0,2

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.

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

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]

Cf. A146892, A151659.

Sequence in context: A050930 A074353 A075055 this_sequence A153372 A028402 A092760

Adjacent sequences: A146888 A146889 A146890 this_sequence A146892 A146893 A146894

nonn

Yasutoshi Kohmoto zbi74583.boat at orange.zero.jp, Apr 17 2009

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