A005574 -id:A005574 - cp's OEIS frontend

A121982 Numbers k such that k^2 + 15 is prime.

2, 4, 8, 14, 16, 22, 26, 32, 34, 38, 44, 46, 52, 64, 68, 76, 86, 88, 98, 104, 106, 124, 134, 158, 172, 178, 184, 196, 202, 206, 212, 236, 238, 242, 248, 256, 262, 272, 284, 296, 298, 304, 316, 322, 326, 328, 338, 356, 362, 364, 374, 386, 388, 394, 398, 452, 472

Offset: 1

Parthasarathy Nambi, Sep 09 2006

			If k=104 then k^2 + 15 = 10831 (prime).

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

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250.

[n: n in [0..6000] | IsPrime(n^2 + 15)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[200], PrimeQ[#1^2 + 15] &] (* G. C. Greubel, Sep 13 2017 *)

is(n)=isprime(n^2+15) \\ Charles R Greathouse IV, Feb 17 2017

A125262 Numbers k such that k^7 + 6 is prime.

Original entry on oeis.org

1, 13, 17, 23, 61, 73, 77, 101, 137, 215, 221, 283, 307, 317, 361, 431, 457, 473, 481, 641, 731, 767, 817, 881, 985, 1015, 1061, 1145, 1235, 1283, 1333, 1337, 1343, 1531, 1693, 1711, 1817, 1847, 1853, 1867, 1903, 1963, 2057, 2093, 2113, 2161, 2201, 2363

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..2374

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259-A125265 (j=4..11).

lst={};k=7;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[2500],PrimeQ[#^7+6]&] (* Harvey P. Dale, May 05 2016 *)

is(n)=isprime(n^7+6) \\ Charles R Greathouse IV, Feb 17 2017

A125265 Numbers k such that k^11 + 10 is prime.

Original entry on oeis.org

1, 7, 19, 21, 33, 69, 153, 157, 193, 253, 379, 391, 439, 543, 549, 559, 579, 609, 879, 937, 939, 993, 1063, 1083, 1107, 1119, 1191, 1209, 1267, 1281, 1287, 1333, 1537, 1617, 1797, 1819, 1923, 1971, 1987, 1989, 2041, 2061, 2073, 2101, 2103, 2343, 2373

Offset: 1

Zak Seidov, Nov 26 2006

nonn

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

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10).

lst={};k=11;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[2500],PrimeQ[#^11+10]&] (* Harvey P. Dale, Jul 01 2015 *)

is(n)=isprime(n^11+10) \\ Charles R Greathouse IV, Feb 17 2017

A134407 Numbers n such that n^2 + 1 is composite.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Jani Melik, Jan 18 2008

			a(1)=3, because 3^2 + 1 = 10 is composite,
a(2)=5, because 5^2 + 1 = 26 is composite,
a(3)=7, because 7^2 + 1 = 50 is composite.

Cf. A002496, A002808, A005574, A018252.

ts_fn2:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false) then ans:=[ op(ans), i ]: fi od: RETURN(ans) end: ts_fn2(200);

Select[Range@100,!PrimeQ[#^2+1]&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

is(n)=!isprime(n^2+1) \\ Charles R Greathouse IV, Sep 15 2014

a(n) ~ n. - Charles R Greathouse IV, Sep 15 2014

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

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

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862 (11), this sequence (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060886, A125258, A085398, A124990.

A246397:=n->`if`(isprime(n^4-n^2+1),n,NULL): seq(A246397(n),n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

for(n=1,10^3,if(isprime(polcyclo(12,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A062325 Numbers k for which phi(prime(k)) is a square.

Original entry on oeis.org

1, 3, 7, 12, 26, 45, 55, 79, 106, 123, 211, 252, 422, 446, 595, 723, 907, 1019, 1101, 1448, 1595, 1687, 1797, 1849, 1949, 2058, 2393, 2516, 2703, 2819, 3146, 3339, 3477, 3626, 4353, 4437, 4590, 5153, 5398, 5653, 5836, 6276, 6543, 6736, 6911, 7207, 7695

Offset: 1

Jason Earls, Jul 05 2001

Also A002496 indexed by A000040.

			79 is in the sequence because the 79th prime is 401 and phi(401) is 400 = 20^2.
595 is in the sequence because the 595th prime is 4357 and phi(4357) is 4356 = 66^2.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A001912, A002496, A005574, A062325, A090693.

Flatten[Position[Table[IntegerQ[Sqrt[Prime[w]-1]], {w, 1, 25000}], True]]
Flatten[Position[EulerPhi[Prime[Range[8000]]],?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Apr 23 2014 *)

for(n=1,1600, if(issquare(eulerphi(prime(n))),print(n)))

{ default(primelimit, 2*10^8); n=m=0; forprime (p=2, 2*10^8, m++; if (issquare(eulerphi(p)), write("b062325.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

a(n) = A000720(A002496(n)).

A000040(a(n)) = A002496(n).

More terms from Labos Elemer, Jul 09 2001

A065876 a(n) is the smallest m > n such that n^2 + 1 divides m^2 + 1.

Original entry on oeis.org

1, 3, 3, 7, 13, 21, 31, 43, 18, 73, 91, 111, 17, 47, 183, 211, 241, 133, 57, 343, 381, 47, 172, 83, 553, 601, 651, 173, 342, 813, 242, 265, 132, 403, 411, 1191, 1261, 237, 327, 1483, 1561, 1641, 748, 857, 850, 1981, 684, 463, 413, 2353, 255, 2551, 593, 1177, 2863, 123, 3081, 307, 1288, 3423

Offset: 0

Benoit Cloitre, Dec 07 2001

nonn

a(n) exists because n^2 + 1 divides (n^2 - n + 1)^2 + 1. The set of n such a(n) = n^2 - n + 1 is S = (2, 3, 4, 5, 6, 7, 9, 11, 14, 15, ...).
a(n) = n^2 - n + 1 whenever n^2 + 1 is prime or twice a prime. Up to n=1000, the only other n for which a(n) = n^2 - n + 1 are 7, 41 and 239. Is it a coincidence that these are NSW primes (A088165)? - Franklin T. Adams-Watters, Oct 17 2006
It appears that the density of even numbers in this sequence approaches a limit near 1/4. It appears that the density of even values for indices where a(n) != n^2 - n + 1 is approaching a number near 1/4 and based on the previous comment the density of n for which a(n) = n^2 - n + 1 is almost certainly 0. - Franklin T. Adams-Watters, Oct 17 2006

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1000

Cf. A002731, A005574, A071557, A088165.

Do[k = 1; While[m = (k^2 + 1)/(n^2 + 1); m < 2 || !IntegerQ[m], k++ ]; Print[k], {n, 1, 40 } ]

a(n) = { my(m=n+1); while ((m^2 + 1)%(n^2 + 1) != 0, m++); m } \\ Harry J. Smith, Nov 03 2009

More terms from Robert G. Wilson v, Dec 11 2001

Further terms from Franklin T. Adams-Watters, Oct 17 2006

A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

Original entry on oeis.org

2, 4, 10, 14, 74, 94, 130, 134, 146, 160, 230, 256, 326, 340, 350, 406, 430, 440, 470, 584, 634, 686, 700, 704, 784, 860, 920, 986, 1054, 1070, 1156, 1210, 1324, 1340, 1354, 1366, 1394, 1420, 1456, 1460, 1564, 1700, 1784, 1816, 1876, 2006, 2080, 2096, 2174

Offset: 1

T. D. Noe, Jan 30 2003

Hardy and Littlewood conjecture that this sequence is infinite. This sequence is the intersection of A005574 (k such that k^2 + 1 is prime) and A049422 (k such that k^2 + 3 is prime).
From Jacques Tramu, Sep 10 2018: (Start)
a(10000)    =     2473624;  C = 2.91596513
a(100000)   =    35866246;  C = 2.70591741
a(1000000)  =   483764726;  C = 2.53454683
a(2000000)  =  1049178316;  C = 2.49209641
a(3000000)  =  1647417724;  C = 2.46880647
a(4000000)  =  2267125384;  C = 2.45259161
a(5000000)  =  2903162576;  C = 2.44036006
a(6000000)  =  3551848640;  C = 2.43024082
a(7000000)  =  4212006124;  C = 2.42214552
a(8000000)  =  4881390700;  C = 2.41510010
a(9000000)  =  5559542740;  C = 2.40915933
a(10000000) =  6245573750;  C = 2.40405768
a(20000000) = 13393786900;  C = 2.36959294
a(30000000) = 20908970800;  C = 2.35131696
a(40000000) = 28659267134;  C = 2.33835867
a(50000000) = 36590858294;  C = 2.32865934
C is the quotient a(n) / (n * log(n) * log(n)). (End)

			10 is in this sequence because 101 and 103 are both prime.

P. Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, 1996, p. 408.

Zak Seidov, Table of n, a(n) for n = 1..32898 (terms < 10^7, first 1000 terms from T. D. Noe)
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A005574, A049422, A145824.

lst={}; Do[If[PrimeQ[m^2+1]&&PrimeQ[m^2+3], AppendTo[lst, m]], {m, 3000}]; lst
okQ[n_]:=Module[{n2=n^2},PrimeQ[n2+1]&&PrimeQ[n2+3]]; Select[Range[2200], okQ]  (* Harvey P. Dale, Apr 21 2011 *)
Select[Range[2500],AllTrue[#^2+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 07 2023 *)

isA080149(n) = isprime(n^2+1) && isprime(n^2+3) \\ Michael B. Porter, Mar 22 2010

Conjecture: a(n) is asymptotic to c*n*log(n)^2 with c around 2.9... - Benoit Cloitre, Apr 16 2004

A083849 a(n) is the largest prime of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

2, 2, 5, 5, 17, 37, 101, 197, 401, 677, 1601, 3137, 8101, 15877, 32401, 62501, 122501, 246017, 512657, 1020101, 2073601, 4137157, 8386817, 16695397, 33339077, 66977857, 133772357, 268304401, 536663557, 1073610757, 2146098277

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that this sequence is increasing, but this has never been proved.

It is easily shown that all terms greater than 5 end in 1 or 7.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083848.

a(n) = my(last = 2^n+1); while ((p = precprime(last-1)) && (! issquare(p-1)), last = p;); p \\ Michel Marcus, Jun 14 2013

a(n)=my(k=sqrtint(2^n-1)); while(!isprime(k^2+1), k--); k^2+1 \\ Charles R Greathouse IV, Nov 29 2013

A090693 Positive numbers n such that n^2 - 2n + 2 is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 15, 17, 21, 25, 27, 37, 41, 55, 57, 67, 75, 85, 91, 95, 111, 117, 121, 125, 127, 131, 135, 147, 151, 157, 161, 171, 177, 181, 185, 205, 207, 211, 225, 231, 237, 241, 251, 257, 261, 265, 271, 281, 285, 301, 307, 315, 327, 341, 351, 385, 387, 397

Offset: 1

Giovanni Teofilatto, Dec 19 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

A002496 gives primes, A062325 gives prime index. Cf. A001912.

A005574(n+1) + 1.

a={};Do[If[PrimeQ[n^2-2n+2],AppendTo[a,n]],{n,1000}];a (* Peter J. C. Moses, Apr 02 2013 *)
Select[Range[400],PrimeQ[#^2-2#+2]&] (* Harvey P. Dale, May 10 2013 *)

# Python 3.2 or higher required.
from itertools import accumulate
from sympy import isprime
A090693_list = [i for i,n in enumerate(accumulate(range(10**5),lambda x,y:x+2*y-3)) if i > 0 and isprime(n+2)] # Chai Wah Wu, Sep 23 2014

a(n) = A005574(n)+1.

Corrected and extended by Ray Chandler, Dec 28 2003

Definition corrected by Chai Wah Wu, Sep 23 2014

cp's OEIS Frontend

A121982 Numbers k such that k^2 + 15 is prime.

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A125262 Numbers k such that k^7 + 6 is prime.

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A125265 Numbers k such that k^11 + 10 is prime.

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A134407 Numbers n such that n^2 + 1 is composite.

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A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

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A062325 Numbers k for which phi(prime(k)) is a square.

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A065876 a(n) is the smallest m > n such that n^2 + 1 divides m^2 + 1.

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A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

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