A040976 -id:A040976 - cp's OEIS frontend

A206038 Values of the difference d for 4 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 3.

6, 12, 18, 42, 48, 54, 84, 96, 126, 132, 252, 348, 396, 426, 438, 474, 594, 636, 642, 648, 678, 804, 858, 1176, 1218, 1272, 1302, 1314, 1362, 1428, 1482, 1566, 1692, 1728, 1896, 1992, 2064, 2106, 2238, 2394, 2442, 2574, 2616, 2688, 2694, 2706, 2832, 2856, 2898

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

			d = 18 then {5, 5 + 1*18, 5 + 2*18, 5 + 3*18} = {5, 23, 41, 59}, which is 4 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083, 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037 - A206045.

t={}; Do[If[PrimeQ[{5, 5 + d, 5 + 2*d, 5 + 3*d}] == {True, True, True, True}, AppendTo[t, d]], {d, 3000}]; t

A206043 Values of the difference d for 9 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 8.

Original entry on oeis.org

32671170, 54130440, 59806740, 145727400, 224494620, 246632190, 280723800, 301125300, 356845020, 440379870, 486229380, 601904940, 676987920, 777534660, 785544480, 789052530, 799786890, 943698210, 1535452800, 1536160080, 1760583300, 1808008020

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

			d = 54130440 then {11, 54130451, 108260891, 162391331, 216521771, 270652211, 324782651, 378913091, 433043531} which is 9 primes in arithmetic progression.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1419
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037, A206038, A206039, A206040, A206041, A206042, A206044, A206045.

a = 11; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d, a + 7*d, a + 8*d}] == {True, True, True, True, True, True, True, True, True}, AppendTo[t,d]], {d, 10^9}]; t

forstep(k=210,1e10,210,forstep(p=k+11,8*k+11,k,if(!isprime(p), next(2)));print1(k", ")) \\ Charles R Greathouse IV, Feb 09 2012

a(20) corrected by Charles R Greathouse IV, Feb 09 2012

A206040 Values of the difference d for 6 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 5.

Original entry on oeis.org

30, 150, 930, 2760, 3450, 4980, 9150, 14190, 19380, 20040, 21240, 28080, 33930, 57660, 59070, 63600, 69120, 76710, 80340, 81450, 97380, 100920, 105960, 114750, 117420, 122340, 134250, 138540, 143670, 150090, 164580, 184470, 184620, 189690, 231360, 237060

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

			d = 150 then {7, 7 + 1*150, 7 + 2*150, 7 + 3*150, 7 + 4*150, 7 + 5*150} = {7, 157, 307, 457, 607, 757} which is 6 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037, A206038, A206039, A206041, A206042, A206043, A206044, A206045.

a = 7; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d}] == {True, True, True, True, True, True}, AppendTo[t,d]], {d, 300000}]; t
Select[Range[250000],AllTrue[7+#*Range[0,5],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2017 *)

A092953 Number of primes of the form n+p, where p is a prime < n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 0, 2, 1, 2, 1, 3, 0, 2, 1, 3, 1, 3, 0, 3, 1, 2, 0, 6, 0, 4, 1, 3, 1, 6, 0, 3, 0, 4, 1, 6, 0, 4, 1, 5, 1, 8, 0, 4, 1, 4, 0, 7, 0, 6, 1, 4, 0, 9, 0, 8, 1, 4, 1, 11, 0, 5, 0, 5, 1, 11, 0, 6, 1, 8, 1, 9, 0, 4, 0, 7, 1, 11, 0, 7, 1, 4, 0, 13, 0, 7, 1, 5, 0, 15, 0, 7, 0, 8, 1, 13, 0, 8, 1, 9, 1, 11

Offset: 1

Amarnath Murthy, Mar 24 2004

nonn

Might be called the additive primability of n.

a(A007921(n))=0; for n > 2: a(A030173(n)) > 0 and a(A040976(n)) = 1. - Reinhard Zumkeller, Nov 10 2012

			a(26) = 4: the primes are 29, 31, 37 and 43.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A092954.

Cf. A061357.

a092953 n = sum $
   zipWith (\u v -> a010051' u * a010051' v) [1 .. n - 1] [n + 1 ..]
-- Reinhard Zumkeller, Nov 10 2012

for(n=1,105,c=0;forprime(p=2,n-1,if(isprime(n+p),c++));print1(c,","))

More terms from Klaus Brockhaus and Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004

A086800 Triangle read by rows in which row n lists differences between prime(n) and prime(k) for 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 5, 4, 2, 0, 9, 8, 6, 4, 0, 11, 10, 8, 6, 2, 0, 15, 14, 12, 10, 6, 4, 0, 17, 16, 14, 12, 8, 6, 2, 0, 21, 20, 18, 16, 12, 10, 6, 4, 0, 27, 26, 24, 22, 18, 16, 12, 10, 6, 0, 29, 28, 26, 24, 20, 18, 14, 12, 8, 2, 0, 35, 34, 32, 30, 26, 24, 20, 18, 14, 8, 6, 0

Offset: 1

Cino Hilliard, Aug 05 2003

Primes in this sequence are of course twin primes.

			2-2=0; 3-2=1, 3-3=0; 5-2=3, 5-3=2, 5=5=5; 7-2=5, 7-3=4, 7-5=2, 7-7=0, ...
Triangle begins:
  0;
  1, 0;
  3, 2, 0;
  5, 4, 2, 0;
  9, 8, 6, 4, 0;
  11, 10, 8, 6, 2, 0;
  15, 14, 12, 10, 6, 4, 0;
  17, 16, 14, 12, 8, 6, 2, 0;

Michel Marcus, Rows n=1..100 of triangle, flattened

Cf. A040976 (1st column).

fn(n) = forprime(x=2,n, forprime(y=2,x,print1(x-y",")))

T(n, k) = prime(n) - prime(k);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 08 2017

Offset corrected by Michel Marcus, Aug 08 2017

A166010 a(n) = prime(n)^2-4.

Original entry on oeis.org

0, 5, 21, 45, 117, 165, 285, 357, 525, 837, 957, 1365, 1677, 1845, 2205, 2805, 3477, 3717, 4485, 5037, 5325, 6237, 6885, 7917, 9405, 10197, 10605, 11445, 11877, 12765, 16125, 17157, 18765, 19317, 22197, 22797, 24645, 26565, 27885, 29925, 32037

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Least common multiple of prime(n)-2 and prime(n)+2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A066830, A084921, A166011.

Cf. A049001, A084920, A182200; A182174.

[NthPrime(n)^2-4: n in [1..41]]; // Bruno Berselli, Apr 17 2012

f[n_]:=LCM[n-2,n+2]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
Prime[Range[5!]]^2 - 4 (* Zak Seidov, Apr 17 2012 *)

a(n)=prime(n)^2-4 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A001248(n)-4 = A040976(n)*A052147(n). [Bruno Berselli, Apr 17 2012]

Definition rewritten by Bruno Berselli, Apr 17 2012

A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.

Original entry on oeis.org

0, 0, 1, -2, 3, 3, 0, 1, 15, 5, -2, 9, 13, 35, 9, -4, 3, 75, 43, 99, 11, -2, 3, 25, 245, 97, 143, 15, 0, 7, 65, 53, 1089, 163, 255, 17, -6, 27, 31, 301, 133, 1859, 253, 323, 21, -4, 5, 375, 73, 1067, 185, 4335, 355, 483, 27, -2, 9, 91, 1715, 151, 2119, 313, 6137, 565, 783, 29, -8, 9, 125, 473, 11979, 229, 4301, 457, 11109, 781, 899, 35

Offset: 1

Antti Karttunen, Dec 14 2024

Each column is strictly increasing.

			The top left corner of the array:
k=  |  1    2    3      4    5      6    7       8      9     10   11      12
2k= |  2    4    6      8   10     12   14      16     18     20   22      24
----+-------------------------------------------------------------------------
  1 |  0,   0,  -2,     0,  -2,    -4,  -2,      0,    -6,    -4,  -2,     -8,
  2 |  1,   3,   1,     9,   3,     3,   7,     27,     5,     9,   9,      9,
  3 |  3,  15,  13,    75,  25,    65,  31,    375,    91,   125,  43,    325,
  4 |  5,  35,  43,   245,  53,   301,  73,   1715,   473,   371,  83,   2107,
  5 |  9,  99,  97,  1089, 133,  1067, 151,  11979,  1261,  1463, 187,  11737,
  6 | 11, 143, 163,  1859, 185,  2119, 229,  24167,  2771,  2405, 295,  27547,
  7 | 15, 255, 253,  4335, 313,  4301, 403,  73695,  4807,  5321, 433,  73117,
  8 | 17, 323, 355,  6137, 457,  6745, 491, 116603,  8165,  8683, 593, 128155,
  9 | 21, 483, 565, 11109, 607, 12995, 733, 255507, 16385, 13961, 817, 298885,

Cf. A000010, A083254, A246278, A379010.
Cf. A040976 (column 1), A378986 (row 1).
Cf. also A378979.

up_to = 11325; \\ = binomial(150+1,2)
A083254(n) = (2*eulerphi(n)-n);
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379011sq(row,col) = A083254(A246278sq(row,col));
A379011list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379011sq(col,(a-(col-1))))); (v); };
v379011 = A379011list(up_to);
A379011(n) = v379011[n];

A(n, k) = 2*A379010(n, k) - A246278(n, k).

A060881 n-th primorial (A002110) + prime(n + 1).

Original entry on oeis.org

3, 5, 11, 37, 221, 2323, 30047, 510529, 9699713, 223092899, 6469693261, 200560490167, 7420738134851, 304250263527253, 13082761331670077, 614889782588491463, 32589158477190044789, 1922760350154212639131

Offset: 0

N. J. A. Sloane, May 05 2001

nonn

Terms are pairwise coprime with very high probability. I didn't find terms which are pairwise noncoprime, although it may be a case of the "strong law of small numbers." - Daniel Forgues, Apr 23 2012

All numbers in the range [primorial(n)+2, a(n)-1] are guaranteed to be a multiple of a prime p whose index is <= n. There are prime(n+1)-2 = A040976(n+1) such numbers. - Jamie Morken and Michel Marcus, Feb 01 2018

			a(2) = 2*3 + 5 = 11.

Harry J. Smith, Table of n, a(n) for n = 0..100
Hisanori Mishima, WIFC (World Integer Factorization Center): PI Pn + NextPrime (n = 1 to 100), PI Pn - NextPrime (n = 1 to 100), PI Pn + 1 (n = 1 to 100), PI Pn - 1 (n = 1 to 100).
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000040, A002110, A060882, A006862, A057588.

a:= n-> mul(ithprime(k), k=1..n)+ithprime(n+1): seq(a(n), n=0..20);  # Muniru A Asiru, Feb 01 2018

Module[{nn=20,pr},pr=Prime[Range[nn+1]];Join[{3},FoldList[ Times,Most[ pr]] + Rest[pr]]] (* Harvey P. Dale, Feb 19 2016 *)
Total /@ Fold[Append[#1, {Prime[#2] #1[[-1, 1]], Prime[#2 + 1]}] &, {{1, 2}}, Range@ 17] (* Michael De Vlieger, Feb 21 2018 *)

{ n=-1; m=1; forprime (p=2, prime(101), write("b060881.txt", n++, " ", m + p); m*=p; ) } \\ Harry J. Smith, Jul 19 2009

a(n) = prod(i=1, n, prime(i)) + prime(n+1); \\ Michel Marcus, Feb 01 2018

a(n) = A002110(n) + A000040(n+1). - Michel Marcus, Feb 01 2018

Name changed by David A. Corneth, Mar 25 2018

A110146 a(n) = n^(n+1) mod (n+2).

Original entry on oeis.org

0, 1, 0, 1, 4, 1, 0, 4, 8, 1, 4, 1, 12, 4, 0, 1, 4, 1, 12, 4, 20, 1, 16, 16, 24, 13, 20, 1, 28, 1, 0, 4, 32, 9, 4, 1, 36, 4, 32, 1, 10, 1, 36, 31, 44, 1, 16, 15, 38, 4, 44, 1, 40, 49, 40, 4, 56, 1, 52, 1, 60, 4, 0, 16, 34, 1, 60, 4, 48, 1, 40, 1, 72, 34, 68, 9

Offset: 0

Zak Seidov, Jul 14 2005

nonn

First occurrence of n such that n^(n+1) (mod n+2) == k for k = 1, 2, 3, ..., or 0 if no such n is known: 1, 20735, 10667, 4, 0, 3761, 3820819, 8, 33, 40, 350849481, 12, 25, ..., .

Congruences not yet occurring for n < 4.6*10^9: 5, 47, 57, 105, 203, 233, 255, 293, 333, 354, 377, 405, 433, ..., .

Carl R. White, Table of n, a(n) for n = 0..10000

Cf. A000918, A040976.

Table[PowerMod[n, n+1, n+2], {n, 0, 120}]

a(n) = lift(Mod(n, n+2)^(n+1)); \\ Michel Marcus, Dec 17 2022

a(A000918(n)) = 0 for n >= 1, a(A040976(n)) = 1 for n >= 2.

a(n-2) = A062173(n) if n is odd or n is power of two, and a(n-2) = n - A062173(n) otherwise. - Thomas Ordowski, Nov 28 2013

A120632 Number of numbers >1 up to 2*prime(n) which are divisible by primes up to prime(n).

Original entry on oeis.org

2, 4, 8, 11, 18, 22, 29, 33, 40, 51, 54, 64, 72, 76, 84, 94, 104, 109, 120, 127, 132, 142, 150, 161, 174, 181, 186, 194, 199, 207, 230, 238, 248, 252, 270, 275, 285, 297, 305, 317, 327, 331, 349, 353, 361, 365, 386, 407, 415, 419, 426, 438, 442, 460, 471, 482

Offset: 1

Lekraj Beedassy, Jun 21 2006

nonn

The first prime(n+1)-2 numbers >1 are divisible by primes up to prime(n).

Complement of A137624; A137621(a(n))=A000040(n); A137621(a(n)+1)=A100484(n). - Reinhard Zumkeller, Jan 30 2008

			a(4)=11 because exactly 11 numbers between 2 and 2*prime(4)=2*7=14, namely: 2,3,4,5,6,7,8,9,10,12,14 are divisible by the first four primes 2,3,5,7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A100484, A137621, A137624.

Cf. A120633, A040976, A076274, A070046.

f:= proc(n) local p;
   p:= ithprime(n); 2*p - numtheory:-pi(2*p)+n-1
end proc:
map(f, [$1..100]); # Robert Israel, Mar 02 2022

a(n) = {nb = 0; for (i = 2, 2*prime(n), for (ip = 1, n, if ( !(i % prime(ip)), nb++; break;););); nb;} \\ Michel Marcus, Oct 26 2013

a(n) = A120633(n) + A040976(n+1) = A076274(n) - A070046(n).

cp's OEIS Frontend

A206038 Values of the difference d for 4 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 3.

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A206043 Values of the difference d for 9 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 8.

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A206040 Values of the difference d for 6 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 5.

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Mathematica

A092953 Number of primes of the form n+p, where p is a prime < n.

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Haskell

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A086800 Triangle read by rows in which row n lists differences between prime(n) and prime(k) for 1 <= k <= n.

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PARI

PARI

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A166010 a(n) = prime(n)^2-4.

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Magma

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A379011 Square array A(n, k) = 2*phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2*phi(n)-n), applied to the prime shift array.

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A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.