# Re April 1 comes late this year

N. J. A. Sloane njas at research.att.com
Sun Nov 27 18:02:43 CET 2005

RKG said:

> When I tried  1, 2, 18, 26  on OEIS I got
> no hits, but  1, 3, 8, 9  yielded my full
> quota of  30.
>
> Alas, I fear that they are both finite
> in the present context, so won't make it
> into the Hall of Fame when Neil returns
> from his well-earned rest (though I don't
> believe that he ever takes one).
>
> They are the integer values of
>
> $\prod_{i=1}^n \frac{\sigma n}{n}$
>
> and the values of  n  which make this
> an integer, where  \sigma n  is the
> sum of divisors function.

Me:  Richard, they are indeed too short to
warrant their own entries.  But I'm adding this pair

Neil

%I A111928
%S A111928 1,3,2,7,21,42,48,18,26,234,2808,6552,7056,12096,96768,187488,3374784,7312032,
%T A111928 29248128,307105344,467970048,8423460864,202163060736,101081530368,3133527441408,
%U A111928 5061852020736,1499808006144,2999616012288,17997696073728,215972352884736
%N A111928 Numerator of f(n) := Product_{i=1..n} sigma(i)/i.
%C A111928 Richard Guy observes (Nov 23, 2005) that it appears that f(n) is an integer iff n = 1, 3, 8, 9, when f(n) = 1, 2, 18, 26 respectively.
%Y A111928 Cf. A111934.
%O A111928 1,2
%K A111928 nonn,frac
%e A111928 1, 3/2, 2, 7/2, 21/5, 42/5, 48/5, 18, 26, 234/5, 2808/55, 6552/55, 7056/55, 12096/55, 96768/275, 187488/275, 3374784/4675, 7312032/4675, 29248128/17765, 307105344/88825, ...
%p A111928 with(numtheory); f:=n->mul(sigma(i)/i,i=1..n);
%A A111928 njas, Nov 27 2005

%I A111934
%S A111934 1,2,1,2,5,5,5,1,1,5,55,55,55,55,275,275,4675,4675,17765,88825,88825,977075,
%T A111934 22472725,4494545,112363625,112363625,22472725,22472725,130341805,651709025,
%U A111934 651709025,651709025,7168799275,121869587675,609347938375,609347938375
%N A111934 Denominator of f(n) := Product_{i=1..n} sigma(i)/i.
%C A111934 Richard Guy observes (Nov 23, 2005) that it appears that f(n) is an integer iff n = 1, 3, 8, 9, when f(n) = 1, 2, 18, 26 respectively.
%Y A111934 Cf. A111928.
%O A111934 1,2
%K A111934 nonn,frac
%e A111934 1, 3/2, 2, 7/2, 21/5, 42/5, 48/5, 18, 26, 234/5, 2808/55, 6552/55, 7056/55, 12096/55, 96768/275, 187488/275, 3374784/4675, 7312032/4675, 29248128/17765, 307105344/88825, ...
%p A111934 with(numtheory); f:=n->mul(sigma(i)/i,i=1..n);
%A A111934 njas, Nov 27 2005