A067186 -id:A067186 - cp's OEIS frontend

A134460 First differences of A067186.

2, 1, 3, 1, 3, 1, 4, 3, 1, 3, 4, 8, 5, 3, 1, 3, 4, 5, 4, 3, 1, 4, 3, 4, 1, 8, 4, 7, 1, 3, 1, 7, 4, 1, 19, 1, 3, 5, 8, 11, 4, 4, 1, 4, 3, 4, 5, 3, 13, 3, 4, 1, 3, 8, 4, 5, 8, 4, 11, 9, 4, 7, 4, 5, 4, 4, 3, 4, 4, 8, 9, 4, 7, 16, 1, 4, 4, 7, 4, 1, 8, 3, 1, 11, 5, 4, 3, 9, 11, 1, 8, 3, 1, 4, 3, 1, 3, 4, 13, 4, 3, 8

Offset: 1

John W. Layman, Oct 26 2007

nonn

Conjecture: Except for the first term a(1)=2, this sequence contains no terms of the form 4m+2. (This has been confirmed for the first several thousand terms.)

Cf. A000217, A067186.

A055469 Primes of the form k(k+1)/2+1 (i.e., central polygonal numbers, or one more than triangular numbers).

Original entry on oeis.org

2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, 631, 821, 947, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2347, 2557, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 8647, 8779, 9181, 9871, 11027, 12721, 13367, 14029, 14197, 14879

Offset: 1

Henry Bottomley, Jun 27 2000

Also primes of the form (n^2 + 7)/8. - Ray Chandler, Oct 08 2005
q=2 and q=5 are the only primes values such that q+1 is a triangular number because 8q+9 is a square for 2 and 5 only. - Benoit Cloitre, Apr 05 2002
n such that A000010(n) = A000217(k). - Giovanni Teofilatto, Jan 29 2010
It is conjectured that this sequence is infinite. - Daniel Forgues, Apr 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A000124, A000217, A067186, A110872, A110873, A129545.

Select[Table[(n^2 + 7)/8, {n, 400}], PrimeQ] (* Ray Chandler, Oct 08 2005 *)
Select[Accumulate[Range[400]]+1,PrimeQ] (* Harvey P. Dale, May 14 2022 *)

forprime(p=2,10^5, if ( issquare(8*p-7), print1(p, ", "))) \\ Joerg Arndt, Jul 14 2012

list(lim)=my(v=List(),p); forstep(s=3,sqrtint(lim\1*8-7),2, if(isprime(p=(s^2+7)/8), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, May 05 2020

a(n) = A000124(A067186(n)) = (A110873(n) + 7)/8. - Ray Chandler, Oct 08 2005

A060371 a(n) = (prime(n) - 1)! + 1.

Original entry on oeis.org

2, 3, 25, 721, 3628801, 479001601, 20922789888001, 6402373705728001, 1124000727777607680001, 304888344611713860501504000001, 265252859812191058636308480000001, 371993326789901217467999448150835200000001

Offset: 1

Jason Earls, Apr 01 2001

nonn

If the prime p is in A055469, that is if p = 2, 7, 11, 29, ... = A055469(j) which is valid for the first, 4th, 5th, 10th,.... entry here with j = 1, 2, 3, ..., then a(n) = A052295[A067186(j)] + 1. - R. J. Mathar, Apr 27 2007

It follows from Wilson's theorem that a(n) is divisible by the n-th prime. - Alonso del Arte, Feb 07 2014

Harry J. Smith, Table of n, a(n) for n = 1..88 (adapted by Vincenzo Librandi, Oct 17 2017)
Takashi Agoh, Karl Dilcher and Ladislav Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861. MR 98h:11003.
C. K. Caldwell, Wilson Primes
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449. MR 97c:11004.

Cf. A052295, A055469, A067186.

Subsequence of A038507. - Michel Marcus, Oct 17 2017

[Factorial(NthPrime(n)-1)+1: n in [1..15]]; // Vincenzo Librandi, Oct 17 2017

Table[(Prime[n] - 1)! + 1, {n, 12}] (* Alonso del Arte, Feb 07 2014 *)

{ n=1; forprime (p=1, 524, write("b060371.txt", n++, " ", (p - 1)! + 1); ) } \\ Harry J. Smith, Jul 04 2009

Corrected offset by Alonso del Arte, Feb 07 2014

A129545 Triangular numbers T such that T+1 is a prime.

Original entry on oeis.org

1, 6, 10, 28, 36, 66, 78, 136, 190, 210, 276, 378, 630, 820, 946, 990, 1128, 1326, 1596, 1830, 2016, 2080, 2346, 2556, 2850, 2926, 3570, 3916, 4560, 4656, 4950, 5050, 5778, 6216, 6328, 8646, 8778, 9180, 9870, 11026, 12720, 13366, 14028, 14196, 14878

Offset: 1

Zak Seidov, May 30 2007

The only triangular numbers T such that T-1 is a (positive) prime are 3 and 6.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000217, A055469, A129755, A130178, A345350.

Select[Accumulate[Range[200]],PrimeQ[#+1]&] (* Harvey P. Dale, Nov 08 2011 *)

from sympy import isprime
def T(n): return n*(n+1)//2
def ok(T): return isprime(T+1)
print(list(filter(ok, (T(n) for n in range(175))))) # Michael S. Branicky, Jun 18 2021

a(n) = A000217(A067186(n)). - R. J. Mathar, Dec 10 2007

a(n) = A055469(n) - 1. - Joerg Arndt, Jun 19 2021

A110872 Numbers n such that (n^2+7)/8 is prime.

Original entry on oeis.org

3, 7, 9, 15, 17, 23, 25, 33, 39, 41, 47, 55, 71, 81, 87, 89, 95, 103, 113, 121, 127, 129, 137, 143, 151, 153, 169, 177, 191, 193, 199, 201, 215, 223, 225, 263, 265, 271, 281, 297, 319, 327, 335, 337, 345, 351, 359, 369, 375, 401, 407, 415, 417, 423, 439, 447

Offset: 1

Giovanni Teofilatto, Sep 18 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A055469, A067186, A110873.

Select[Range[300], PrimeQ[(#^2 + 7)/8] &] (* Ray Chandler, Oct 08 2005 *)

is(n)=isprime((n^2+7)/8) \\ Charles R Greathouse IV, May 22 2017

a(n) = 2*A067186(n) + 1 = Sqrt(A110873(n)).

Extended by Ray Chandler, Oct 08 2005

A110873 Squares of the form 8p - 7, where p is prime.

Original entry on oeis.org

9, 49, 81, 225, 289, 529, 625, 1089, 1521, 1681, 2209, 3025, 5041, 6561, 7569, 7921, 9025, 10609, 12769, 14641, 16129, 16641, 18769, 20449, 22801, 23409, 28561, 31329, 36481, 37249, 39601, 40401, 46225, 49729, 50625, 69169, 70225, 73441, 78961

Offset: 1

Giovanni Teofilatto, Sep 18 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A055469, A067186, A110872.

Select[Table[n^2, {n, 300}], PrimeQ[(# + 7)/8] &] (* Ray Chandler, Oct 01 2005 *)
Select[8#-7&/@Prime[Range[1300]],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Apr 11 2018 *)

for(i=1,1000,n=i^2+7;if(n%8==0&&isprime(n/8),print1(n-7,","))) (Klasen)

a(n) = A110872(n)^2 = 8*A055469(n) - 7.

a(n) = 8*A055469(n)-7.

Extended by Lambert Klasen (lambert.klasen(AT)gmx.net) and Ray Chandler, Oct 01 2005

A217755 Numbers n such that ((n^2 + n)/2)^2 + 1 is prime.

Original entry on oeis.org

1, 3, 4, 8, 11, 15, 20, 24, 28, 31, 39, 43, 48, 64, 75, 76, 79, 80, 111, 116, 120, 135, 148, 155, 160, 168, 179, 184, 200, 203, 208, 211, 216, 223, 224, 235, 243, 251, 263, 264, 288, 303, 319, 320, 324, 348, 351, 356, 364, 371, 375, 388, 416, 419, 420, 424, 428, 439, 443, 451, 456, 459

Offset: 1

Luca Brigada Villa, Mar 26 2013

nonn

Also, numbers n such that the sum of the first n cubes precedes a prime.

			1 is in the sequence because ((1^2 + 1)/2)^2 + 1 = 2 and 2 is prime;
3 is in the sequence because ((3^2 + 3)/2)^2 + 1 = 37 and 37 is prime.

Cf. A067186.

[n: n in [1..500] | IsPrime(s+1) where s is (n^2+n)^2 div 4]; // Bruno Berselli, Mar 27 2013

Select[Range[400], PrimeQ[((#^2 + #)/2)^2 + 1] &] (* Alonso del Arte, Mar 26 2013 *)

for(n=1,10^3, if(isprime(((n^2 + n)/2)^2 + 1), print1(n,", "))); /* Joerg Arndt, Mar 27 2013 */

cp's OEIS Frontend

A134460 First differences of A067186.

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A055469 Primes of the form k(k+1)/2+1 (i.e., central polygonal numbers, or one more than triangular numbers).

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A060371 a(n) = (prime(n) - 1)! + 1.

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A129545 Triangular numbers T such that T+1 is a prime.

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A110872 Numbers n such that (n^2+7)/8 is prime.

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Mathematica

PARI

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Extensions

A110873 Squares of the form 8p - 7, where p is prime.

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Mathematica

PARI

Formula

Extensions

A217755 Numbers n such that ((n^2 + n)/2)^2 + 1 is prime.

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Magma

Mathematica

PARI