A107141 -id:A107141 - cp's OEIS frontend

A107132 Primes of the form 2x^2 + 13y^2.

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A124923 a(n) = n^(n-1) + 1.

Original entry on oeis.org

2, 3, 10, 65, 626, 7777, 117650, 2097153, 43046722, 1000000001, 25937424602, 743008370689, 23298085122482, 793714773254145, 29192926025390626, 1152921504606846977, 48661191875666868482, 2185911559738696531969

Offset: 1

Alexander Adamchuk, Nov 12 2006

Prime p divides a(p-1). n divides a(n-1) for all prime n and all odd composite n.
p divides a((p+1)/2) for prime p = {3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, ...} = A003629 (Primes congruent to {3,5} mod 8).
p divides a((p+3)/4) for prime p = {13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457, 541, 709, 733, 757, 829, ...} = A107141 (Primes of the form 4x^2+9y^2).
p divides a((p+5)/6) for prime p = {43, 61, 79, 109, 151, 163, 181, 193, 313, 337, 433, 523, 577, 631, 643, 673, 787, 829, 907, 991, ...}.
p divides a((p+7)/8) for prime p = {113, 137, 569, 641, 673, 1129, 1289, 1297, 1481, 1801, ...}.
p divides a((3p-1)/2) for prime p = {5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, ...} = A003628 (Primes congruent to {5, 7} mod 8).
p^2 divides a((3p-1)/2) for prime p = {5, 13, 173, 5501, ...} = A124924.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000169, A003628, A003629, A107141, A124924.

List([1..20], n-> n^(n-1) + 1); # G. C. Greubel, Nov 19 2019

[n^(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Aug 14 2012

seq(n^(n-1) + 1, n=1..20); # G. C. Greubel, Nov 19 2019

Mathematica
```
Table[n^(n-1)+1, {n,20}]
```

vector(20, n, n^(n-1) + 1) \\ G. C. Greubel, Nov 19 2019

[n^(n-1) + 1 for n in (1..20)] # G. C. Greubel, Nov 19 2019

a(n) = n^(n-1) + 1.
a(n) = A000169(n) + 1.
E.g.f.: -1 + exp(x) - W(-x), where W(x) is the Lambert w-function. - G. C. Greubel, Nov 19 2019

cp's OEIS Frontend

A107132 Primes of the form 2x^2 + 13y^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI