A239394 -id:A239394 - cp's OEIS frontend

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

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

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

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

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).

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.

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

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

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

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

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

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

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

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)

A239393 Nonnegative prime Lipschitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Mar 21 2014

A Lipschitz quaternion has all integer components. The norms of quaternions are (rational) primes 2, 3, 5, 7, 11,... A quaternion is commonly written a + b*i + c*j + d*k, where 1, i, j, and k are units.

			The first six nonnegative prime Lipschitz quaternions are 1+i, 1+j, 1+k, i+j, i+k, and j+k.

T. D. Noe, Table of n, a(n) for n = 1..2586 (4-vectors)
Wikipedia , Hurwitz quaternion

Cf. A239394 (number of Lipschitz quaternions having norm prime(n)).

Cf. A239395 (Hurwitz quaternions).

(* first << Quaternions` *) mx = 5; lst = Flatten[Table[{a, b, c, d}, {a, 0, mx}, {b, 0, mx}, {c, 0, mx}, {d, 0, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; Sort[q, Norm[#1] < Norm[#2] &]

A240068 Number of prime Lipschitz quaternions having norm prime(n).

Original entry on oeis.org

24, 32, 48, 64, 96, 112, 144, 160, 192, 240, 256, 304, 336, 352, 384, 432, 480, 496, 544, 576, 592, 640, 672, 720, 784, 816, 832, 864, 880, 912, 1024, 1056, 1104, 1120, 1200, 1216, 1264, 1312, 1344, 1392, 1440, 1456, 1536, 1552, 1584, 1600, 1696, 1792

Offset: 1

T. D. Noe, Apr 01 2014

nonn

This sequence counts all prime Lipschitz quaternions having a given norm; A239394 counts only the prime nonnegative Lipschitz quaternions.

Wikipedia, Hurwitz quaternion

Cf. A239393 (prime Lipschitz quaternions), A239394.

Cf. A055669 (number of prime Hurwitz quaternions of norm prime(n)).

(* first << Quaternions` *)
mx = 17; lst = Flatten[Table[{a, b, c, d}, {a, -mx, mx}, {b, -mx, mx}, {c, -mx, mx}, {d, -mx, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; q2 = Sort[q, Norm[#1] < Norm[#2] &]; Take[Transpose[Tally[(Norm /@ q2)^2]][[2]], mx]

a(n) = 8 * (prime(n) + 1) = 8 * A008864(n).

cp's OEIS Frontend

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

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A239393 Nonnegative prime Lipschitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

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Mathematica

A240068 Number of prime Lipschitz quaternions having norm prime(n).

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Mathematica

Formula