curious eta-GFs and two news seqs

[Joerg Arndt]

sym(j)=eval(Str("t" j))  \\ return symbol tj
 1 + t1*x + (t1^2 + t2)*x^2 + (t1^3 + t2*t1 + t3)*x^3 + \
 (t1^4 + t2*t1^2 + t3*t1 + (t2^2 + t4))*x^4 + [...]
 1 + t2*x + t6*x^3 + t6*t2*x^4 + t10*x^5 + t10*t2*x^6 + \
 t14*x^7 + (t14*t2 + t10*t6)*x^8 + t10*t6*t2*x^9 + t14*t6*x^10 + \
 (t14*t6*t2 + t22)*x^11 + O(x^12)
 1 + t1*x + t3*x^3 + t3*t1*x^4 + t5*x^5 + t5*t1*x^6 + \
 t7*x^7 + (t7*t1 + t5*t3)*x^8 + t5*t3*t1*x^9 + t7*t3*x^10 + \
 (t7*t3*t1 + t11)*x^11 + O(x^12)
 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + 2*x^11 + O(x^12)
 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + 2*x^11 + O(x^12)
Return-Path: <njas at research.att.com>
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Date: Sat, 17 May 2008 12:55:20 -0400
From: "N. J. A. Sloane" <njas at research.att.com>
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To: seqfan at ext.jussieu.fr
Subject: a question about A134865
Cc: njas at research.att.com
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From: Jimmy Lowell <jhbubby at mindspring.com>
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Subject: Extending A134865
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Date: Sat, 17 May 2008 14:03:05 -0400
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Subject: Re: a question about A134865
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> There are 2 numbers, 332640 and 352800, that I want to check and see if they qualify for A134865.

A134865  	 	Numbers n meeting the following criterion: if n is a
multiple of d, then it is also a multiple of the smallest number with
same number of divisors as d.

I offer the following unoptimized PARI code:

isA134865(n)={ n%2 & return(n==1); fordiv(n,d, bigomega(d)>1|next;
nd=numdiv(d); for(k=4,d,numdiv(k)==nd | next; n%k & return;break));1
}
(13:45) gp > for(i=1,999,isA134865(i)&print1(i", "))
1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720,
(13:45) gp > isA134865(332640 )
%1 = 1
(13:47) gp > isA134865(352800)
%2 = 1

i.e. both are in this seq.

Maximilian


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