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%I #14 Jun 10 2018 22:37:19
%S 31,7,975425,85939741154,936552965614980263201,
%T 933486208332286775628057914016814688052592
%N Denominators a(n) of the fractions Sum_{n>=1} {n/a(n)} = 1/a(1) + 2/a(2) + 3/a(3) + ... such that the sum has the concatenation of these denominators as decimal part. Case a(1) = 31.
%C It appears that fractions of this kind exist only for a(1) equal to 3 (A304288), 10 (A304289), 11 (A305661), 14 (A305662), and 31 (this sequence).
%C For each of the other cases, the terms of the sequence are in increasing order, but this sequence begins with 31, 7, ... (see Example).
%C a(7) has 84 digits. - _Giovanni Resta_, Jun 08 2018
%e 1/31 = 0.03225... At the beginning instead of 31 we have 03 as first decimal digits. Adding the second term this is fixed.
%e 1/31 + 2/7 = 0.317972...
%e 1/31 + 2/7 + 3/975425 = 0.317975425812...
%e The sum is 0.31 7 975425...
%p P:=proc(q,h) local a,b,d,n,t; a:=1/h; b:=ilog10(h)+1; d:=h; print(d);
%p t:=2; for n from 1 to q do if trunc(evalf(a+t/n, 100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+t/n; t:=t+1; print(n); fi; od; end: P(10^20,31);
%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305664, A305665, A305666.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Jun 08 2018
%E a(4)-a(6) from _Giovanni Resta_, Jun 08 2018
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