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.

User: Duc Van Khanh Tran

Duc Van Khanh Tran's wiki page.

Duc Van Khanh Tran has authored 26 sequences. Here are the ten most recent ones:

A375583 a(n) is the number of ways n can be written as a sum of a practical number and two 11-gonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 3, 2, 1, 2, 2, 3, 2, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 4, 3, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 6, 4, 2, 1, 2, 3, 3, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 4, 5, 5, 5, 2, 4, 4, 6, 4, 3, 1, 4, 5, 4, 4, 2, 3, 4, 4, 6, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 6, 7, 4, 4, 1, 4, 4, 6, 6
Offset: 1

Views

Author

Duc Van Khanh Tran, Aug 19 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n.
Conjecture (checked up to 10^8): a(n) = 0 if and only if n = 11.

Links

Crossrefs

Cf. A005153, A051682.

A375582 a(n) is the number of ways n can be written as a sum of a practical number and two decagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 1, 2, 1, 3, 1, 4, 4, 4, 3, 4, 2, 3, 2, 3, 1, 4, 3, 4, 3, 3, 4, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 5, 6, 4, 5, 3, 5, 3, 3, 2, 5, 4, 5, 4, 5, 3, 5, 2, 3, 3, 5, 2, 4, 3, 3, 2, 6, 4, 6, 4, 5, 4, 5, 4, 5, 4, 6, 6, 6, 4, 6
Offset: 1

Views

Author

Duc Van Khanh Tran, Aug 19 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A001107, A005153.

A374415 a(n) is the number of ways n can be written as a sum of a practical number and two octagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 3, 3, 1, 4, 2, 3, 2, 4, 3, 3, 2, 4, 3, 4, 2, 4, 4, 3, 1, 3, 4, 3, 2, 4, 5, 5, 3, 5, 4, 4, 2, 5, 6, 5, 2, 4, 3, 4, 1, 6, 5, 5, 2, 4, 4, 5, 3, 6, 5, 5, 4, 5, 6, 5, 3, 7, 5, 5, 3, 4, 6, 5, 3, 6
Offset: 1

Views

Author

Duc Van Khanh Tran, Jul 08 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A000567, A005153.

A374416 a(n) is the number of ways n can be written as a sum of a practical number and two nonagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 2, 3, 0, 2, 3, 5, 2, 3, 3, 5, 2, 3, 3, 4, 1, 3, 4, 3, 2, 3, 4, 5, 2, 2, 3, 2, 1, 5, 6, 5, 3, 4, 3, 5, 3, 5, 5, 4, 2, 4, 4, 2, 4, 5, 5, 7, 4, 2, 2, 2, 3, 5, 5, 5, 4, 5, 3, 7, 5, 5
Offset: 1

Views

Author

Duc Van Khanh Tran, Jul 08 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n.
Conjecture (checked up to 10^8): a(n) = 0 if and only if n = 23.

Links

Crossrefs

Cf. A001106, A005153.

A374344 a(n) is the number of ways n can be written as a sum of a practical number and two heptagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 3, 3, 2, 1, 2, 2, 2, 2, 3, 1, 3, 3, 6, 2, 3, 2, 3, 3, 5, 3, 2, 2, 4, 4, 4, 2, 3, 3, 5, 6, 6, 3, 4, 3, 7, 6, 4, 2, 3, 4, 4, 5, 5, 2, 2, 4, 6, 5, 7, 5, 5, 4, 5, 6, 5, 4, 7, 4, 6, 6, 7, 2, 5, 4, 7, 6, 9, 4, 5, 3, 5, 6, 8
Offset: 1

Views

Author

Duc Van Khanh Tran, Jul 08 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A000566, A005153.

A373688 a(n) is the number of ways n can be written as a sum of a practical number and two hexagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 2, 4, 2, 2, 1, 2, 3, 3, 1, 3, 3, 5, 3, 4, 2, 4, 3, 4, 3, 2, 3, 3, 3, 6, 5, 6, 4, 5, 4, 7, 4, 4, 3, 5, 3, 6, 3, 5, 4, 6, 6, 7, 6, 4, 4, 6, 4, 6, 5, 4, 7, 7, 3, 6, 7, 7, 6, 7, 3, 8, 5, 7, 6, 7, 6, 8, 6, 7, 7, 8, 4, 8, 7, 6
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 13 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A000384, A005153.

A373687 a(n) is the number of ways n can be written as a sum of a practical number and two pentagonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 3, 3, 2, 2, 2, 3, 3, 5, 1, 3, 3, 6, 3, 3, 4, 3, 3, 5, 6, 5, 1, 6, 6, 7, 4, 4, 5, 4, 5, 6, 6, 6, 3, 7, 7, 10, 5, 5, 5, 6, 5, 8, 4, 5, 4, 9, 8, 8, 7, 7, 5, 8, 8, 9, 4, 6, 6, 11, 8, 8, 7, 7, 6, 6, 9, 12, 5, 8, 8, 12, 10, 8, 9, 7
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 13 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A000326, A005153.

A373686 a(n) is the number of ways n can be written as a sum of a practical number and two squares.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 3, 4, 3, 4, 3, 4, 2, 4, 5, 5, 4, 6, 6, 6, 2, 6, 5, 7, 4, 7, 7, 6, 5, 7, 9, 7, 4, 8, 9, 10, 2, 9, 10, 9, 6, 9, 9, 8, 5, 8, 10, 9, 5, 10, 11, 9, 7, 12, 11, 11, 6, 9, 11, 10, 3, 11, 14, 13, 9, 12, 13, 11, 7, 10, 15, 14, 4, 13, 13, 8, 8, 15
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 13 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n and conjectured that a(n) > 0 for all n > 0. The conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A000290, A005153.

A373664 Positive integers that cannot be written as a sum of a practical number and a 12-gonal number.

Original entry on oeis.org

10, 11, 15, 22, 23, 26, 27, 38, 46, 47, 50, 58, 59, 62, 71, 74, 77, 83, 86, 95, 98, 103, 110, 114, 115, 119, 122, 131, 134, 139, 143, 146, 149, 155, 166, 167, 170, 175, 178, 179, 182, 187, 191, 194, 202, 203, 206, 207, 215, 227, 230, 238, 239, 242, 248, 250
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 12 2024

Keywords

Comments

Somu and Tran (2024) proved that there are infinitely many such integers. More generally, infinitely many positive integers cannot be written as a sum of a practical number and an s-gonal number if s is a multiple of 12.

Links

Crossrefs

Cf. A005153, A051624.

A373647 Positive integers that cannot be written as a sum of a practical number and a square.

Original entry on oeis.org

14, 23, 35, 47, 59, 62, 71, 74, 86, 95, 98, 107, 110, 119, 131, 134, 138, 143, 155, 158, 167, 179, 182, 183, 191, 194, 195, 203, 206, 215, 218, 230, 239, 242, 251, 254, 263, 266, 275, 278, 282, 287, 299, 302, 311, 314, 318, 323, 327, 335, 338, 347, 350, 359
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 12 2024

Keywords

Comments

Somu et al. (2023) proved that there are infinitely many such integers.
Somu and Tran (2024) proved a more general result, which states that infinitely many positive integers cannot be written as a sum of a practical number and an s-gonal number if s is congruent to 4 modulo 12.

Links

Crossrefs

Cf. A000290, A005153.

Programs

  • Mathematica
    Lim=360;sqlim=Sqrt[Lim];
PracticalQ[nn_] := Module[{f, p, e, prod=1, ok=True}, If[nn<1 || (nn>1 && OddQ[n]), False, If[nn==1, True, f=FactorInteger[nn]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1,prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i,Length[p]}]; ok]]];prac= Select[Range[Lim],PracticalQ] ;
seq={};Do[sq=i^2;sqi=prac+sq;AppendTo[seq,sqi],{i,0,sqlim}] (* sums of squares and practical numbers *);
Complement[Range[Lim],Union[Flatten[seq]]] (* James C. McMahon, Jun 15 2024 *)

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