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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A273498 Numbers that are, at the same time, the sum of: two positive squares, a positive square and a positive cube, and two positive cubes. In other words, intersection of A000404, A003325 and A055394.

Original entry on oeis.org

2, 65, 72, 128, 468, 730, 793, 1241, 1332, 1458, 2000, 2745, 3528, 4097, 4160, 4608, 4825, 5096, 5840, 5913, 6344, 8125, 8192, 9000, 9325, 9928, 12168, 13357, 13498, 14824, 15626, 15633, 15689, 16354, 17640, 18369, 18737, 19721, 19773, 21953, 22681, 27792, 29449
Offset: 1

Views

Author

Altug Alkan, May 23 2016

Keywords

Comments

Numbers n such that n = x^a + y^b where x,y > 0, is soluble for all 1 < a <= b < 4.
Perfect power terms are 128, 8192, 97344, 140625, 524288, 1500625, ...

Examples

			793 is a term because 793 = 3^2 + 28^2 = 8^2 + 9^3 = 4^3 + 9^3.

Links

Crossrefs

Cf. A000404, A003325, A055394.

Programs

  • PARI
    isA003325(n)=for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))
isA000404(n) = for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))
isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
lista(nn) = for(n=1, nn, if(isA003325(n) && isA000404(n) && isA055394(n), print1(n, ", ")));
  • PARI
    isA000404(n)=my(f=factor(n)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); n>1 && (vecmin(f[,1]%4)==1 || (f[1, 1]==2 && f[1,2]%2))
isA055394(n) = for(k=1, sqrtnint(n-1,3), if(issquare(n-k^3), return(1))); 0
list(lim)=my(v=List(),n3,t); lim\=1; for(n=1,sqrtnint(lim-1,3), n3=n^3; for(m=1,sqrtnint(lim-n3,3), t=n3+m^3; if(isA000404(t) && isA055394(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, May 31 2016

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","))

A024515 Positions of even numbers in A000404 (sums of 2 nonzero squares).

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 12, 13, 15, 18, 19, 21, 24, 25, 27, 28, 29, 32, 34, 35, 37, 38, 41, 43, 45, 46, 47, 50, 51, 55, 56, 57, 59, 61, 62, 66, 68, 69, 71, 72, 73, 76, 78, 80, 82, 83, 85, 87, 91, 92, 95, 97, 98, 100, 101, 103, 105, 107, 109, 112, 113, 114, 117, 118, 119, 122, 125, 126
Offset: 1

Views

Author

Clark Kimberling

Keywords

A024516 Positions of odd numbers in A000404 (sums of 2 nonzero squares).

Original entry on oeis.org

2, 5, 6, 9, 11, 14, 16, 17, 20, 22, 23, 26, 30, 31, 33, 36, 39, 40, 42, 44, 48, 49, 52, 53, 54, 58, 60, 63, 64, 65, 67, 70, 74, 75, 77, 79, 81, 84, 86, 88, 89, 90, 93, 94, 96, 99, 102, 104, 106, 108, 110, 111, 115, 116, 120, 121, 123, 124, 127, 130, 132, 134, 135, 138, 140, 141
Offset: 1

Views

Author

Clark Kimberling

Keywords

A024518 a(n) = position of 1 + n^2 in A000404 (sums of 2 nonzero squares).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 23, 29, 36, 43, 49, 59, 67, 76, 86, 97, 108, 119, 132, 143, 157, 169, 182, 198, 215, 230, 245, 263, 280, 297, 316, 334, 354, 375, 394, 415, 436, 457, 480, 503, 527, 549, 574, 598, 624, 650, 674, 703, 730, 758, 787, 815, 844, 871, 902, 934, 965, 994
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Jul 13 2019

A024519 Position of n^2 + (n+1)^2 in A000404 (sums of 2 nonzero squares).

Original entry on oeis.org

2, 5, 9, 16, 22, 30, 40, 49, 63, 74, 89, 104, 120, 138, 155, 174, 195, 217, 238, 262, 287, 312, 339, 366, 396, 425, 453, 489, 520, 553, 586, 623, 659, 697, 735, 776, 816, 857, 898, 943, 985, 1033, 1078, 1122, 1172, 1220, 1270, 1321, 1374, 1427, 1480, 1536
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from James Sellers, May 03 2000

A024520 Positions of primes in A000404 (sums of 2 nonzero squares).

Original entry on oeis.org

1, 2, 5, 6, 11, 14, 16, 20, 22, 26, 31, 33, 36, 39, 40, 48, 52, 54, 60, 63, 65, 67, 77, 79, 81, 86, 90, 93, 94, 99, 104, 106, 111, 115, 116, 123, 127, 130, 132, 135, 138, 141, 145, 148, 150, 163, 167, 172, 178, 181, 182, 189, 191, 195, 196
Offset: 1

Views

Author

Clark Kimberling

Keywords

A084889 Duplicate of A000404.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68
Offset: 1

Views

Author

Keywords

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","))
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