A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.
0, 1, 5, 4, 23, 71, 527, 316, 117, 283, 3183, 5737, 75736, 170777, 186793, 100904, 1730383, 1295397, 24782713, 13522987, 42878411, 91488457, 2113934201, 1149922463, 234446350, 494634185, 169835681, 89698402, 2608690087, 84946052281, 2639797313941, 1370038779503, 1412581913773
Offset: 1
Examples
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, ...
Links
- Robert Israel, Table of n, a(n) for n = 1..2346
Crossrefs
Programs
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Maple
N:= 100: # for a(1) .. a(N) P:= select(isprime,[$1..N]): f:= proc(n) local k; numer(add(floor(n/P[k])/P[k],k=1..numtheory:-pi(n))) end proc: map(f, [$1..N]); # Robert Israel, Jan 26 2025
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Mathematica
Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Numerator Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Numerator 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 -
PARI
a(n) = my(vp=primes(primepi(n))); numerator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025
Formula
G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).
a(n) is the numerator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).
Comments