cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A155568 Intersection of A001481 inter A020670: N = a^2 + b^2 = c^2 + 7d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 25, 29, 32, 36, 37, 49, 53, 64, 72, 81, 100, 109, 113, 116, 121, 128, 137, 144, 148, 149, 169, 193, 196, 197, 200, 212, 225, 232, 233, 256, 261, 277, 281, 288, 289, 296, 317, 324, 333, 337, 361, 373, 389, 392, 400, 401, 421, 424, 436, 441, 449
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Contains A155578 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Crossrefs

Cf. A001481, A002479, A003136, A002481, A020668 ff.

Programs

  • PARI
    isA155568(n,/* use optional 2nd arg to get other analogous sequences */c=[7,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155568(n) & print1(n","))

A155569 Intersection of A002479 inter A002481: N = a^2 + 2b^2 = c^2 + 6d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 22, 24, 25, 33, 36, 49, 54, 64, 73, 81, 88, 96, 97, 100, 118, 121, 132, 144, 150, 166, 169, 177, 193, 196, 198, 214, 216, 225, 241, 249, 256, 262, 289, 292, 294, 297, 313, 321, 324, 337, 352, 358, 361, 384, 388, 393, 400, 409, 433, 438, 441
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Contains A155709 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Crossrefs

Cf. A001481, A002479, A003136, A002481, A020668 ff.

Programs

  • PARI
    isA155569(n,/* use optional 2nd arg to get other analogous sequences */c=[6,2]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155569(n) & print1(n","))

A155570 Intersection of A003136 and A020669: N = a^2 + 3b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 16, 21, 25, 36, 49, 61, 64, 81, 84, 100, 109, 121, 129, 144, 169, 181, 189, 196, 201, 225, 229, 241, 244, 256, 289, 301, 309, 324, 336, 349, 361, 381, 400, 409, 421, 436, 441, 469, 484, 489, 516, 525, 529, 541, 549, 576, 601, 625, 661, 669, 676, 709
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Contains A155710 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Crossrefs

Cf. A001481, A002479, A003136, A002481, A020668 ff.

Programs

  • PARI
    isA155570(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,800, isA155570(n) & print1(n","))

A280084 1 together with the Pythagorean primes.

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 25 2016

Keywords

Comments

Positive noncomposite numbers of the form 4k + 1.
Positive noncomposite numbers in A020668.
Essentially the same as A002313 and A002144. - R. J. Mathar, Jan 04 2017

Crossrefs

Cf. A002144, A006005, A008578, A020668.

Programs

  • Magma
    [1] cat [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ];
  • PARI
    is(n)=if(isprime(n), n%4==1, n==1) \\ Charles R Greathouse IV, Oct 10 2018

Formula

A002144 UNION {1}. - R. J. Mathar, Jan 04 2017

A298950 Numbers k such that 5*k - 4 is a square.

Original entry on oeis.org

1, 4, 8, 17, 25, 40, 52, 73, 89, 116, 136, 169, 193, 232, 260, 305, 337, 388, 424, 481, 521, 584, 628, 697, 745, 820, 872, 953, 1009, 1096, 1156, 1249, 1313, 1412, 1480, 1585, 1657, 1768, 1844, 1961, 2041, 2164, 2248, 2377, 2465, 2600, 2692, 2833, 2929, 3076, 3176, 3329, 3433
Offset: 1

Views

Author

Bruno Berselli, Jan 30 2018

Keywords

Comments

a(n) is a member of A140612. Proof: a(n) = n^2 + (n/2-1)^2 for even n, otherwise a(n) = (n-1)^2 + ((n+1)/2)^2; also, a(n) + 1 = (n-1)^2 + (n/2+1)^2 for even n, otherwise a(n) + 1 = n^2 + ((n-3)/2)^2. Therefore, both a(n) and a(n) + 1 belong to A001481.
Primes in sequence are listed in A245042.
Squares in sequence are listed in A081068.

Links

Crossrefs

Cf. A195162: numbers k such that 5*k + 4 is a square.
Subsequence of A001481, A020668, A036404, A140612.
Cf. A036666, A081068, A106833 (first differences), A245042.

Programs

  • GAP
    List([1..60], n -> (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8);
  • Magma
    [(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8: n in [1..60]];
  • Mathematica
    Table[(10 n (n - 1) + (2 n - 1) (-1)^n + 9)/8, {n, 1, 60}]
LinearRecurrence[{1,2,-2,-1,1},{1,4,8,17,25},60] (* Harvey P. Dale, Sep 16 2022 *)
  • Maxima
    makelist((10*n*(n-1)+(2*n-1)*(-1)^n+9)/8, n, 1, 60);
  • PARI
    Vec((1+x^2)*(1+3*x+x^2)/((1-x)^3*(1+x)^2)+O(x^60))
  • PARI
    vector(60, n, nn; (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8)
  • Python
    [(10*n*(n-1)+(2*n-1)*(-1)**n+9)/8 for n in range(1, 60)]
  • Sage
    [(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8 for n in (1..60)]

Formula

G.f.: x*(1 + x^2)*(1 + 3*x + x^2)/((1 - x)^3*(1 + x)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (10*n*(n-1) + (2*n-1)*(-1)^n + 9)/8.
a(n) = A036666(n) + 1.
Previous Showing 11-15 of 15 results.

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