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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) = 14.
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%I #10 Jun 10 2018 22:37:02

%S 14,28,619829,64119408562,89683259163592378754652,

%T 97322840981260080521305808534009505107316888799

%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) = 14.

%C It appears that fractions of this kind exist only for a(1) equal to 3 (A304288), 10 (A304289), 11 (A305661), 14 (this sequence), and 31 (A305663).

%C a(7) has 94 digits. - _Giovanni Resta_, Jun 08 2018

%e 1/14 = 0.07142... At the beginning instead of 14 we have 07 as first decimal digits. Adding the second term this is fixed.

%e 1/14 + 2/28 = 0.14285771...

%e 1/14 + 2/28 + 3/619829 = 0.1428619829017...

%e The sum is 0.14 28 619829...

%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(10^20,14);

%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305663, 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