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

[Maximilian Hasler]

>> 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","))

38,58,66,87,118,178,205,217,275,295,298,395,451,478,492,517,538,575,660,718,766,775,838,839,870,898,1018,1138,1175,1195,1318,1671,1678,1775,1795,1975,2050,2163,2170,2295,2395,2518,2578,2638,2665,2750,2818,2875,2950,2995,2998,3118,3175,3567,3635,3775,3837,3857,3875,3894,3898,3950,3998,4163,4175,4195,4198,4378,4510,4618,4645,4667,4678,4692,4775,4798,4862,4875,4918,4920,4994,5098,5170,5218,5578,5609,5638,5750,5786,5875,5878,5896,5907,5938,5944,5975,5998,6178,6418,6505,6557,6598,6600,6775,6778,7195,7438,7660,7678,7750,7978,8057,8058,8098,8278,8390,8578,8616,8668,8700,8818,8975,9185,9298,9591,9635,9646,9682,9725,9745,9778,9838,9879,


> 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.

Maximilian