A226774 Integers a(n) = Sum_{i=1..q} 1/d(i) where d(i) are the divisors of A225110(n) for some q.

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Michel Lagneau, Jun 17 2013

Keywords

• nonn

Comments

The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 10, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 24, 4, 4, 4, 4, 4, 4, ...

By convention, a(1)=1. For a majority of n, a(n) = 2.

a(n) = 3 for n = 11, 16, 52, 145, 634, ... where A225110(n) = 120, 180, 672, 1890, 8460, ...

a(n) = 4 for n = 2284, 2476, 6871, ... where A225110(n) = 30240, 32760, 90720, ...

Examples

			a(16) = 3 because the divisors of A225110(16) = 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 and 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/30 + 1/36 + 1/45 = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A225110.

Programs

Maple

with(numtheory): for n from 1 to 2000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to
n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 then ii:=1: printf(`%d, `, s):else fi:od:od:

PARI

either_A226774_or_0(n) = { if(1==n,return(1)); my(divs=divisors(n),s=0); for(i=1,#divs,s += (1/divs[i]); if((1==denominator(s))&&(i>1),return(s))); return(0); };
up_to = 16384; i=0; n=0; while(iA226774_or_0(n); if(s, i++; write("b226774.txt", i, " ", s)));
\\ Antti Karttunen, Dec 16 2017