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.

Showing 1-10 of 140 results. Next

A085366 Semiprimes that are the sum of two positive cubes. Common terms of A003325 and A046315.

Original entry on oeis.org

9, 35, 65, 91, 133, 217, 341, 407, 559, 737, 793, 1027, 1241, 1339, 1343, 1843, 1853, 2071, 2413, 2771, 2869, 3197, 3383, 3439, 3473, 4097, 4439, 5129, 5833, 6119, 6641, 7471, 7859, 8027, 8587, 9773, 10261, 10649, 10991, 11377, 12679, 12913, 14023
Offset: 1

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Author

Hugo Pfoertner, Jun 25 2003

Keywords

Comments

Sum of two positive cubes x^3 + y^3 such that both x+y and x^2 - x*y + y^2 are primes.
The only square is 9. Also, all terms have a unique representation as a sum of two distinct positive cubes. - Zak Seidov, Jun 02 2011

Examples

			a(2) = 35 because 3^3 + 2^3 = 5*7.
a(5) = 133 = 5^3 + 2^3 = (5+2)*(5^2 - 5*2 + 2^2) = 7*19.

Links

Crossrefs

Cf. A003325, A046388, A001358, A085367.

A267702 Numbers that are the sum of 3 nonzero squares (A000408) and the sum of 2 positive cubes (A003325).

Original entry on oeis.org

9, 35, 54, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 370, 432, 468, 513, 539, 576, 637, 686, 728, 730, 737, 756, 793, 854, 945, 1001, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1729, 1736, 1755, 1843, 1853
Offset: 1

Views

Author

Altug Alkan, Jan 23 2016

Keywords

Comments

Intersection of A000408 and A003325.
Sequence focuses on the solutions of equation x^3 + y^3 = a^2 + b^2 + c^2 where x, y, a, b, c > 0.

Examples

			9 is a term because 9 = 1^3 + 2^3 = 1^2 + 2^2 + 2^2.
35 is a term because 35 = 2^3 + 3^3 = 1^2 + 3^2 + 5^2.
54 is a term because 54 = 3^3 + 3^3 = 3^2 + 3^2 + 6^2.

Links

Crossrefs

Cf. A000408, A003325.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
S3:= {seq(seq(seq(a^2+b^2+c^2, c = b .. floor(sqrt(N-a^2-b^2))),
b=a .. floor(sqrt((N-a^2)/2))), a = 1 .. floor(sqrt(N/3)))}:
C2:= {seq(seq(a^3+b^3, b = a .. floor((N-a^3)^(1/3))),a = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S3 intersect C2, list)); # Robert Israel, Jan 25 2016
  • PARI
    isA000408(n) = {my(a, b); a=1; while(a^2+1A003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=3, 1e4, if(isA000408(n) && isA003325(n), print1(n, ", ")));

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

A282872 Numbers in A003325 whose 4th power is the sum of two positive cubes in a nontrivial way.

Original entry on oeis.org

2457, 4914, 4977, 8001, 8216, 10773, 15561, 16263, 19656, 39816, 64008, 66339, 80236, 86184, 124336, 124488, 127062, 130104, 132678, 132867, 157248, 166887, 201717, 221832, 238329, 252035, 290871, 307125, 318528, 338821, 358036, 406952, 411021, 420147, 421876
Offset: 1

Views

Author

Chai Wah Wu, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A001235, A003325, A051387, A282873.

Formula

A003325 INTERSECT A051387.

A024667 a(n) = position of 2*n^3 in A003325.

Original entry on oeis.org

1, 3, 6, 11, 18, 25, 33, 44, 57, 68, 81, 99, 116, 134, 152, 177, 200, 223, 246, 276, 304, 331, 360, 397, 433, 465, 501, 541, 579, 617, 662, 707, 749, 793, 845, 895, 944, 995, 1051, 1105, 1161, 1214, 1279, 1337, 1397, 1456, 1528, 1591, 1657, 1722, 1799, 1870
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A003325.

Programs

  • Maple
    M:= 200: # to get a(1) .. a(M)
N:= 2*N^3:
A:=sort(convert({seq(seq(x^3 + y^3, y = 1 .. floor((N-x^3)^(1/3))),x=1..floor(N^(1/3)))},list)):
filter:= proc(n) local F;
if n::odd then return false fi;
  F:= ifactors(n/2)[2][..,2] mod 3;
  andmap(`=`,F,0)
end proc:
select(t -> filter(A[t]), [$1..nops(A)]); # Robert Israel, Sep 19 2024

Formula

It appears that a(n) ~ 2^(2/3) * Pi^2 * n^2/(9 * Gamma(2/3)^3). - Robert Israel, Sep 19 2024

A145732 Terms in A003325 which are sum of two subsequent terms in A003325.

Original entry on oeis.org

217, 341, 1458, 2457, 3059, 12005, 27216, 27683, 39520, 41965, 53128, 72296, 115505, 250559, 251378, 251775, 344728, 425024, 476747, 520000, 578368, 584136, 827099, 843661, 1033676, 1061333, 1185499, 1222039, 1228123, 1299942, 1395000
Offset: 1

Views

Author

Zak Seidov, Oct 17 2008

Keywords

Examples

			a(1)=217=91+126=A003325(15)=A003325(9)+A003325(10), a(2)=341=152 + 189=A003325(20)=A003325(13)+A003325(14), a(3)=1458=728 + 730=A003325(15)=A003325(34)+A003325(35), a(69)=9776312=4887584 + 4888728=A003325(20016)=A003325(12602)+A003325(12603).

Crossrefs

A003325

A145755 Terms in A003325 which are sum of three subsequent terms in A003325.

Original entry on oeis.org

1674, 3059, 5488, 24696, 29744, 50661, 67375, 69095, 109655, 148608, 164502, 247589, 248976, 407511, 421876, 421911, 684216, 762048, 877058, 884763, 942920, 1265544, 1725542, 1817179, 1975545, 2240000, 3133809, 3819905, 4120389
Offset: 1

Views

Author

Zak Seidov, Oct 17 2008

Keywords

Examples

			a(1)=1674=539+ 559+ 576, or A003325(60)=A003325(29)+A003325(30)+A003325(31); a(2)=3059=1008+ 1024+ 1024, or A003325(91)=A003325(43)+A003325(44)+A003325(45); a(36)=7968512=2656064+ 2656151+ 2656297, or A003325(17465)=A003325(8385)+A003325(8386)+A003325(8387).

Links

Crossrefs

Cf. A145732 Terms in A003325 which are sum of two subsequent terms in A003325, A003325 Numbers that are the sum of 2 positive cubes.

A197719 Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.

Original entry on oeis.org

61, 110, 248, 328, 445, 499, 510, 561, 697, 708, 1001, 1004, 1145, 1226, 1309, 1342, 1470, 1563, 1565, 1785, 2012, 2042, 2065, 2259, 2372, 2515, 2540, 2795, 2800, 2806, 2840, 2958, 3076, 3390, 3448, 3779, 3896, 4022, 4031, 4135, 4235, 4320, 4345, 4396, 4412
Offset: 1

Views

Author

Zak Seidov, Oct 17 2011

Keywords

Examples

			First taxi-cab number A001235(1)=1729 is A003325(61) hence a(1)=61; 2nd taxi-cab number A001235(2)=4104 is A003325(110) hence a(2)=110.

Links

Crossrefs

Cf. A001235, A003325.

Formula

A001235(n) = A003325(a(n)).

A271717 Integers k such that both k and k^3-1 are the sum of two positive cubes (see A003325).

Original entry on oeis.org

9, 11664, 36864, 38134, 345744, 1750329, 4782969, 20820969, 47775744, 65804544, 95004009, 150994944, 448084224, 733055625, 1093955625, 1416167424, 2197265625, 4318066944, 5194805625, 6198727824, 7169347584, 10771948944, 13013105625, 19591041024, 32427005625
Offset: 1

Views

Author

Altug Alkan, Apr 12 2016

Keywords

Comments

Values of a^3 + b^3 such that (a^3 + b^3)^3 - 1 is of the form x^3 + y^3 where a, b, x, y > 0.
38134 = 2*23*829 is the first term that is nonsquare. What are the next square terms of this sequence?
n is a member of A007412 and n^3 is a member of A003072, obviously.

Examples

			9 is a term because 9 = 1^3 + 2^3 and 9^3 - 1 = 6^3 + 8^3.

Links

Crossrefs

Cf. A003325, A050787, A068601.

Programs

  • PARI
    isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
for(n=1, 1e7, if(isA003325(n) && isA003325(n^3-1), print1(n, ", ")));

Extensions

a(8)-a(16) from Chai Wah Wu, Apr 17 2016
a(17)-a(25) from Chai Wah Wu, Jul 21 2025

A024665 Positions of even numbers in A003325.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 16, 18, 19, 21, 23, 25, 26, 28, 31, 33, 34, 35, 37, 39, 43, 44, 46, 47, 49, 51, 54, 56, 57, 58, 60, 62, 64, 67, 68, 69, 71, 73, 74, 76, 80, 81, 83, 85, 87, 89, 90, 94, 95, 97, 99, 101, 102, 104, 105, 110, 112, 114, 116, 118, 119, 120, 122, 124, 127, 128, 131, 134
Offset: 1

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 140 results. Next

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