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.

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A155714 Least number expressible as a^2 + p b^2 with positive integers a,b, for each prime p <= prime(n) = A000040(n).

Original entry on oeis.org

3, 12, 36, 144, 144, 4356, 4356, 4356, 7056, 17424, 176400, 2547216, 2547216, 6290064, 6780816, 6780816, 6780816, 6780816, 93315600, 93315600, 271986064, 271986064, 271986064, 271986064, 271986064, 308213136, 308213136, 308213136
Offset: 1

Views

Author

M. F. Hasler, Feb 10 2009

Keywords

Comments

a(n) > 10^9 for n >= 33. [From Donovan Johnson, Sep 29 2009]

Links

Crossrefs

Cf. A155715, A028372, A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

Programs

  • PARI
    A155714(k,n=1) = { local(p); until( !n++, p=prime(k); until( !p=precprime(p-1), for( b=1, sqrtint((n-1)\p), issquare(n-p*b^2) & next(2)); next(2)); break);n}
t=1; for(k=1,30, print1(t=A155714(k,t),","))

Extensions

a(12)-a(32) and b-file from Donovan Johnson, Sep 29 2009

A294774 a(n) = 2*n^2 + 2*n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905
Offset: 0

Views

Author

Bruno Berselli, Nov 08 2017

Keywords

Comments

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:
2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Links

Crossrefs

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

Programs

  • Maple
    seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017
  • Mathematica
    Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)
  • PARI
    Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

Formula

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A155574 Intersection of A154777 and A092572: N = a^2 + 2b^2 = c^2 + 3d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

12, 19, 36, 43, 48, 57, 67, 73, 76, 97, 108, 129, 139, 144, 147, 163, 171, 172, 192, 193, 201, 211, 219, 228, 241, 268, 283, 291, 292, 300, 304, 307, 313, 324, 331, 337, 361, 379, 387, 388, 409, 417, 432, 433, 441, 457, 475, 484, 489, 499, 507, 513, 516, 523
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155564 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

Programs

  • PARI
    isA155574(n,/* optional 2nd arg allows us to get other sequences */c=[3,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155574(n) & print1(n","))

A155575 Intersection of A000404 and A154778: N = a^2 + b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

29, 41, 45, 61, 89, 101, 109, 116, 145, 149, 164, 180, 181, 205, 225, 229, 241, 244, 245, 261, 269, 281, 305, 349, 356, 369, 389, 401, 404, 405, 409, 421, 436, 445, 449, 461, 464, 505, 509, 521, 541, 545, 549, 569, 580, 596, 601, 641, 656, 661, 701, 709, 720
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155565 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

Programs

  • PARI
    isA155575(n,/* optional 2nd arg allows us to get other sequences */c=[5,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155575(n) & print1(n","))

A155576 Intersection of A000404 and A155716: N = a^2 + b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

10, 25, 40, 58, 73, 90, 97, 100, 106, 145, 160, 193, 202, 225, 232, 241, 250, 265, 292, 298, 313, 337, 346, 360, 388, 394, 400, 409, 424, 433, 457, 490, 505, 522, 538, 577, 580, 586, 601, 625, 634, 640, 657, 673, 730, 745, 769, 772, 778, 808, 810, 841, 865
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155566 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

Programs

  • PARI
    isA155576(n,/* optional 2nd arg allows us to get other sequences */c=[6,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155576(n) & print1(n","))

A155577 Intersection of A154777 and A154778: N = a^2 + 2b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

6, 9, 24, 36, 41, 54, 81, 86, 89, 96, 129, 134, 144, 150, 164, 166, 201, 214, 216, 225, 241, 246, 249, 281, 294, 321, 324, 326, 344, 356, 369, 384, 401, 409, 441, 449, 454, 486, 489, 516, 521, 534, 536, 566, 569, 576, 600, 601, 614, 641, 656, 664, 681, 694
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155567 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

Programs

  • PARI
    isA155577(n,/* optional 2nd arg allows us to get other sequences */c=[5,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155577(n) & print1(n","))

A155709 Intersection of A154777 and A155716: N = a^2 + 2b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

22, 33, 73, 88, 97, 118, 121, 132, 150, 166, 177, 193, 198, 214, 225, 241, 249, 262, 292, 294, 297, 313, 321, 337, 352, 358, 388, 393, 409, 433, 438, 441, 454, 457, 472, 484, 502, 528, 537, 550, 577, 582, 600, 601, 649, 657, 664, 673, 681, 694, 708, 726, 753
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155569 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

Programs

  • PARI
    isA155709(n,/* optional 2nd arg allows us to get other sequences */c=[6,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155709(n) & print1(n","))

A155710 Intersection of A092572 and A154778: N = a^2 + 3b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

21, 36, 49, 61, 84, 109, 129, 144, 181, 189, 196, 201, 229, 241, 244, 301, 309, 324, 336, 349, 381, 409, 421, 436, 441, 469, 489, 516, 525, 541, 549, 576, 601, 661, 669, 709, 721, 724, 756, 769, 784, 804, 829, 849, 889, 900, 916, 921, 964, 976, 981, 1009, 1021
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Comments

Subsequence of A155570 (where a,b,c,d may be zero).

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

Programs

  • PARI
    isA155710(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1111, isA155710(n) & print1(n","))

A155571 Intersection of A000404, A092572 and A154778: N = a^2 + b^2 = c^2 + 3d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

61, 109, 181, 229, 241, 244, 349, 409, 421, 436, 541, 549, 601, 661, 709, 724, 769, 829, 900, 916, 964, 976, 981, 1009, 1021, 1069, 1129, 1201, 1225, 1249, 1321, 1381, 1396, 1429, 1489, 1521, 1525, 1549, 1609, 1621, 1629, 1636, 1669, 1684, 1741, 1744, 1789
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

Programs

  • PARI
    isA155571(n,/* optional 2nd arg allows us to get other sequences */c=[5,3,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155571(n) & print1(n","))

A155572 Intersection of A000404, A154777 and A154778: N = a^2 + b^2 = c^2 + 2d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

41, 89, 164, 225, 241, 281, 356, 369, 401, 409, 449, 521, 569, 601, 641, 656, 761, 769, 801, 809, 881, 900, 929, 964, 1009, 1025, 1049, 1124, 1129, 1201, 1249, 1289, 1321, 1361, 1409, 1424, 1476, 1481, 1489, 1521, 1601, 1604, 1609, 1636, 1681, 1721, 1796
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2009

Keywords

Crossrefs

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

Programs

  • PARI
    isA155572(n,/* optional 2nd arg allows us to get other sequences */c=[5,2,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155572(n) & print1(n","))
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