A039701 -id:A039701 - cp's OEIS frontend

A379401 Rectangular array, read by descending antidiagonals: the Type 1 runlength index array of A039701 (primes mod 3); see Comments.

1, 2, 10, 3, 12, 17, 4, 16, 22, 56, 5, 19, 33, 75, 57, 6, 24, 38, 97, 134, 98, 7, 37, 41, 115, 165, 274, 109, 8, 40, 48, 162, 181, 299, 275, 166, 9, 47, 55, 180, 220, 466, 318, 276, 241, 11, 52, 68, 201, 273, 554, 467, 363, 279, 256, 13, 59, 92, 264, 294

Offset: 1

Clark Kimberling, Jan 15 2025

We begin with a definition of Type 1 runlength array, U(s), of a sequence s:
Suppose s is a sequence (finite or infinite), and define rows of U(s) as follows:
(row 0) = s
(row 1) = sequence of 1st terms of runs in (row 0); c(1) = complement of (row 1) in (row 0)
For n>=2,
(row n) = sequence of 1st terms of runs in c(n-1); c(n) = complement of (row n) in (row n-1),
where the process stops if and when c(n) is empty for some n.
***
The corresponding Type 1 runlength index array, UI(s) is now contructed from U(s) in two steps:
(1) Let U*(s) be the array obtaining by repeating the construction of U(s) using (n,s(n)) in place of s(n).
(2) Then UI(s) results by retaining only n in U*.
Thus, loosely speaking, (row n) of UI(s) shows the indices in s of the numbers in (row n) of U(s).
The array UI(s) includes every positive integer exactly once.
***
Regarding the present array, each row of U(s) splits an increasing sequence of primes according to remainder modulo 3; e.g., in row 2, the remainders of primes in positions 10,12,16,19,24,37 are 2,1,2,1,2,1,2,1, respectively.

			Corner:
      1    2    3     4    5      6     7     8     9    11    13    14
     10   12   16    19   24     37    40    47    52    59    72    74
     17   22   33    38   41     48    55    68    92   101   104   112
     56   75   97   115   162   180   201   264   293   328   359   440
     57  134  165   181   220   273   294   341   360   451   545   623
     98  274  299   466   554   624   661   742   786   836   898   941
    109  275  318   467   555   631   704   749   823   839   903  1046
    166  276  363   500   600   758   824   856   912  1059  1176  1212
    241  279  364   505   601   861   913  1076  1177  1229  1258  1368
    256  510  608   866   964  1077  1180  1533  1645  2006  2156  2215
    421  521  709  1088  1181  2007  2163  2248  2551  2690  2919  3138
    424  522  710  1089  1184  2008  2174  2785  2920  3141  3466  3938
Starting with s = A039701, we have for U*(s):
(row 1) = ((1,1), (2,0), (3,2), (4,2), (5,2), (6,1), (7,2), (8,1), (9,2), ...)
c(1) = ((10,2), (12,1), (16,2), (17,2), (14,1), (17,1), (19,1), (22,1), (24,2), ...)
(row 2) = ((10,2), (12,1), (16,2), (19,1), (24,2), (23,1), (27,2), (29,1), (36,2), ...)
c(2) = ((17,2), (22,1), (33,2), ...)
(row 3) = ((17,2), (22,1), ...)
so that UI(s) has
(row 1) = (1,2,3,4,5,6,7,8,9,11,13, ...)
(row 2) = (10,12,16.19,24, ...)
(row 3) = (17,22,33,...)

Cf. A000040, A039701, A379046, A379402, A379403, A379404.

r[seq_] := seq[[Flatten[Position[Prepend[Differences[seq[[All, 1]]], 1], _?(# != 0 &)]], 2]];  (* Type 1 *)
row[0] = Mod[Prime[Range[4000]], 3];(* A039701 *)
row[0] = Transpose[{#, Range[Length[#]]}] &[row[0]];
k = 0; Quiet[While[Head[row[k]] === List, row[k + 1] = row[0][[r[
     SortBy[Apply[Complement, Map[row[#] &, Range[0, k]]], #[[2]] &]]]]; k++]];
m = Map[Map[#[[2]] &, row[#]] &, Range[k - 1]];
p[n_] := Take[m[[n]], 12]
t = Table[p[n], {n, 1, 12}]
Grid[t]  (* array *)
w[n_, k_] := t[[n]][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Dec 04 2024 *)

A379402 Rectangular array, read by descending antidiagonals: the Type 2 runlength index array of A039701 (primes mod 3); see Comments.

Original entry on oeis.org

1, 2, 9, 3, 11, 15, 4, 16, 18, 54, 5, 21, 23, 58, 91, 6, 32, 36, 102, 110, 205, 7, 37, 39, 129, 160, 272, 194, 8, 40, 46, 161, 167, 419, 271, 139, 10, 47, 55, 174, 238, 499, 416, 260, 86, 12, 56, 73, 245, 273, 597, 496, 359, 257, 357, 13, 67, 96, 274, 292

Offset: 1

Clark Kimberling, Jan 15 2025

We begin with a definition of Type 2 runlength array, V(s), of any sequence s for which all the runs referred to have finite length:
Suppose s is a sequence (finite or infinite), and define rows of V(s) as follows:
(row 0) = s
(row 1) = sequence of last terms of runs in (row 0); c(1) = complement of (row 1) in (row 0)
For n>=2,
(row n) = sequence of last terms of runs in c(n-1); c(n) = complement of (row n) in (row n-1),
where the process stops if and when c(n) is empty for some n.
***
The corresponding Type 2 runlength index array,  The runlength index array, VI(s) is now contructed from V(s) in two steps:
(1) Let V*(s) be the array obtaining by repeating the construction of V(s) using (n,s(n)) in place of s(n).
(2) Then VI(s) results by retaining only n in V*.
Thus, loosely speaking, (row n) of VI(s) shows the indices in s of the numbers in (row n) of V(s).
The array VI(s) includes every positive integer exactly once.
***
Regarding the present array, each row of U(s) splits a sequence of primes according to remainder modulo 3; e.g., in row 2, the remainders of primes in positions 9,11,16,21,32,37,40,47 are 2,1,2,1,2,1,2,1, respectively.
Conjecture: every column is eventually increasing.

			Corner:
      1    2    3    4      5     6     7     8    10    12    13    14
      9   11   16   21     32    37    40    47    56    67    71    74
     15   18   23   36     39    46    55    73    96    99   107   111
     54   58  102  129    161   174   245   274   311   326   423   515
     91  110  160  167    238   273   292   321   420   508   598   621
    205  272  419  499    597   618   703   733   813   835   896   932
    194  271  416  496    576   617   702   730   776   834   989  1128
    139  260  359  489    699   713   771   831   988  1127  1173  1190
     86  257  358  464    698   830   987  1124  1164  1185  1251  1298
    357  461  697  829    942  1107  1412  1498  1717  2059  2138  2179
    356  438  889  1062  1714  2046  2137  2176  2551  2820  2927  3270
    291  437  882  1055  1711  2033  2550  2741  2926  3269  3699  3918
Starting with s = A039701, we have for U*(s):
(row 1) = ((1,1), (2,0), (3,2), (4,2), (5,2), (6,1), (7,2), (8,1), (10,2), ...)
c(1) = ((9,2), (11,1), (15,2), (16,2), (18,1), (21,1), (23,1), (32,2), ...)
(row 2) = ((9,2), (11,1), (16,2), (21,1), (36,1), ...)
c(2) = ((15,2), (37,1), ...)
(row 3) = ((15,2), (18,1), (23,2), ...)
so that UI(s) has
(row 1) = (1,2,3,4,5,6,7,8,10,12,13, ...)
(row 2) = (9,11,16.21,32, ...)
(row 3) = (15,18,23,...)

Cf. A000040, A039701, A379046, A379401, A379403, A379404.

r[seq_] := seq[[Flatten[Position[Append[Differences[seq[[All, 1]]], 1], _?(# != 0 &)]], 2]];  (* Type 2 *)
row[0] = Mod[Prime[Range[4000]], 3];(* A039701 *)
row[0] = Transpose[{#, Range[Length[#]]}] &[row[0]];
k = 0; Quiet[While[Head[row[k]] === List, row[k + 1] = row[0][[r[
     SortBy[Apply[Complement, Map[row[#] &, Range[0, k]]], #[[2]] &]]]]; k++]];
m = Map[Map[#[[2]] &, row[#]] &, Range[k - 1]];
p[n_] := Take[m[[n]], 12]
t = Table[p[n], {n, 1, 12}]
Grid[t]  (* array *)
w[n_, k_] := t[[n]][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Dec 04 2024 *)

A118683 Triangle, T(n,k) = A039701(n) + A039701(k) - A039701(n)*A039701(k), read by rows.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, May 19 2006

			Triangle begins as:
  0;
  2, 0;
  0, 2, 0;
  1, 1, 1, 1;
  0, 2, 0, 1, 0;
  1, 1, 1, 1, 1, 1;
  0, 2, 0, 1, 0, 1, 0;
  1, 1, 1, 1, 1, 1, 1, 1;
  0, 2, 0, 1, 0, 1, 0, 1, 0;
  0, 2, 0, 1, 0, 1, 0, 1, 0, 0;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A039701, A099618 (right diagonal).

A039701:= func< n | NthPrime(n) mod 3 >;
A118683:= func< n,k | A039701(n)+A039701(k)-A039701(n)*A039701(k) >;
[A118683(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2024

A039701[n_]:= Mod[Prime[n],3];
T[n_, k_]:= A039701[n] +A039701[k] -A039701[n]*A039701[k];
Table[T[n,k], {n,12}, {k,n}]//Flatten

def A039701(n): return nth_prime(n)%3
def A118683(n,k): return A039701(n)+A039701(k)-A039701(n)*A039701(n)
flatten([[A039701(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Apr 01 2024

Offset corrected, definition clarified, sequence extended by Assoc. Eds. of the OEIS, Jun 15 2010

A002476 Primes of the form 6m + 1.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

N. J. A. Sloane

Equivalently, primes of the form 3m + 1.
Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003
Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003
Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014
Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006
A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - Jonathan Vos Post, Sep 03 2006
A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007
Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008
Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012
Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016
For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016
For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - R. J. Mathar, Apr 16 2024

			Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).)
Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.
17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (-3/p)
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18. See p. 13.
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.
Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index to sequences related to decomposition of primes in quadratic fields

Cf. A045331, A242660.
For values of m see A024899. Primes of form 3n - 1 give A003627.
These are the primes arising in A024892, A024899, A034936.
A091178 gives prime index.
Cf. A006512, A007528.
Subsequence of A016921 and of A050931.
Cf. A004611 (multiplicative closure).

Filtered(List([0..110],k->6*k+1),n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019

a002476 n = a002476_list !! (n-1)
a002476_list = filter ((== 1) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

(#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018

[n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

select(p->p%3==1,primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n >= 1} 1/a(n)^2 = A175644.
Sum_{n >= 1} 1/a(n)^3 = A175645. (End)
a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.
Product_{k>=1} (1 + 1/a(k)^2) = A334481.
Product_{k>=1} (1 - 1/a(k)^3) = A334478.
Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)
Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A003627 Primes of the form 3n-1.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

A007652 Final digit of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 3, 7, 1, 7, 9, 9, 1, 7, 3, 7, 3, 9, 1, 1, 3, 7, 9, 1, 3, 7, 9, 3, 9, 1, 1, 7, 3, 9, 1, 7, 1, 3, 3, 7, 1, 3, 7, 1, 7, 7, 9, 3, 9, 7, 3, 9, 3, 9, 7, 1, 9, 9, 1, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 7, 1, 9, 3, 9, 1, 3, 1, 7, 7, 3, 9, 1

Offset: 1

N. J. A. Sloane, Simon Plouffe

Primes modulo 10.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Alberto Fraile et al., Prime numbers and random walks in a square grid, Phys. Rev. E, 104(5) (2021), 054114.
Evelyn Lamb, "Peculiar Pattern Found in 'Random' Prime Numbers", Nature, March 14, 2016, republished by Scientific American.
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv preprint arXiv:1603.03720 [math.NT], 2016.
Index entries for sequences related to final digits of numbers.

Cf. A000040, A010879, A110923, A039701-A039706, A038194, A039709-A039715, A290450.

[Intseq(p)[1]: p in PrimesUpTo(600)]; // Bruno Berselli, Feb 14 2013

[p mod 10: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 10, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 10], {n, 120}] (* Ray Chandler, Oct 01 2005 *)
Mod[Prime[Range[100]], 10] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%10 \\ Charles R Greathouse IV, Oct 13 2016

primes(100)%10 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = A010879(A000040(n)). - Michel Marcus, May 06 2014

Sum_k={1..n} a(k) ~ 5*n. - Amiram Eldar, Dec 11 2024

Extended by Ray Chandler, Oct 01 2005

A039702 a(n) = n-th prime modulo 4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Except for the first term, A100672(n) = (a(n)-1)/2 = parity of A005097. - Jeremy Gardiner, May 17 2008

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A010873.
Cf. A005097, A100672.
Cf. A039701, A039703-A039706, A100672, A005097, A038194, A007652, A039709-A039715.

a039702 = (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 20 2012

[p mod 4: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 4, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 4], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 4] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%4 \\ Charles R Greathouse IV, Jun 13 2013

Sum_k={1..n} a(k) ~ 2*n. - Amiram Eldar, Dec 11 2024

A039706 a(n) = n-th prime modulo 8.

Original entry on oeis.org

2, 3, 5, 7, 3, 5, 1, 3, 7, 5, 7, 5, 1, 3, 7, 5, 3, 5, 3, 7, 1, 7, 3, 1, 1, 5, 7, 3, 5, 1, 7, 3, 1, 3, 5, 7, 5, 3, 7, 5, 3, 5, 7, 1, 5, 7, 3, 7, 3, 5, 1, 7, 1, 3, 1, 7, 5, 7, 5, 1, 3, 5, 3, 7, 1, 5, 3, 1, 3, 5, 1, 7, 7, 5, 3, 7, 5, 5, 1, 1, 3, 5, 7, 1, 7, 3, 1, 1, 5, 7, 3, 7, 7, 3, 3, 7, 5, 1, 3, 5, 3, 5, 3, 1, 3

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039705, A038194, A007652, A039709-A039715.

[p mod(8): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 8, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 8], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 8] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%8 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ 4*n. - Amiram Eldar, Dec 11 2024

A039703 a(n) = n-th prime modulo 5.

Original entry on oeis.org

2, 3, 0, 2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 3, 2, 1, 2, 4, 4, 1, 2, 3, 2, 3, 4, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 4, 1, 1, 2, 3, 4, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 4, 3, 4, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 2, 1, 4, 3, 4, 1, 3, 1, 2, 2, 3, 4, 1

Offset: 1

Clark Kimberling

a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A010874, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.

[p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)

primes(105) % 5 \\ Zak Seidov, Apr 09 2013

Sum_k={1..n} a(k) ~ (5/2)*n. - Amiram Eldar, Dec 11 2024

A039715 Primes modulo 17.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 0, 2, 6, 12, 14, 3, 7, 9, 13, 2, 8, 10, 16, 3, 5, 11, 15, 4, 12, 16, 1, 5, 7, 11, 8, 12, 1, 3, 13, 15, 4, 10, 14, 3, 9, 11, 4, 6, 10, 12, 7, 2, 6, 8, 12, 1, 3, 13, 2, 8, 14, 16, 5, 9, 11, 4, 1, 5, 7, 11, 8, 14, 7, 9, 13, 2, 10, 16, 5

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709-A039714.

[p mod(17): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 17, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 17], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 17] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%17 \\ Charles R Greathouse IV, Apr 16 2012

[mod(p, 17) for p in primes(500)] # Bruno Berselli, May 05 2014

By the Prime Number Theorem in Arithmetic Progressions, all nonzero residue classes are equiprobable. In particular, Sum_{k=1..n} a(k) ~ 8.5n. - Charles R Greathouse IV, Apr 16 2012

cp's OEIS Frontend

A379401 Rectangular array, read by descending antidiagonals: the Type 1 runlength index array of A039701 (primes mod 3); see Comments.

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A379402 Rectangular array, read by descending antidiagonals: the Type 2 runlength index array of A039701 (primes mod 3); see Comments.

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A118683 Triangle, T(n,k) = A039701(n) + A039701(k) - A039701(n)*A039701(k), read by rows.

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A002476 Primes of the form 6m + 1.

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J

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A003627 Primes of the form 3n-1.

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A007652 Final digit of prime(n).

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