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 1-6 of 6 results.

A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

Original entry on oeis.org

34, 63, 89, 99, 139, 164, 174, 193, 204, 245, 314, 399, 424, 454, 464, 489, 504, 524, 549, 714, 1049, 1149, 1174, 1439, 1504, 1539, 1639, 1799, 1814, 1919, 2164, 2239, 2313, 2374, 2414, 2439, 2764, 2789, 3079, 3319, 3414, 3669, 3774, 3814, 4019, 4114
Offset: 1

Views

Author

Giovanni Resta, Jan 15 2006

Keywords

Comments

There are 718 such numbers up to 2*10^9, the last one in this range being 99570649.
It is known that each natural number can be written as the sum of two squares and a triangular number. I believe that the sequence only has 718 terms as found by _Giovanni Resta and listed in the b-file. - Zhi-Wei Sun, Apr 15 2020

Links

Crossrefs

Cf. A022552, A014158, A115160, A115161, A115162, A115163, A262959, A334086, A334113.

Programs

  • Mathematica
    TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-x^4-y^2],Goto[aa]],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]}];tab=Append[tab,n];Label[aa],{n,0,4114}];Print[tab] (From Zhi-Wei Sun)

Extensions

Definition corrected by Giovanni Resta, Aug 17 2011

A115162 Positive numbers that are not the sum of a triangular number, a square and a cube, all of them greater than or equal to 1.

Original entry on oeis.org

1, 2, 4, 7, 9, 14, 21, 28, 35, 70, 126, 152, 161, 481
Offset: 1

Views

Author

Giovanni Resta, Jan 15 2006

Keywords

Comments

Probably finite. No other terms up to 10^9.

Crossrefs

Cf. A022552, A014158, A115159, A115160, A115161, A115163.

A115161 Numbers that are not the sum of a triangular number, a nonnegative cube and a fourth power.

Original entry on oeis.org

13, 35, 40, 41, 50, 51, 59, 76, 77, 112, 124, 139, 149, 150, 157, 165, 168, 175, 176, 178, 182, 183, 189, 193, 197, 205, 208, 215, 224, 229, 241, 243, 249, 273, 288, 305, 306, 314, 329, 332, 366, 373, 383, 397, 412, 413, 418, 420, 429, 438, 453, 455, 457, 461
Offset: 1

Views

Author

Giovanni Resta, Jan 15 2006

Keywords

Comments

There are 15682461 such numbers up to 10^9, the last one in this range being 999999923.

Crossrefs

Cf. A022552, A014158, A115159, A115160, A115162, A115163.

A115163 Numbers that are not the sum of two triangular numbers and a nonnegative cube.

Original entry on oeis.org

41, 104, 188, 923
Offset: 1

Views

Author

Giovanni Resta, Jan 15 2006

Keywords

Comments

Probably finite. No other terms up to 10^9.

Crossrefs

Cf. A022552, A014158, A115159, A115160, A115161, A115162.

A334086 Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

Original entry on oeis.org

19, 82, 109, 118, 145, 149, 271, 280, 296, 349, 350, 371, 392, 454, 491, 643, 670, 692, 754, 755, 923, 937, 986, 989, 1021, 1031, 1150, 1189, 1210, 1294, 1346, 1372, 1610, 1682, 1699, 1720, 1819, 1913, 2050, 2065, 2141, 2227, 2479, 2524, 2753, 2996, 3184, 3451, 3590, 3805, 3968, 4129, 4139, 4199, 4261, 4706
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 14 2020

Keywords

Comments

Conjecture: The sequence has totally 216 terms as listed in the b-file.
As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers.
Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.
It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.
a(217) > 10^9, if it exists. - Giovanni Resta, Apr 14 2020

Examples

			a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power.

Links

Crossrefs

Cf. A000217, A000583, A115160, A290491, A306227, A334113.

Programs

  • Mathematica
    TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-2x^4-y(y+1)/2],Goto[aa]],{x,0,(n/2)^(1/4)},{y,0,(Sqrt[4(n-2x^4)+1]-1)/2}];tab=Append[tab,n];Label[aa],{n,0,5000}];Print[tab]

A334113 Positive numbers not of the form 4*x^4 + y*(y+1)/2 + z*(z+1)/2, where x,y,z are nonnegative integers.

Original entry on oeis.org

23, 44, 54, 63, 117, 138, 149, 162, 180, 188, 243, 251, 261, 270, 287, 294, 398, 401, 458, 512, 611, 657, 684, 693, 734, 797, 842, 863, 914, 932, 936, 945, 987, 1029, 1047, 1098, 1323, 1401, 1449, 1472, 1484, 1494, 1574, 1608, 1637, 1769, 1792, 1799, 1823, 1839, 1902, 1995
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 14 2020

Keywords

Comments

Conjecture: The sequence only has 602 terms as listed in the b-file.
Our computation indicates that after the 602-th term 31737789 there are no other terms below 10^8.
It is known that each n = 0,1,2,... can be written as the sum of an even square and two triangular numbers.

Links

Crossrefs

Cf. A000217, A000583, A115160, A306227, A334086.

Programs

  • Mathematica
    TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-4x^4-y(y+1)/2],Goto[aa]],{x,0,(n/4)^(1/4)},{y,0,(Sqrt[4(n-4x^4)+1]-1)/2}];tab=Append[tab,n];Label[aa],{n,0,2000}];Print[tab]
Showing 1-6 of 6 results.

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