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 11-17 of 17 results.

A155567 Intersection of A002479 and A020669 : N = a^2 + 2b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 24, 25, 36, 41, 49, 54, 64, 81, 86, 89, 96, 100, 121, 129, 134, 144, 150, 164, 166, 169, 196, 201, 214, 216, 225, 241, 246, 249, 256, 281, 289, 294, 321, 324, 326, 344, 356, 361, 369, 384, 400, 401, 409, 441, 449, 454, 484, 486, 489, 516, 521
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Contains A155577 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
    isA155567(n,/* use optional 2nd arg to get other analogous sequences */c=[5,2]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,600, isA155567(n) & print1(n","))

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","))

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","))

A370268 Intersection of A189715 and A370267.

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214, 216, 217, 220, 223, 225, 231, 232, 240, 241, 247
Offset: 1

Views

Author

Peter Munn, Feb 13 2024

Keywords

Comments

A189715 and A370267 are closely related in that they may be generated by the same process, but starting from numbers of the form 6m+1 and 8m+1 respectively - see A370267 for details.
Independent definition: numbers with an even number of prime factors not of the form 3m+1 and an even number of prime factors not of the form 8m+-1 (counting repetitions).
The sequence starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). After the absence of 0, this sequence next differs from A002481 by including 247, 391, 442, ... . From these early intermittent differences, the densities of the two sequences diverge progressively, driven by the absence from A002481 of many of the squarefree composite numbers that are present here though their prime factors are not. (Both sequences are closed under multiplication.) Asymptotic densities are 1/4 and 0 respectively.
Likewise, if we list the even terms halved, we find a similar relationship to the nonzero terms of A002480. The first 66 terms match, then we find we have generated intermittent extra terms: 221, 299, 323, ... .
Numbers whose squarefree part is congruent to {1,7} mod 24, {10,22} mod 48, {15,33} mod 72, or {6,42} mod 144. (Each congruence describes a coset of A334832 under A059897(.,.) as described in A334832. This sequence corresponds to the subgroup of the quotient group generated by {6,7,10}.)

Crossrefs

Intersection of A189715 and A370267.
A002481\{0}, A334832 are subsequences.
Cf. A002480, A042999, A045309, A059897.

Programs

  • PARI
    isok(k) = {c = core(k); c%24 == 1 || c%24 == 7 || c%48 == 10 || c%48 == 22 || c%72 == 15 || c%72 == 33 || c%144 == 6 || c%144 == 42}

Formula

{a(n) : n >= 1} = {A059897(i,j*k) : i in A334832, j in {1,7}, k in {1,6,10,15}}.

A135658 Nonprimes of the form 4x^2-4xy+7y^2.

Original entry on oeis.org

4, 15, 16, 24, 28, 36, 40, 55, 60, 63, 64, 87, 88, 96, 100, 112, 124, 132, 135, 144, 159, 160, 168, 175, 196, 216, 220, 231, 232, 240, 247
Offset: 1

Views

Author

Artur Jasinski, Nov 25 2007

Keywords

Comments

Because 4x^2-4*x*y+7*y^2 = (2*x-y)^2+6*y^2, this is a subsequence of A002481. - R. J. Mathar, Jan 18 2021

Crossrefs

Cf. A107006, A124477, A135657.

Programs

  • Mathematica
    Do[Do[w = 4x^2 - 4x y + 7y^2; If[w > 0, If[PrimeQ[w],[null], AppendTo[a, w]]], {x, 0, 100}], {y, 0, 100}]; Union[a]

A216513 Number of nonnegative integer solutions to the equation x^2 + 6*y^2 = n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0
Offset: 0

Views

Author

V. Raman, Sep 08 2012

Keywords

Crossrefs

Cf. A002481.

Programs

  • Mathematica
    r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 6 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Jun 24 2017 *)

Extensions

Verified by N. J. A. Sloane, Sep 10 2012
a(96) and a(97) corrected by Jean-François Alcover, Jun 24 2017

A247979 Numbers of the form x^2 + 7y^2 with x, y integers, or x^2/4 + 7y^2/4 with x, y odd integers.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 9, 11, 14, 16, 18, 21, 22, 23, 25, 28, 29, 32, 36, 37, 43, 44, 46, 49, 50, 53, 56, 58, 63, 64, 67, 71, 72, 74, 77, 79, 81, 86, 88, 92, 98, 99, 100, 106, 107, 109, 112, 113, 116, 121, 126, 127, 128, 134, 137, 142, 144, 148, 149, 151, 154, 158, 161, 162, 163, 169, 172
Offset: 1

Views

Author

Alonso del Arte, Sep 28 2014

Keywords

Comments

Norms of numbers in O_Q(sqrt(-7)).
A033207 and A045386 are subsets of this sequence. - Colin Barker, Sep 29 2014

Examples

			1/4 + 7/4 = 2, so 2 is in the sequence. (This also means 2 is composite in O_Q(sqrt(-7))).
2^2 + 7 * 0^2 = 4, so 4 is in the sequence.
There is no way to express 5 as x^2 + 7y^2, nor as x^2/4 + 7y^2/4 if x and y are constrained to odd integers, hence 5 is not in the sequence. (This also means 5 is prime in O_Q(sqrt(-7)) and its norm is 25).

Crossrefs

Cf. A002481, A033207, A045386.
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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