cp's OEIS Frontend

A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.

%I A380314 #18 Jan 26 2025 14:52:56
%S A380314 0,1,5,4,23,71,527,316,117,283,3183,5737,75736,170777,186793,100904,
%T A380314 1730383,1295397,24782713,13522987,42878411,91488457,2113934201,
%U A380314 1149922463,234446350,494634185,169835681,89698402,2608690087,84946052281,2639797313941,1370038779503,1412581913773
%N A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.
%C A380314 Prime divisors counted without multiplicity.
%H A380314 Robert Israel, <a href="/A380314/b380314.txt">Table of n, a(n) for n = 1..2346</a>
%F A380314 G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).
%F A380314 a(n) is the numerator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).
%e A380314 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, ...
%p A380314 N:= 100: # for a(1) .. a(N)
%p A380314 P:= select(isprime,[$1..N]):
%p A380314 f:= proc(n) local k;
%p A380314   numer(add(floor(n/P[k])/P[k],k=1..numtheory:-pi(n)))
%p A380314 end proc:
%p A380314 map(f, [$1..N]); # _Robert Israel_, Jan 26 2025
%t A380314 Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Numerator
%t A380314 Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Numerator
%t A380314 nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
%o A380314 (PARI) a(n) = my(vp=primes(primepi(n))); numerator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ _Michel Marcus_, Jan 26 2025
%Y A380314 Cf. A000720, A007947, A013939, A024451, A024924, A028235, A284648, A379367, A380315 (denominators).
%K A380314 nonn,frac
%O A380314 1,3
%A A380314 _Ilya Gutkovskiy_, Jan 20 2025