cp's OEIS Frontend

A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.

%I A380315 #10 Jan 26 2025 17:43:42
%S A380315 1,2,6,3,15,30,210,105,35,70,770,1155,15015,30030,30030,15015,255255,
%T A380315 170170,3233230,1616615,4849845,9699690,223092870,111546435,22309287,
%U A380315 44618574,14872858,7436429,215656441,6469693230,200560490130,100280245065,100280245065
%N A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.
%C A380315 Prime divisors counted without multiplicity.
%C A380315 Differs from A379370 first at n=15.
%F A380315 G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).
%F A380315 a(n) is the denominator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).
%e A380315 0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...
%t A380315 Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
%t A380315 Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
%t A380315 nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
%o A380315 (PARI) a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ _Michel Marcus_, Jan 26 2025
%Y A380315 Cf. A000720, A007947, A013939, A024924, A028235, A284650, A379370, A379368, A380314 (numerators).
%K A380315 nonn,frac
%O A380315 1,2
%A A380315 _Ilya Gutkovskiy_, Jan 20 2025