A008864 a(n) = prime(n) + 1.

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy

Comments

Sum of divisors of prime(n). - Labos Elemer, May 24 2001

For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - T. D. Noe, Mar 31 2014

These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - Hieronymus Fischer, Apr 10 2014

Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800, A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - R. J. Mathar, Mar 15 2018

1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - Amiram Eldar, Jan 23 2021

References

• C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• R. P. Boas and N. J. A. Sloane, Correspondence, 1974
• N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021).

Crossrefs

Cf. A000040, A060800, A131991, A131992, A131993, A141468.

Cf. A007953, A079696, A187813, A239703, A239708.

Column 1 of A341605, column 2 of A286623 and of A328464.

Partial sums of A125266.

Programs

Haskell

a008864 = (+ 1) . a000040
-- Reinhard Zumkeller, Sep 04 2012, Oct 08 2012

Magma

[NthPrime(n)+1: n in [1..70]]; // Vincenzo Librandi, Jul 30 2016

Maple

A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2014

Mathematica

Prime[Range[70]]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

PARI

forprime(p=2,1e3,print1(p+1", ")) \\ Charles R Greathouse IV, Jun 16 2011

PARI

A008864(n) = (1+prime(n)); \\ Antti Karttunen, Mar 14 2021

Sage

[nth_prime(n) +1 for n in (1..70)] # G. C. Greubel, May 20 2019

Formula

a(n) = prime(n) + 1 = A000040(n) + 1.

a(n) = A000005(A034785(n)) = A000203(A000040(n)). - Labos Elemer, May 24 2001

a(n) = A084920(n) / A006093(n). - Reinhard Zumkeller, Aug 06 2007

A239703(a(n)) <= 1. - Hieronymus Fischer, Apr 10 2014

From Ilya Gutkovskiy, Jul 30 2016: (Start)

a(n) ~ n*log(n).

Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)