 A146894
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KEYWORD
| nonn
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AUTHOR
| Yasutoshi Kohmoto zbi74583.boat at orange.zero.jp, Apr 17 2009
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 24 2009
Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009
Description of relation between a(n) and c(k) corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009
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