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.

A053614 Numbers that are not the sum of distinct triangular numbers.

Original entry on oeis.org

2, 5, 8, 12, 23, 33
Offset: 1

Views

Author

Jud McCranie, Mar 19 2000

Keywords

Comments

The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - T. D. Noe, Jul 24 2006, Jan 05 2009

Examples

			a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.

References

  • Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
  • David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".

Links

  • Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Crossrefs

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.

Programs

  • Mathematica
    nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

Formula

Complement of A061208.

Extensions

Entry revised by N. J. A. Sloane, Jul 23 2006

A115160 Numbers that are not the sum of two triangular numbers and a fourth power.

Original entry on oeis.org

33, 63, 75, 125, 365, 489, 492, 684, 693, 723, 954, 1043, 1185, 1505, 1623, 1629, 1736, 1775, 1899, 1904, 1925, 2015, 2051, 2679, 2883, 3534, 3774, 3936, 4332, 4461, 4739, 4923, 5445, 5721, 5847, 6285, 6348, 6474, 6783, 7034, 7478, 8604, 9576, 9686, 9863
Offset: 1

Views

Author

Giovanni Resta, Jan 15 2006

Keywords

Comments

There are 88 such numbers up to 2*10^9, the last one in this range being 1945428.

Links

Crossrefs

Cf. A022552, A014158, A115159, A115161, A115162, A115163.

Programs

  • PARI
    sumset_lim(a,b,lim)=my(v=[],u,t);if(a==b,for(i=1,#a,u=List();for(j=i,#b,t=a[i]+b[j];if(t>lim,break);listput(u,t));v=vecsort(concat(v,Vec(u)),,8)),for(i=1,#a,u=List();for(j=1,#b,t=a[i]+b[j];if(t>lim,break);listput(u,t));v=vecsort(concat(v,Vec(u)),,8)));v
makev(lim)=my(n=floor(sqrt(2*lim)-1/2),v);sumset_lim(v=vector(n,k,k*(k-1)/2),v,lim)
is(n)=for(i=1,#v,if(ispower(n-v[i],4),return(0));if(v[i]>n,return(1)))
v=makev(1e5);
for(n=1,1e5,if(is(n),print1(n", "))) \\ Charles R Greathouse IV, Aug 17 2011

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.
Showing 1-6 of 6 results.

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