A066872 a(n) = prime(n)^2 + 1.

5, 10, 26, 50, 122, 170, 290, 362, 530, 842, 962, 1370, 1682, 1850, 2210, 2810, 3482, 3722, 4490, 5042, 5330, 6242, 6890, 7922, 9410, 10202, 10610, 11450, 11882, 12770, 16130, 17162, 18770, 19322, 22202, 22802, 24650, 26570, 27890, 29930, 32042

Offset: 1

Author

Joseph L. Pe, Jan 21 2002

Keywords

• easy
• nonn

Comments

From R. J. Mathar, Aug 28 2011: (Start)

There are at least three "natural" embeddings of this function into multiplicative functions b(n), c(n) and d(n):

(i) The first is b(n) = 1, 5, 10, 0, 26, 0, 50, ... (n>=1) with b(p) = p^2+1, b(p^e)=0 if e>=2, substituting zero for all composite n.

(ii) The second is c(n) = 1, 5, 10, 9, 26, 50, 50, 17, 28, 130, ... (n>=1) with c(p^e)= p^(e+1)+1.

(iii) The third is d(n) = 1, 5, 10, 5, 26, 50, 50, 5, 10, 130, ... (n>=1) with d(p^e) = p^2+1 if e>=1. (End)

For n > 1, a(n)/2 is of the form 4*k+1. - Altug Alkan, Apr 08 2016

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)
• R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Crossrefs

Cf. A002522, A000010, A000040, A000203, A001248, A182448, A323599.

Programs

Magma

[p^2+1: p in PrimesUpTo(300)]; // Vincenzo Librandi, Oct 31 2014

Mathematica

Table[Prime[n]^2 + 1, {n, 41}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

A066872(n) = (prime(n)^2 + 1); \\ Harry J. Smith, Apr 02 2010

Formula

a(n) = A002522(A000040(n)). - Altug Alkan, Apr 08 2016

a(n) = A000010(A000040(n)^2) + A323599(A000040(n)^2). - Torlach Rush, Jan 25 2019

Product_{n>=1} (1 - 1/a(n)) = Pi^2/15 (A182448). - Amiram Eldar, Nov 07 2022

From Antti Karttunen, Dec 24 2024: (Start)

a(n) = 1 + A001248(n).

a(n) = A000203(A000040(n)^3) / A000203(A000040(n)). (End)