A356006 -id:A356006 - cp's OEIS frontend

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883

Offset: 1

Alexander Adamchuk, Jul 01 2006

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...

a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Robert Israel, Table of n, a(n) for n = 1..1495
Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A128646 (denominators), A119686, A006093, A000040.
Cf. A135212, A078456.
Cf. A007814, A169611, A356006.

R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025

Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)

a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period {0 1 1 1 1 2 0 1 1 2 0 2 0 1 2 1 0 2 0 2 1 1 0 2 1 1 1 1 0 3} of length 30. There are 26 coincidences on the interval [1,30] with A156542.

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000005, A001221, A156542, A171182, A178143, A178144, A178142.
Cf. A059841, A079978, A079998, A355582, A356006.
Number of distinct prime factors <= p: A171182 (p=3), this sequence (p=5), A210679 (p=7).

Rest@ CoefficientList[Series[-x^2*(3*x^6 + 6*x^5 + 7*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)/((x - 1)*(x + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* G. C. Greubel, May 16 2017 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{0,1,1,1,1,2,0,1},120] (* Harvey P. Dale, Sep 29 2021 *)
a[n_] := PrimeNu[GCD[n, 30]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

my(x='x+O('x^50)); concat([0], Vec(-x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1)/((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)))) \\ G. C. Greubel, May 16 2017

a(n) = omega(gcd(n, 30)); \\ Amiram Eldar, Sep 16 2023

a(n) = a(n-2) + a(n-3) - a(n-7) - a(n-8) + a(n-10), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2, a(7) = 0, a(8) = 1, a(9) = 1, a(10) = 2.
G.f.: -x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Mar 13 2013
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 5, and 0 otherwise.
a(n) = A059841(n) + A079978(n) + A079998(n).
a(n) = A001221(gcd(n, 30)).
a(n) = A001221(A355582(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/30. (End)

Name edited by Amiram Eldar, Sep 16 2023

cp's OEIS Frontend

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions