A001912 -id:A001912 - cp's OEIS frontend

A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, 142573, 147853, 169933

Offset: 1

N. J. A. Sloane

nonn

Primes p such that p = 1 + 3*m^2 for some integer m (A111051). - Michael Somos, Sep 15 2005

			193 is a term since 193 = (9^3 - 7^3)/(9 - 7) is a prime.

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..5000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Eric Weisstein's World of Mathematics, Cuban Prime.
Wikipedia, Cuban prime.

Cf. A002407, A111051 (values of m).

A subsequence of A007645.

[a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011

Select[Table[3n^2+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

a(n) = 3*A111051(n)^2 + 1. - Paul F. Marrero Romero, Nov 03 2023

Entry revised by N. J. A. Sloane, Jan 29 2013

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

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

A002646 Half-quartan primes: primes of the form p = (x^4 + y^4)/2.

Original entry on oeis.org

41, 313, 353, 1201, 3593, 4481, 7321, 8521, 10601, 14281, 14321, 14593, 21601, 26513, 32633, 41761, 41801, 42073, 42961, 49081, 56041, 66361, 67073, 72481, 90473, 97241, 97553, 104561, 106921, 111521, 139921, 141121, 165233, 195353, 198593

Offset: 1

N. J. A. Sloane

The 1001-digit number ((10^250 + 5659)^4 + (10^250 + 5661)^4)/2 is currently the largest known half-quartan prime. - Paul Muljadi, Mar 03 2011
The largest known is now ((2*3960926^2048 + 1)^4 + 1^4)/2 with 54051 digits. - Jens Kruse Andersen, Mar 20 2011
Primes of the form p = a^2 + b^2 with a > b > 0 such that a + b and a - b are squares. - Thomas Ordowski, Jul 07 2016
Primes p = a^2 + b^2 with a > b > 0 such that a^2 - b^2 is a square. - Thomas Ordowski, Feb 14 2017
Primes p > 5 such that the Diophantine equation X^4 + Y^2 = p^2 has a solution X,Y with nonzero X. Then X must be odd. - Thomas Ordowski and Robert G. Wilson v, Nov 29 2017

			41 is in the sequence since it is prime and 41 = (3^4 + 1^4)/2. - _Michael B. Porter_, Jul 07 2016

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Tim Evink, Jaap Top, and Jakob Dirk Top, A remark on prime (non)congruent numbers, arXiv:2105.01450 [math.NT], 2021. See p. 12.

Cf. A000583, A002645, A005408, A010051.

a002646 n = a002646_list !! (n-1)
a002646_list = [hqp | x <- [1, 3 ..], y <- [1, 3 .. x - 1],
                      let hqp = div (x ^ 4 + y ^ 4) 2, a010051' hqp == 1]
-- Reinhard Zumkeller, Jul 15 2013

N:= 10^6: # to get all terms <= N
sort(select(isprime, convert({seq(seq((x^4+y^4)/2, y=x..floor((2*N-x^4)^(1/4)),2),x=1..floor((2*N-1)^(1/4)),2)},list))); # Robert Israel, Jul 11 2016

nmax = 200000; jmax = Floor[(nmax/8)^(1/4)]; s = {}; Do[n = ((2 j + 1)^4 + (2 k + 1)^4)/2; If[n <= nmax && PrimeQ[n], AppendTo[s, n]], {j, 0, jmax}, {k, j,  jmax}]; Union[s] (* Jean-François Alcover, Mar 23 2011 *)
Sort[Select[Total/@(Union[Sort/@Tuples[Range[0,50],2]]^4)/2,PrimeQ]] (* Harvey P. Dale, Feb 12 2012 *)

More terms from Len Smiley

A214517 Differences between the numbers n such that 4n^2 + 1 is prime.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 5, 2, 7, 1, 5, 4, 5, 3, 2, 8, 3, 2, 2, 1, 2, 2, 6, 2, 3, 2, 5, 3, 2, 2, 10, 1, 2, 7, 3, 3, 2, 5, 3, 2, 2, 3, 5, 2, 8, 3, 4, 6, 7, 5, 17, 1, 5, 2, 3, 7, 5, 3, 2, 2, 10, 1, 2, 2, 8, 3, 20, 4, 6, 7, 3, 4, 5, 20, 1, 4, 1, 4, 10, 3, 3, 2, 3

Offset: 1

T. D. Noe, Aug 06 2012

nonn

Sequence A001912 has the values of n. This sequence is the first differences of A001912.

T. D. Noe, Table of n, a(n) for n = 1..10000

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

Differences[Select[Range[100], PrimeQ[1 + 4*#^2] &, 101]]

a(n) = A214516(n)/2 for n > 1.

A002649 Quintan primes: p = (x^5 - y^5)/(x - y).

Original entry on oeis.org

5, 31, 211, 1031, 2801, 4651, 5261, 6841, 8431, 14251, 17891, 20101, 21121, 22621, 22861, 26321, 30941, 33751, 36061, 41141, 46021, 48871, 51001, 58411, 61051, 88741, 92821, 103801, 109141, 114641, 118061, 125591, 170101, 176641, 209801

Offset: 1

N. J. A. Sloane

nonn

5 is a term because x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4 = 5 when x=y=1. - N. J. A. Sloane, May 12 2014

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 2, p. 200.
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. A002650.

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

a(26)-a(35) from Sean A. Irvine, May 08 2014

A002641 Numbers k such that (k^2 + k + 1)/7 is prime.

Original entry on oeis.org

4, 9, 11, 23, 32, 39, 44, 51, 53, 60, 65, 72, 86, 93, 95, 114, 123, 156, 170, 179, 186, 200, 207, 212, 219, 228, 233, 240, 249, 261, 270, 303, 317, 333, 338, 345, 375, 389, 401, 443, 452, 473, 480, 492, 515, 534, 548, 564, 578, 585, 597, 599, 611, 641, 660, 662

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]

Subsequence of A047348.

/* See Crossrefs: */
A047348:=[m: m in [1..700] | m mod 7 in [2,4]];
[n: n in A047348 | IsPrime( (n^2+n+1) div 7 )];
// Bruno Berselli, Sep 26 2012

Select[Range[700], PrimeQ[(#^2 + # + 1)/7] &] (* Harvey P. Dale, Sep 16 2011 *)

is(n)=isprime((n^2+n+1)/7) \\ Charles R Greathouse IV, May 22 2017

More terms from Jon E. Schoenfield, Mar 24 2010

A002642 Numbers k such that (k^2 + k + 1)/13 is prime.

Original entry on oeis.org

9, 29, 35, 42, 48, 113, 120, 126, 152, 185, 204, 224, 237, 243, 276, 302, 308, 321, 341, 386, 399, 419, 432, 477, 503, 510, 516, 542, 549, 588, 633, 659, 666, 705, 731, 770, 776, 783, 789, 815, 848, 854, 887, 906, 932, 945, 965, 978

Offset: 1

N. J. A. Sloane

All terms are congruent to 3 or 9 (mod 13). [Bruno Berselli, Sep 26 2012]

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]

I:=[m: m in [1..1000] | m mod 13 in [3,9]];
[n: n in I | IsPrime( (n^2 + n + 1) div 13 )];
// Bruno Berselli, Sep 26 2012

Select[Range[1000], PrimeQ[(#^2 + # + 1)/13]&] (* Vincenzo Librandi, Sep 25 2012 *)

forstep(n=9,1e4,[7,6],if(isprime((n^2+n+1)/13),print1(n", "))) \\ Charles R Greathouse IV, Sep 25 2012

A325437 Final digit of primes of the form k^2 + 1.

Original entry on oeis.org

2, 5, 7, 7, 1, 7, 7, 1, 7, 7, 7, 1, 7, 7, 7, 7, 7, 1, 7, 1, 7, 1, 7, 7, 1, 7, 7, 1, 7, 1, 1, 7, 1, 7, 7, 7, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 1, 7, 1, 7, 7, 7, 1, 1, 7, 7, 7, 1, 7, 1, 1, 7, 1, 7, 7, 7, 1, 7, 1, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 1, 7, 1, 7

Offset: 1

Martin Renner, Apr 27 2019

This sequence is presumably infinite. See 1st comment of A002496.
For k > 2, i.e., primes > 5 the final digit is always 1 or 7. Proof: Let k = 2*m - 1 odd. Then k^2 + 1 is divisible by 2, hence prime only for m = 1. Let k = 2*m even. Then k^2 + 1 = 4*m^2 + 1. The final digit of multiples of four is 4, 8, 2, 6, 0, 4, 8, 2, 6, 0, ... and of squares 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, ... (cf. A008959), hence the last digit of the product 4*m^2 is 4, 6, 6, 4, 0, ... or of the sum 4*m^2 + 1 is 5, 7, 7, 5, 1, ... (cf. A053755) and therefore for primes > 5 the final digit is 1 or 7.
Accordingly, for large k approximately one-third of the primes of the form k^2 + 1 end in 1, two-thirds end in 7.

Robert Israel, Table of n, a(n) for n = 1..10000
Edmund Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion, Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228, here p. 224.
Eric Weisstein's World of Mathematics, Landau's Problems., Nr. 4.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A001912, A002496, A005574.

seq(k mod 10,k=select(isprime,[2,seq(4*i^2+1,i=1..10000)]));

Mod[#,10]&/@Select[Range[1000]^2+1,PrimeQ] (* Harvey P. Dale, Jul 05 2023 *)

lista(nn) = {forprime(p=2, nn, if (issquare(p-1), print1(p % 10, ", ")););} \\ Michel Marcus, May 07 2019

a(n) = A002496(n) mod 10.

A002640 Numbers k such that (k^2 + k + 1)/3 is prime.

Original entry on oeis.org

4, 7, 10, 13, 19, 28, 31, 34, 40, 43, 52, 70, 73, 76, 82, 85, 91, 97, 103, 112, 115, 124, 127, 136, 145, 148, 157, 166, 175, 187, 190, 199, 202, 223, 241, 244, 259, 265, 271, 274, 280, 286, 316, 325, 358, 370, 376, 385, 388, 409, 421, 427, 442, 460, 469, 472

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]
Hajrudin Fejzić, Dan Rinne, and Bob Stein, On Sums of Cubes, The College Mathematics Journal, Vol. 36, No. 3 (May, 2005), pp. 226-228. See p. 228.

Cf. A002384.

[n: n in [4..500] | IsPrime((n^2+n+1) div 3)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[500], PrimeQ[(#^2 + # + 1)/3] &] (* Vincenzo Librandi, Sep 25 2012 *)

isok(k) = my(x=k^2+k+1); !(x%3) && isprime(x/3); \\ Michel Marcus, Aug 22 2025

cp's OEIS Frontend

A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

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

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

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A002646 Half-quartan primes: primes of the form p = (x^4 + y^4)/2.

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A214517 Differences between the numbers n such that 4n^2 + 1 is prime.

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A002649 Quintan primes: p = (x^5 - y^5)/(x - y).

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PARI

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A002641 Numbers k such that (k^2 + k + 1)/7 is prime.

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