A066872 - OEIS

%I #56 Dec 24 2024 18:25:41

%S 5,10,26,50,122,170,290,362,530,842,962,1370,1682,1850,2210,2810,3482,

%T 3722,4490,5042,5330,6242,6890,7922,9410,10202,10610,11450,11882,

%U 12770,16130,17162,18770,19322,22202,22802,24650,26570,27890,29930,32042

%N a(n) = prime(n)^2 + 1.

%C From _R. J. Mathar_, Aug 28 2011: (Start)

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

%C (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.

%C (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.

%C (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)

%C For n > 1, a(n)/2 is of the form 4*k+1. - _Altug Alkan_, Apr 08 2016

%H Antti Karttunen, <a href="/A066872/b066872.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Harry J. Smith)

%H R. P. Boas & N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>

%F a(n) = A002522(A000040(n)). - _Altug Alkan_, Apr 08 2016

%F a(n) = A000010(A000040(n)^2) + A323599(A000040(n)^2). - _Torlach Rush_, Jan 25 2019

%F Product_{n>=1} (1 - 1/a(n)) = Pi^2/15 (A182448). - _Amiram Eldar_, Nov 07 2022

%F From _Antti Karttunen_, Dec 24 2024: (Start)

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

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

%t Table[Prime[n]^2 + 1, {n, 41}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%o (PARI) A066872(n) = (prime(n)^2 + 1); \\ _Harry J. Smith_, Apr 02 2010

%o (Magma) [p^2+1: p in PrimesUpTo(300)]; // _Vincenzo Librandi_, Oct 31 2014

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

%K easy,nonn

%O 1,1

%A _Joseph L. Pe_, Jan 21 2002