A007645 -id:A007645 - cp's OEIS frontend

A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

0, 2, 1, 4, 2, 5, 4, 7, 8, 5, 2, 7, 10, 1, 10, 8, 2, 7, 4, 13, 1, 14, 8, 14, 11, 7, 14, 13, 16, 8, 11, 16, 17, 7, 2, 19, 4, 17, 19, 11, 1, 14, 5, 10, 22, 16, 4, 23, 20, 8, 23, 13, 10, 5, 16, 22, 20, 19, 25, 4, 11, 22, 25, 8, 26, 13, 1, 28, 28, 26, 23, 29, 28

Offset: 1

N. J. A. Sloane

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
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).
B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Cf. A001480, A007645, A000196, A010052, A033428.

a001479 n = a000196 $ head $
   filter ((== 1) . a010052) $ map (a007645 n -) $ tail a033428_list
-- Reinhard Zumkeller, Jul 11 2013

nmax = 56; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := x /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 0; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[1])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

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

A001480 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = y.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 1, 4, 5, 4, 1, 6, 3, 5, 7, 6, 7, 2, 8, 1, 7, 3, 6, 8, 5, 6, 3, 9, 8, 5, 4, 10, 11, 2, 11, 6, 4, 10, 12, 9, 12, 11, 1, 9, 13, 2, 7, 13, 4, 12, 13, 14, 11, 7, 9, 10, 4, 15, 14, 9, 6, 15, 5, 14, 16, 1, 3, 7, 10, 2, 5, 14, 17, 13, 9, 16, 17

Offset: 1

N. J. A. Sloane

a(n) = A000196((A007645(n) - A000290(A001479(n))) / 3). - Reinhard Zumkeller, Jul 11 2013

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
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).
B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Cf. A001479, A007645.

a001480 n = a000196 $ (`div` 3) $ (a007645 n) - (a001479 n) ^ 2
-- Reinhard Zumkeller, Jul 11 2013

nmax = 63; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := y /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 1; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[2])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

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

A002367 Let p = A007645(n) be the n-th generalized cuban prime and write p^2 = x^2 + 3*y^2 with y > 0; a(n) = x.

Original entry on oeis.org

1, 11, 13, 23, 13, 11, 37, 61, 23, 71, 1, 97, 107, 73, 11, 143, 59, 131, 157, 191, 193, 83, 169, 13, 143, 121, 61, 229, 179, 71, 181, 241, 251, 359, 349, 347, 181, 313, 179, 431, 47, 407, 263, 481, 13, 491, 517, 253, 443, 481, 263, 407, 563, 107, 337, 157, 61

Offset: 2

N. J. A. Sloane

nonn

A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
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 = 2..1000
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927. [Annotated scanned copy of selected pages]

Cf. A002368, A007645.

Name improved and offset changed by T. D. Noe, Apr 13 2010

A002368 Let p = A007645(n) be the n-th generalized cuban prime and write p^2 = x^2 + 3*y^2 with y > 0; a(n) = y.

Original entry on oeis.org

4, 4, 8, 12, 20, 24, 28, 16, 40, 20, 56, 20, 12, 60, 80, 28, 84, 56, 52, 16, 28, 112, 84, 132, 112, 140, 156, 96, 144, 176, 160, 136, 140, 44, 76, 88, 204, 152, 220, 24, 252, 120, 220, 44, 288, 104, 92, 280, 208, 184, 312, 260, 140, 352, 308, 360, 380, 200

Offset: 2

N. J. A. Sloane

nonn

A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
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 = 2..1000
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927. [Annotated scanned copy of selected pages]

Cf. A002367, A007645.

Name improved and offset changed by T. D. Noe, Apr 13 2010

A107890 Semiprimes that are the product of two members of A007645.

Original entry on oeis.org

9, 21, 39, 49, 57, 91, 93, 111, 129, 133, 169, 183, 201, 217, 219, 237, 247, 259, 291, 301, 309, 327, 361, 381, 403, 417, 427, 453, 469, 471, 481, 489, 511, 543, 553, 559, 579, 589, 597, 633, 669, 679, 687, 703, 721, 723, 763, 793, 813, 817, 831, 849, 871

Offset: 1

Jonathan Vos Post, Jun 12 2005

Conway, J. H. and Guy, R. K., The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Eisenstein Integer.
Eric Weisstein's World of Mathematics, Eisenstein Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A007645, A108164.

N:= 1000: # for terms <= N
P:= [3,op(select(isprime, [seq(i,i=1..N/3,6)]))]:
R:= NULL:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P,N/P[i]+1/2);
  R:= R, seq(P[i]*P[j],j=i..m);
od:
sort([R]); # Robert Israel, Aug 28 2020

{a(n)} = {p*q: p and q both elements of A007645} = {p*q: p and q both of form 3*m^2 * n^2 for integers m, n}.

Edited by Ray Chandler, Oct 15 2005

Definition corrected by N. J. A. Sloane, Feb 06 2008

A172113 Partial sums of the generalized Cuban primes A007645.

Original entry on oeis.org

3, 10, 23, 42, 73, 110, 153, 214, 281, 354, 433, 530, 633, 742, 869, 1008, 1159, 1316, 1479, 1660, 1853, 2052, 2263, 2486, 2715, 2956, 3227, 3504, 3787, 4094, 4407, 4738, 5075, 5424, 5791, 6164, 6543, 6940, 7349, 7770, 8203, 8642, 9099, 9562, 10049

Offset: 1

Jonathan Vos Post, Jan 25 2010

Partial sums of primes of the form 3*m+1/2+-1/2. - Juri-Stepan Gerasimov, Jan 29 2010. E.g. a(1)=3*1+1/2-1/2=3, a(2)=3+3*2+1/2+1/2=10.

The primes in this sequence begin: a(1) = 3, a(3) = 23, a(5) = 73, a(9) = 281, a(11) = 433. Of these, the subset of generalized cuban primes which are partial sums of generalized cuban primes begins: 3, 73, 433.

			a(30) = 3 + 7 + 13 + 19 + 31 + 37 + 43 + 61 + 67 + 73 + 79 + 97 + 103 + 109 + 127 + 139 + 151 + 157 + 163 + 181 + 193 + 199 + 211 + 223 + 229 + 241 + 271 + 277 + 283 + 307 = 4094.

Cf. A000040, A007645, A038349.

Contribution from R. J. Mathar, Apr 24 2010: (Start)
A007645 := proc(n) if n <= 2 then op(n,[3,7]) ; ; else for a from procname(n-1)+2 by 2 do if isprime(a) and (a mod 3) <> 2 then return a ; end if; end do: end if; end proc:
A172113 := proc(n) add( A007645(i),i=1..n) ; end proc: seq(A172113(n),n=1..80) ; (End)

a(n) = SUM[i=1..n] A007645(i) = SUM[i=1..n] {primes of the form x^2 + xy + y^2} = SUM[i=1..n] {primes of form x^2 + 3*y^2} = SUM[i=1..n] {primes == 0 or 1 mod 3}.

a(5) corrected and more terms appended by R. J. Mathar, Feb 07 2010

Edited by N. J. A. Sloane, Sep 26 2010, Jan 29 2013.

A217035 Generalized cuban primes (A007645) which are also Class 1- (or Pierpont) primes (A005109).

Original entry on oeis.org

3, 7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993, 1769473

Offset: 1

Jonathan Vos Post, Sep 24 2012

Is this the union of A058383 and {3}? - R. J. Mathar, Sep 28 2012
Yes, it is, because the only Fermat prime == 0 or 1 mod 3 is 3. - Robert Israel, Mar 02 2018
Generalized cuban primes are primes of the form x^2 + xy + y^2; or: primes of form x^2 + 3*y^2; or: primes == 0 or 1 mod 3. Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

Ray Chandler, Table of n, a(n) for n = 1..8379 (terms < 10^1000)

Cf. A007645, A005109, A058383.

nn = 100000; t1 = Join[{3}, Select[Prime[Range[nn]], MemberQ[{1}, Mod[#, 3]] &]]; t2 = Select[Prime[Range[nn]], Max @@ First /@ FactorInteger[# - 1] < 5 &]; Intersection[t1, t2] (* T. D. Noe, Sep 26 2012 *)

A007645 INTERSECTION A005109.

A144919 Duplicate of A007645.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

dead

A173147 Numbers n such that exactly one of prime(n-1) and prime(n+1) is a generalized cuban prime (A007645).

Original entry on oeis.org

9, 10, 11, 12, 15, 17, 18, 19, 21, 22, 23, 24, 32, 33, 36, 38, 39, 41, 46, 48, 51, 52, 54, 57, 58, 59, 67, 68, 71, 72, 73, 75, 76, 77, 84, 85, 86, 87, 91, 92, 96, 98, 99, 101, 102, 104, 105, 106, 107, 109, 110, 112, 114, 115, 118, 120, 121, 122, 123, 124, 129, 131

Offset: 1

Juri-Stepan Gerasimov, Feb 11 2010

			a(2)=9 because prime(9-1)=19 is a generalized cuban prime and prime(9+1)=29 is not.

Cf. A007645, A003627.

84 inserted, 88 removed - R. J. Mathar, Mar 01 2010

Further corrections and edits from Charles R Greathouse IV, Mar 25 2010

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

Original entry on oeis.org

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

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.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

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cp's OEIS Frontend

A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

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A001480 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = y.

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A002367 Let p = A007645(n) be the n-th generalized cuban prime and write p^2 = x^2 + 3*y^2 with y > 0; a(n) = x.

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A002368 Let p = A007645(n) be the n-th generalized cuban prime and write p^2 = x^2 + 3*y^2 with y > 0; a(n) = y.

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A107890 Semiprimes that are the product of two members of A007645.

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A172113 Partial sums of the generalized Cuban primes A007645.

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A217035 Generalized cuban primes (A007645) which are also Class 1- (or Pierpont) primes (A005109).

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A144919 Duplicate of A007645.

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A173147 Numbers n such that exactly one of prime(n-1) and prime(n+1) is a generalized cuban prime (A007645).

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