A111174 -id:A111174 - cp's OEIS frontend

A114914 Terms in A114913 that are not in A111174.

76, 101, 149, 198, 201, 251, 326, 351, 368, 394, 426, 451, 476, 489, 492, 514, 601, 626, 639, 683, 688, 701, 801, 835, 879, 884, 933, 973, 976, 1051, 1076, 1098, 1168, 1176, 1178, 1201, 1215, 1227, 1251, 1301, 1351, 1359, 1374, 1376, 1459, 1551, 1570

Offset: 1

Christian G. Bower, Jan 06 2006

nonn

If 24*k+1 = 25*p for some prime p, then k will be in this sequence, implying that this sequence is infinite. - Dean Hickerson, Jan 19 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111174, A114913.

q[n_] := Module[{f = FactorInteger[n], f1, f2}, f1 = Select[f, MemberQ[{1, 5, 7, 11}, Mod[First[#], 24]] &]; f2 = Select[f, MemberQ[{13, 17, 19, 23}, Mod[First[#], 24]] &]; AllTrue[f2[[;;, 2]], EvenQ] && Count[f1[[;;, 2]], ?OddQ] == 1]; Select[Range[1600], CompositeQ[24 * # + 1] && q[24 * # + 1] &] (* _Amiram Eldar, Aug 24 2024 *)

A107008 Primes of the form x^2 + 24*y^2.

Original entry on oeis.org

73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713

Offset: 1

T. D. Noe, May 09 2005

Presumably this is the same as primes congruent to 1 mod 24, so a(n) = 24*A111174(n) + 1. - N. J. A. Sloane, Jul 11 2008. Checked for all terms up to 2 million. - Vladimir Joseph Stephan Orlovsky, May 18 2011.
Discriminant = -96.
Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - T. D. Noe, May 19 2008
Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020
More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020
Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
P. L. Clark, J. Hicks, H. Parshall, K. Thompson, GONI: primes represented by binary quadratic forms, INTEGERS 13 (2013) #A37
D. A. Cox, Primes of the form x^2 + n*y^2, A Wiley-Interscience publication, 1989
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007) 1589-1617

Subset of A033199 (2y here = y there).
Is this the same as A141375?
Cf. A000926, A002476, A007519, A111174, A139642.
See also the cross-references in A140633.

QuadPrimes[1, 0, 24, 10000] (* see A106856 *)

is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020

Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014

A005123 Numbers k such that 8k + 1 is prime.

Original entry on oeis.org

2, 5, 9, 11, 12, 14, 17, 24, 29, 30, 32, 35, 39, 42, 44, 50, 51, 54, 56, 57, 65, 71, 72, 74, 75, 77, 80, 84, 95, 96, 101, 107, 110, 116, 117, 119, 122, 126, 129, 131, 137, 141, 144, 149, 150, 152, 156, 161, 162, 165, 170, 176, 179, 185, 186, 194, 200, 201

Offset: 1

N. J. A. Sloane

nonn

4*a(n) is the degree of the minimal polynomial of 2*cos(Pi/A007519(n)), called C(A007519(n), x) in A187360. - Wolfdieter Lang, Oct 24 2013

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A007519, A111174, A115249.

[n: n in [0..200] | IsPrime(8*n+1)]; // Vincenzo Librandi, Jun 25 2014

Select[Range[0, 300], PrimeQ[8 # + 1] &] (* Vincenzo Librandi, Jun 25 2014 *)

is(n)=isprime(8*n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = (A007519(n)-1)/8. - Zak Seidov, Sep 26 2007

More terms from Wesley Ivan Hurt, Jun 25 2014

A114913 Numbers k such that A114912(k) = 1. Numbers k such that A000009(k) == 2 (mod 4).

Original entry on oeis.org

3, 4, 8, 10, 13, 14, 17, 18, 19, 24, 25, 28, 32, 39, 42, 43, 47, 48, 50, 52, 54, 55, 62, 67, 69, 73, 74, 75, 76, 78, 83, 84, 87, 88, 89, 90, 95, 99, 101, 103, 105, 108, 109, 112, 113, 118, 119, 123, 125, 127, 130, 132, 134, 138, 140, 143, 144, 147, 149, 153, 154, 157

Offset: 1

Christian G. Bower, Jan 06 2006

nonn

All the terms are the sum of a generalized pentagonal number A001318 and a square A000290.

Let 24*k+1 = p_1^e_1 * ... * p_r^e_r * q_1^f_1 * ... * q_s^f_s, where the p_i's are distinct primes == 1, 5, 7, or 11 (mod 24) and the q_i's are distinct primes == 13, 17, 19, or 23 (mod 24). Then k belongs to the sequence iff all of the f_i's are even and all but one of the e_i's are even and the one e_i which is odd is == 1 (mod 4). - Dean Hickerson, Jan 19 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, Partition Identities Involving Gaps and Weights, Transactions of the American Mathematical Society, Vol. 349, No. 12 (Dec 1997), pp. 5001-5019.

Cf. A000009, A000290, A001318, A114912, A114914.
A111174 is a subsequence.
See comments in A113780 for explanation.

q[n_] := Module[{f = FactorInteger[n], f1, f2}, f1 = Select[f, MemberQ[{1, 5, 7, 11}, Mod[First[#], 24]] &]; f2 = Select[f, MemberQ[{13, 17, 19, 23}, Mod[First[#], 24]] &]; AllTrue[f2[[;;, 2]], EvenQ] && Count[f1[[;;, 2]], ?OddQ] == 1]; Select[Range[160], q[24 * # + 1] &] (* _Amiram Eldar, Aug 24 2024 *)

A153384 Numbers n such that 24*n+1 is not prime.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 9, 11, 12, 15, 16, 20, 21, 22, 23, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 44, 45, 46, 49, 51, 53, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 68, 70, 71, 72, 76, 77, 79, 80, 81, 82, 85, 86, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102

Offset: 1

Vincenzo Librandi, Dec 25 2008

Contains all numbers == 1 (mod 5), ==2 (mod 7), ==5 (mod 11), == 7 (mod 13), == 12 (mod 17), == 15 (mod 19), == 22 (mod 23), == 6 (mod 29) etc, so it is the union of A016861, A017005, A017449, A269044, etc. - R. J. Mathar, Jun 10 2020

Even terms of A153383, halved. - R. J. Mathar, Jun 10 2020

			Triangle begins:
*;
*,1;
*,*,2;
*,*,*,*;
*,*,*,*,5;
*,*,*,*,*,7;
*,*,*,*,*,*,*;
*,*,*,*,*,*,*,12;
*,*,*,*,*,*,*,*,15;
*,*,*,*,*,*,*,*,*,*;
*,*,*,*,*,*,*,*,*,*,22; etc.
where * marks the non-integer values of (2*h*k + k + h)/12 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013

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

Cf. A001318, A111174 (complement).

[n: n in [0..150] | not IsPrime(24*n + 1)]; // Vincenzo Librandi, Jan 14 2013

Select[Range[0, 200], !PrimeQ[24 # + 1] &] (* Vincenzo Librandi, Jan 14 2013 *)

0 added by Arkadiusz Wesolowski, Aug 03 2011

A255218 Numbers k such that 12k+1, 24k+1, 36k+1 and 72k+1 are all prime.

Original entry on oeis.org

28, 103, 190, 253, 355, 848, 1328, 1783, 1898, 1958, 1988, 2170, 2213, 3003, 3533, 3808, 3913, 3988, 4450, 4488, 4593, 4800, 5460, 5808, 5853, 6448, 6545, 6903, 7103, 7238, 7295, 7400, 7483, 7693, 8533, 9310, 9780, 10260, 10885, 12185, 12628, 15513, 16163

Offset: 1

Vincenzo Librandi, Feb 26 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), p. 14.

Subsequence of A110801 and A111174.

Cf. A255578.

[n: n in [0..20000] | IsPrime(12*n+1) and IsPrime(24*n+1) and IsPrime(36*n+1) and IsPrime(72*n+1)];

[n: n in [0..20000] | forall{i: i in Divisors(6) | IsPrime(12*i*n+1)}];

Select[Range[10000], PrimeQ[12 # + 1] && PrimeQ[24 # + 1] && PrimeQ[36 # + 1] && PrimeQ[72 # + 1] &]
Select[Range[17000],AllTrue[{12,24,36,72}#+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 16 2016 *)

A139527 Numbers n such that numbers 24n+5 are primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 12, 13, 16, 19, 21, 23, 27, 28, 29, 32, 33, 34, 39, 42, 44, 46, 49, 51, 53, 54, 57, 62, 67, 68, 71, 72, 78, 79, 81, 82, 83, 86, 89, 92, 93, 96, 97, 98, 99, 103, 106, 109, 112, 114, 116, 118, 119, 121, 123, 134, 141, 142, 144, 147, 148, 149, 153, 154

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Numbers n such that:
24n+1 is prime see A111174, primes 24n+1 see A107008
24n+5 is prime see A139527, primes 24n+5 see A107003
24n+7 is prime see A139483, primes 24n+7 see A107006
24n+11 is prime A139528, primes 24n+11 see A107007
24n+13 is prime see A139529, primes 24n+13 see A139530
24n+17 is prime see A139531, primes 24n+17 see A107181
24n+19 is prime see A139532, primes 24n+19 see A141373
24n+23 is prime see A131210, primes 24n+23 see A134517

Harvey P. Dale, Table of n, a(n) for n = 1..1000

a = {}; Do[If[PrimeQ[24 n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Table[(Prime[n]-5)/24,{n,800}],IntegerQ] (* Harvey P. Dale, Feb 25 2016 *)

cp's OEIS Frontend

A114914 Terms in A114913 that are not in A111174.

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A107008 Primes of the form x^2 + 24*y^2.

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A005123 Numbers k such that 8k + 1 is prime.

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A114913 Numbers k such that A114912(k) = 1. Numbers k such that A000009(k) == 2 (mod 4).

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A153384 Numbers n such that 24*n+1 is not prime.

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A255218 Numbers k such that 12*k+1, 24*k+1, 36*k+1 and 72*k+1 are all prime.

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Magma

Magma

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A139527 Numbers n such that numbers 24n+5 are primes.

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Mathematica

A255218 Numbers k such that 12k+1, 24k+1, 36k+1 and 72k+1 are all prime.