A001912 -id:A001912 - cp's OEIS frontend

A002644 Numbers k such that (k^2 + k + 1)/21 is prime.

16, 25, 37, 46, 58, 88, 109, 130, 142, 151, 184, 193, 205, 247, 268, 298, 310, 319, 331, 340, 382, 394, 403, 415, 424, 457, 478, 487, 541, 550, 604, 613, 688, 697, 709, 730, 739, 751, 760, 793, 844, 865, 886, 907, 970, 1012, 1045, 1054, 1066, 1117, 1138

Offset: 1

N. J. A. Sloane

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

Select[Range[1200], PrimeQ[(#^2 + # + 1)/21] &] (* Vincenzo Librandi, Sep 25 2012 *)

is(n)=isprime((n^2+n+1)/21) \\ Charles R Greathouse IV, Jun 06 2017

More terms from Larry Reeves (larryr(AT)acm.org), Sep 21 2000

A002650 Quintan primes: p = (x^5 + y^5)/(x + y).

Original entry on oeis.org

11, 61, 181, 421, 461, 521, 991, 1621, 1871, 3001, 4441, 4621, 6871, 9091, 9931, 12391, 13421, 14821, 19141, 25951, 35281, 35401, 55201, 58321, 61681, 62071, 72931, 74731, 91331, 92921, 95881, 108421, 117911, 117991, 131041, 132661, 141961

Offset: 1

N. J. A. Sloane

nonn

(x^5 + y^5)/(x + y) = x^4 - y*x^3 + y^2*x^2 - y^3*x + y^4. - Jens Kruse Andersen, Jul 14 2014

			(3^5 + 1^5)/(3 + 1) = 61. This is prime and therefore in the sequence. - _Jens Kruse Andersen_, Jul 14 2014

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 2, p. 201.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

Cf. A002649.

Take[Select[Union[(#[[1]]^5+#[[2]]^5)/Total[#]&/@Tuples[Range[200],2]], #>0&& PrimeQ[#]&],50] (* Harvey P. Dale, May 21 2012 *)

m=10^6; v=[]; for(x=1, (2*m)^(1/4), for(y=1, x, n=(x^5+y^5)/(x+y); if(n<=m && isprime(n), v=concat(v,n)))); vecsort(v) \\ Jens Kruse Andersen, Jul 14 2014

More terms from Sean A. Irvine, May 08 2014

A115349 Numbers k such that (4*k^5 + 1) is prime.

Original entry on oeis.org

1, 12, 15, 19, 27, 40, 49, 57, 60, 90, 93, 102, 132, 133, 147, 148, 153, 177, 190, 219, 240, 249, 258, 265, 274, 277, 280, 294, 313, 324, 337, 342, 363, 382, 394, 435, 448, 453, 462, 483, 489, 502, 522, 534, 538, 550, 580, 588, 609, 613, 634, 643, 648, 649

Offset: 1

Parthasarathy Nambi, Mar 07 2006

			If k=90 then (4*90^5 + 1) = 23619600001, which is prime.
If k=133 then (4*133^5 + 1) = 166463183573, which is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A115104, A001912.

[n: n in [0..1500] | IsPrime(4*n^5 + 1)]; // Vincenzo Librandi, Jan 31 2011

select(t -> isprime(4*t^5+1), [$1..1000]); # Robert Israel, Jun 19 2018

Do[If[PrimeQ[4*n^5 + 1], Print[n]], {n, 0, 1000}]
Select[Range[700],PrimeQ[4#^5+1]&] (* Harvey P. Dale, Aug 25 2023 *)

is(n)=isprime((4*n^5+1)) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Craig Baribault (csb166(AT)psu.edu), Mar 14 2006 and Jessica M. Cornwall (jmc510(AT)psu.edu), Mar 22 2006

A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.

Original entry on oeis.org

5, 37, 677, 1297, 2917, 8837, 13457, 50177, 147457, 156817, 246017, 341057, 414737, 746497, 1136357, 1726597, 1833317, 2119937, 2802277, 2808977, 3013697, 3549457, 3865157, 3896677, 4104677, 4384837, 5354597, 5410277, 5779217, 6031937, 6635777, 7001317

Offset: 1

Zak Seidov, Aug 28 2006

nonn

Intersection of A001912 and A121326. Except for the first term all other terms are == 7 (mod 10). Also all the primes 4*p^2+1 are == 7 (mod 10). - Zak Seidov, Mar 05 2015

Cf. A002496, A052291, A121326.

fpQ[n_]:=PrimeQ[n]&&PrimeQ[4n^2+1]; Select[4Range[2000]^2+1,fpQ] (* Harvey P. Dale, Nov 07 2016 *)

isA121834(n)={ if( (n-1) %4, return(0) ; ) ; if( issquare((n-1)/4), if( isprime(4*n^2+1), return(1), return(0) ), return(0) ; ) ; } { for(i=1,1000000, p=prime(i) ; if( isA121834(p), print1(p,",") ; ) ; ) ; } /* R. J. Mathar, Sep 01 2006*/

More terms from R. J. Mathar, Sep 01 2006

A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

Original entry on oeis.org

13, 9833, 41647, 151607, 264757, 356123, 361223, 446863, 449093, 457813, 531383, 641057, 655927, 841697, 855947, 899263, 913687, 1052813, 1081757, 1379383, 1506493, 1575757, 1685087, 1821013, 1821377, 1981517, 2054233, 2142037

Offset: 1

Zak Seidov, Oct 20 2006

nonn

Next terms up to 400000th prime are 2286877, 2524157, 2595247, 2621737, 2931583, 3023437, 3425843, 3428567, 3538517, 3705187, 3777883, 3799717, 3875143, 3913727, 3973553, 4019833, 4167073, 4249523, 4488167, 4651873, 4822193, 4914937, 5054167, 5108293, 5140147, 5465303, 5520007, 5542003. - Zak Seidov, Jan 16 2009

Subsequence of A122424. - Pierre CAMI, Jul 21 2014

			13 is there because 13, 677, 1833317 and 13444204889957 are prime.

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p.74.

Pierre CAMI, Table of n, a(n) for n = 1..3636

Cf. A052291, A005574, A001912, A122424.

Reap[Do[p=Prime[n];q=4p^2+1;r=4q^2+1;s=4r^2+1;If[PrimeQ[{q,r,s}]=={True, True,True},Sow[p]],{n,15000}]][[2,1]]
Select[Prime[Range[200000]],AllTrue[NestList[4#^2+1&,#,3],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

f(x)=4*x^2+1;
forprime(p=1, 10^8, if(isprime(f(p))&&isprime(f(f(p)))&&isprime(f(f(f(p)))), print1(p, ", "))) \\ Derek Orr, Jul 31 2014

More terms from Don Reble, Oct 24 2006

Edited by R. J. Mathar, Nov 02 2009

A157935 Primes of the form m^2+1 such that m^2-7 = prevprime(m^2) (= A007917(m^2)).

Original entry on oeis.org

2917, 4357, 8101, 24337, 57601, 72901, 93637, 224677, 324901, 331777, 404497, 562501, 608401, 1166401, 1742401, 1822501, 4137157, 4639717, 5788837, 7290001, 7617601, 10265617, 10497601, 10929637, 12110401, 12362257, 14107537, 14243077

Offset: 1

M. F. Hasler, Mar 18 2009

nonn

A subsequence of A005574. The values for m are listed in A157934, the next lower prime in A157183.

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

Cf. A157183, A028883, A028882, A005574, A001912, A157934.

Select[Range[4000]^2+1,PrimeQ[#]&&NextPrime[#,-1]==#-8&] (* Harvey P. Dale, Jun 14 2020 *)

forstep(m=2,9999,2, isprime(m^2+1) & precprime(m^2)==m^2-7 & print1(m^2+1,","))

a(n) = A157934(n)^2+1 = A157183(n)+8

A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 10, 17, 20, 23, 44, 50, 56, 65, 76, 106, 144, 165, 173

Offset: 1

T. D. Noe, Aug 06 2012

			a(1) = 1 because 4*1^2 + 1 = 5 and 4*2^2 + 1 = 17 are primes.
a(2) = 2 because 4*3^2 + 1 = 37 is prime, 4*4^2 + 1 = 65 is composite, and 4*5^2 + 1 = 101 is prime.
a(3) = 5 because 4*13^2 + 1 is prime, 4*n^2 + 1 is composite for n = 14..17, and 4*18^2 + 1 is prime.

Cf. A121326 (primes of the form 1+4*n^2), A001912 (values of n).

Cf. A214517 (differences), A214519 (where record differences occur).

n = 1; last = 1; t = {1}; While[Length[t] < 15, n++; p = 1 + 4*n^2; If[PrimeQ[p], If[n - last > t[[-1]], AppendTo[t, n - last]]; last = n]]; t

A230312 Squares which cannot be written as the sum of a smaller nonzero square and twice a triangular number.

Original entry on oeis.org

1, 4, 9, 25, 49, 64, 100, 144, 169, 324, 400, 729, 784, 1089, 1369, 1764, 2025, 2209, 3025, 3364, 3600, 3844, 3969, 4225, 4489, 5329, 5625, 6084, 6400, 7225, 7744, 8100, 8464, 10404, 10609, 11025, 12544, 13225, 13924, 14400, 15625, 16384, 16900

Offset: 1

Kieren MacMillan, Dec 20 2013

The conjecture a(n) = A001912(n)^2 (mentioned in the formula part) is easy. In fact, any prime divisor of 4*n^2 + 1 is congruent to 1 modulo 4 and hence it can be written as a sum of two squares. Thus 4*n^2 + 1 = (2*n)^2 + 1^2 is composite if and only if it can be written as a sum of two squares in at least two ways. So the conjecture follows immediately. - Zhi-Wei Sun, Feb 23 2014

Positive squares that are the sum of two triangular numbers in exactly one way. Note that each positive square is the sum of two consecutive triangular numbers since A000217(n) + A000217(n+1) = n*(n+1)/2 + (n+1)*(n+2)/2 = (n+1)^2. - Altug Alkan, Jul 06 2016

			16 is not in the sequence because it can be expressed as 2^2 + 2 * 6.
But there is no such expression for 25 and hence it is in the sequence.

Cf. A001912.

A230312 = Reap[For[k = 1, k < 200, k++, n = k^2; If[Reduce[a > 0 && b > 0 && n == a^2 + b * (b + 1), {a, b}, Integers] == False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 03 2014 *)

lista(nn) = for(n=1, nn, if(isprime(4*n^2+1), print1(n^2, ", "))); \\ Altug Alkan, Jul 06 2016

Conjecture: a(n) = A001912(n)^2, that is, squares of numbers n such that 4n^2 + 1 is prime. - Alonso del Arte, Dec 20 2013

A255634 Numbers n such that 1 + 16n^2 is prime.

Original entry on oeis.org

1, 4, 5, 6, 9, 10, 14, 21, 29, 30, 31, 39, 40, 44, 45, 46, 51, 56, 59, 60, 64, 65, 66, 70, 71, 75, 85, 96, 99, 100, 105, 109, 110, 111, 116, 124, 134, 136, 139, 144, 146, 159, 161, 170, 174, 175, 176, 179, 185, 190, 191, 195, 196, 204, 215, 216, 230, 234, 240, 251, 259, 265, 270, 274, 281

Offset: 1

Zak Seidov, Feb 28 2015

nonn

Note the sets of 3 consecutive numbers starting with 4, 29, 44, 64, 109, 174, ..., these numbers are congruent to 4 mod 5; cf. A255635.

Cf. A001912, A005574, A255635.

[n: n in [0..300] | IsPrime(16*n^2+1)]; // Vincenzo Librandi, Mar 03 2015

A255634:=n->`if`(isprime(1+16*n^2), n, NULL): seq(A255634(n), n=1..300); # Wesley Ivan Hurt, Feb 28 2015

Select[Range[400], PrimeQ[16 #^2 + 1] &] (* Vincenzo Librandi, Mar 03 2015 *)

select(n->isprime(1+16*n^2), vector(300, n, n)) \\ Colin Barker, Mar 01 2015

A255635 Numbers n such that 1+16n^2, 1+16(n+1)^2 and 1+16(n+2)^2 are prime.

Original entry on oeis.org

4, 29, 44, 64, 109, 174, 329, 614, 1044, 1694, 1879, 2044, 2254, 2474, 2709, 3814, 5024, 5039, 5154, 5364, 5634, 5784, 6244, 6624, 6779, 6804, 6949, 7964, 8079, 8509, 8624, 9034, 9324, 9394, 9729, 10719, 11114, 11504, 11954, 12149, 13064, 13319, 13354, 13554, 14019

Offset: 1

Zak Seidov, Feb 28 2015

nonn

Numbers n, n+1 and n+2 are terms in A255634.
The corresponding primes for 1+16n^2 are 257, 13457, 30977, 65537, 190097, 484417, ... (all == 7 mod 10);
The corresponding primes for 1+16(n+1)^2 are 401, 14401, 32401, 67601, 193601, 490001, ... (all == 1 mod 10);
The corresponding primes for 1+16(n+2)^2 are 577, 15377, 33857, 69697, 197137, 495617, ... (all == 7 mod 10).

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

Cf. A001912, A005574, A255634.

[n: n in [0..15000] | forall{16*n^2+i: i in [1, 32*n+17, 64*n+65] |  IsPrime(16*n^2+i)}]; // Vincenzo Librandi, Mar 04 2015

A255635:=n->`if`(isprime(1+16*n^2) and isprime(1+16*(n+1)^2) and isprime(1+16*(n+2)^2), n, NULL): seq(A255635(n), n=1..2*10^4); # Wesley Ivan Hurt, Feb 28 2015

Select[Range[15000],AllTrue[{16#^2,16(#+1)^2,16(#+2)^2}+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 09 2015 *)

select(n->isprime(1+16*n^2) && isprime(1+16*(n+1)^2) && isprime(1+16*(n+2)^2), vector(15000, n, n)) \\ Colin Barker, Mar 01 2015

cp's OEIS Frontend

A002644 Numbers k such that (k^2 + k + 1)/21 is prime.

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A002650 Quintan primes: p = (x^5 + y^5)/(x + y).

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A115349 Numbers k such that (4*k^5 + 1) is prime.

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Magma

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A121834 Primes p of the form 4*n^2 + 1 such that 4*p^2+1 is also prime.

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A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

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A157935 Primes of the form m^2+1 such that m^2-7 = prevprime(m^2) (= A007917(m^2)).

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Mathematica

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A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

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Mathematica

A230312 Squares which cannot be written as the sum of a smaller nonzero square and twice a triangular number.

A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.