[seqfan] Re: Fwd: A nice (decimal) property of 78

Maximilian Hasler maximilian.hasler at gmail.com
Sat Nov 8 21:09:55 CET 2008

>> phi(31977)=phi(31)*phi(97)*phi(7)
>> phi(437388)=phi(43)*phi(73)*phi(88)
>> phi(773976)=phi(7)*phi(73)*phi(976)
>> phi(778998)=phi(77)*phi(89)*phi(98)

> And why stop there? We should do sigma as well.

there is already the single digit version:
A098771		 Numbers n such that
sigma(n)=sigma(d_1)*sigma(d_2)*...*sigma(d_k) where d_1 d_2 ... d_k is
the decimal expansion of n.
1, 2, 3, 4, 5, 6, 7, 8, 9, 38, 58, 66, 87,

the ( a || b ) version can be computed as follows:

is_sig(n)={ local(p=1, s=sigma(n)); while( n>p*=10, n%p | next;
s==sigma( n\p )*sigma( n%p ) & return(1))}
for(n=1,9999, is_sig(n) & print1(n","))


> The problem is, in order to prove there are an infinitude of primitive
> solutions, you generally have to exhibit and infinite family of primitive
> solutions. But then only one member of the family is primitive, the others
> become nonprimitive members of the family.

well, for pythagorean triples there are definitions that make sense.
but I agree that here it does not make much sense to disallow trailing
zeroes of the 2nd half, b, but allow them for the 1st part, a.


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