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

1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 30030, 15015, 255255, 170170, 3233230, 1616615, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 14872858, 7436429, 215656441, 6469693230, 200560490130, 100280245065, 100280245065

Offset

1,2

Comments

Prime divisors counted without multiplicity.

Differs from A379370 first at n=15.

Links

Table of n, a(n) for n=1..33.

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).

a(n) is the denominator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

Example

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, ...

Mathematica

Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
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

Prog

(PARI) a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025

Crossrefs

Cf. A000720, A007947, A013939, A024924, A028235, A284650, A379370, A379368, A380314 (numerators).

Sequence in context: A302033 A337643 A379370 * A071301 A145642 A056195

Adjacent sequences: A380312 A380313 A380314 * A380316 A380318 A380319

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Jan 20 2025

Status

approved