A005097 -id:A005097 - cp's OEIS frontend

A104635 Odd n such that 2*n+1 is prime.

1, 3, 5, 9, 11, 15, 21, 23, 29, 33, 35, 39, 41, 51, 53, 63, 65, 69, 75, 81, 83, 89, 95, 99, 105, 111, 113, 119, 125, 131, 135, 141, 153, 155, 165, 173, 179, 183, 189, 191, 209, 215, 219, 221, 231, 233, 239, 243, 245, 249, 251, 261, 273, 281, 285, 293, 299

Offset: 1

Zak Seidov, Mar 18 2005

Also: Numbers k such that 2k+1 is in A002145, i.e., a Gaussian prime. - M. F. Hasler, Feb 25 2011

Also: Number of quadratic residues modulo A002145(n). - M. F. Hasler, Feb 25 2011

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

Cf. A002145, A005097, A104636.

[n: n in [1..500 by 2] | IsPrime(2*n+1)]; // Vincenzo Librandi, Aug 14 2018

Select[Range[1,301,2],PrimeQ[2#+1]&] (* Harvey P. Dale, May 08 2012 *)

forstep( k=1,250,2, isprime(2*k+1) && print1(k", ")) \\ M. F. Hasler, Feb 25 2011

forprime( p=1,500, p%4==3 || next; print1(p\2", ")) \\ M. F. Hasler, Feb 25 2011

a(n) = floor(A002145(n)/2). - M. F. Hasler, Feb 25 2011

A124041 Numbers k such that 2k+1, 4k+1 and 8*k+1 are primes.

Original entry on oeis.org

9, 39, 165, 219, 249, 309, 414, 534, 639, 765, 1044, 1065, 1089, 1155, 1395, 1509, 1530, 1554, 1590, 1884, 2079, 2115, 2130, 2310, 2319, 2430, 2475, 2709, 2874, 3060, 3105, 3354, 3420, 3684, 3705, 3780, 3819, 4104, 4314, 4554, 4599, 4659, 4869, 5160

Offset: 1

Artur Jasinski, Nov 02 2006

nonn

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

Cf. A005097, A123998, A124412-A124417.

Select[3*Range[2000], And @@ PrimeQ /@ ({2, 4, 8}*# + 1) &] (* Ray Chandler, Dec 06 2006 *)

A124412 Numbers k such that 2k+1, 4k+1, 8k+1 and 16k+1 are primes.

Original entry on oeis.org

765, 1065, 1155, 1530, 3105, 3420, 3705, 5160, 6840, 7695, 8325, 9060, 11265, 11505, 12195, 14835, 15390, 15885, 16650, 17655, 20745, 22185, 23205, 27300, 28155, 28995, 30165, 30690, 33300, 33825, 39015, 41715, 42690, 44370, 48465, 49935

Offset: 1

Artur Jasinski, Nov 02 2006

nonn

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

Cf. A005097, A123998, A124041, A124413-A124417.

Select[15*Range[3500], And @@ PrimeQ /@ ({2, 4, 8, 16}*# + 1) &] (* Ray Chandler, Nov 21 2006 *)

A177771 a(n) = (prime(n) - 1)!.

Original entry on oeis.org

1, 2, 24, 720, 3628800, 479001600, 20922789888000, 6402373705728000, 1124000727777607680000, 304888344611713860501504000000, 265252859812191058636308480000000

Offset: 1

Giovanni Teofilatto, May 13 2010

nonn

By Wilson's theorem, a(n) = -1 (mod p) where p is the n-th prime. - Charles R Greathouse IV, Sep 04 2013

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..80

Cf. A006093, A039716.

[Factorial(p-1): p in PrimesUpTo(20)]; // Vincenzo Librandi, May 31 2013

(Prime[Range[12]]-1)! (* Harvey P. Dale, Jul 12 2011 *)

apply(p->(p-1)!,primes(10)) \\ Charles R Greathouse IV, Sep 04 2013

a(n) = A010050(A005097(n-1)). a(n)^2 = A177926(n). - R. J. Mathar, May 24 2010

A193773 Number of ways to write n as 2xy - x - y with 1 <= x <= y.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jan 02 2013

nonn

a(A005097(n)) = 1; for n > 1: a(A047845(n)) > 1. - Reinhard Zumkeller, Jan 02 2013

Number of ways to write 2*n+1 as a difference of two squares. Note that 2*(2*x*y - x - y) + 1 = (2*x - 1) * (2*y - 1) = (y + x - 1)^2 - (y - x)^2. - Michael Somos, Dec 23 2018

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... - _Michael Somos_, Dec 23 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000005, A005097, A047845, A125203.

a193773 n = length [() | x <- [1 .. n + 1],
                         let (y,m) = divMod (x + n) (2 * x - 1),
                         x <= y, m == 0]

a[ n_] := If[ n < 0, 0, Ceiling[ DivisorSigma[0, 2 n + 1] / 2]]; (* Michael Somos, Dec 23 2018 *)

{a(n) = if(n < 0, 0, (numdiv(2*n+1) + 1)\2)}; /* Michael Somos, Dec 23 2018 */

a(n) = ceiling(A000005(2*n+1) / 2). - Michael Somos, Dec 23 2018

A104636 Even n such that 2n+1 is prime.

Original entry on oeis.org

2, 6, 8, 14, 18, 20, 26, 30, 36, 44, 48, 50, 54, 56, 68, 74, 78, 86, 90, 96, 98, 114, 116, 120, 128, 134, 138, 140, 146, 156, 158, 168, 174, 176, 186, 194, 198, 200, 204, 210, 216, 224, 228, 230, 254, 260, 270, 278, 284, 288, 296, 300, 306, 308, 320, 326, 330

Offset: 1

Zak Seidov, Mar 18 2005

If q = 2*n + 1 is prime, and n is even, q divides (n!)^2 + 1. - Arkadiusz Wesolowski, Sep 06 2012

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

Cf. A005097, A104635.

Select[Range[2, 300, 2], PrimeQ[2*# + 1] &]

a(48)-a(57) from Arkadiusz Wesolowski, Sep 06 2012

A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

Original entry on oeis.org

1, 3, 18, 105, 135, 153, 165, 168, 300, 363, 585, 618, 648, 765, 828, 1110, 1140, 1278, 1518, 1530, 1533, 2130, 2223, 2400, 2475, 2613, 2790, 2925, 3075, 3180, 3345, 3420, 3483, 3810, 3840, 3843, 3933, 4008, 4083, 4095, 4143, 4260, 4263, 4323, 4470, 4545

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A005097, A005098, A024899, A123998, A124409, A124410, A124411, A071576.

Select[Range[4600], And @@ PrimeQ /@ ({2, 4, 6}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 3, isprime(2*j*k+1)) == 3; \\ Jinyuan Wang, Aug 04 2019

A124409 Numbers k such that 2k+1, 4k+1, 6k+1 and 8k+1 are primes.

Original entry on oeis.org

165, 765, 1530, 2130, 2475, 3420, 5415, 7695, 9060, 11505, 12705, 13020, 15885, 16650, 20055, 20745, 22530, 24915, 26940, 29670, 32925, 35070, 36885, 39270, 44370, 47730, 48465, 54735, 55860, 56310, 58860, 65655, 66600, 67365, 67650

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..2000

Cf. A005097, A005098, A024899, A005123, A123998, A124408, A124410, A124411, A071576.

Select[Range[68000], And @@ PrimeQ /@ ({2, 4, 6, 8}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 4, isprime(2*j*k+1)) == 4; \\ Jinyuan Wang, Aug 04 2019

A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

Original entry on oeis.org

5415, 12705, 13020, 44370, 82950, 98280, 105525, 112200, 115140, 123855, 134250, 134460, 187740, 188745, 210165, 225705, 247170, 256410, 296310, 302085, 367875, 375645, 382890, 399585, 404040, 476340, 487830, 526845, 532095, 566430, 578085

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A005097, A005098, A024899, A005123, A024912, A123998, A124408, A124409, A124411, A071576.

Select[Range[600000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 5, isprime(2*j*k+1)) == 5; \\ Jinyuan Wang, Aug 04 2019

A124411 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1, 10k+1 and 12k+1 are primes.

Original entry on oeis.org

12705, 13020, 105525, 256410, 966840, 1707510, 1944495, 2310000, 2478630, 3132675, 3836070, 3976770, 4112430, 4532325, 5499585, 5920005, 6610485, 7390845, 8552250, 10739505, 11120340, 12231450, 12338130, 13243230, 16467255

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..100

Cf. A005097, A005098, A024899, A005123, A024912, A110801, A123998, A124408, A124409, A124410, A071576.

Select[Range[10^7], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 6, isprime(2*j*k+1)) == 6; \\ Jinyuan Wang, Aug 04 2019

Extended by Ray Chandler, Nov 20 2006

cp's OEIS Frontend

A104635 Odd n such that 2*n+1 is prime.

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A124041 Numbers k such that 2*k+1, 4*k+1 and 8*k+1 are primes.

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A124412 Numbers k such that 2*k+1, 4*k+1, 8*k+1 and 16*k+1 are primes.

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A177771 a(n) = (prime(n) - 1)!.

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A193773 Number of ways to write n as 2*x*y - x - y with 1 <= x <= y.

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A104636 Even n such that 2n+1 is prime.

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A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

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PARI

A124409 Numbers k such that 2k+1, 4k+1, 6k+1 and 8k+1 are primes.

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A124041 Numbers k such that 2k+1, 4k+1 and 8*k+1 are primes.

A124412 Numbers k such that 2k+1, 4k+1, 8k+1 and 16k+1 are primes.

A193773 Number of ways to write n as 2xy - x - y with 1 <= x <= y.