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 1-3 of 3 results.

A062739 Odd powerful numbers.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 675, 729, 841, 961, 1089, 1125, 1225, 1323, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3087, 3125, 3249, 3267, 3375, 3481, 3721, 3969, 4225
Offset: 1

Views

Author

Labos Elemer, Jul 12 2001

Keywords

Comments

Smallest term of this sequence not also in A075109 is 675, followed by 1125. - Alonso del Arte, Nov 22 2011

Examples

			Consecutive-odd examples from Sentance: {25,27},{70225,70227},{189750625,189750627}

References

  • R. K. Guy, Unsolved Problems in Number Theory, B16

Links

Crossrefs

Cf. A001694, A060355, A060859, A060860, A082695.
Cf. A076445 (consecutive odd powerful numbers).

Programs

  • Mathematica
    Powerful[n_Integer] := (n ==1) || Min[Transpose[FactorInteger[n]][[2]]]>=2; Select[Range[5000],OddQ[ # ]&&Powerful[ # ]&] (* T. D. Noe, May 04 2006 *)
Join[{1},Select[Range[3,4301,2],Min[FactorInteger[#][[All,2]]]>1&]] (* Harvey P. Dale, Jan 08 2021 *)

Formula

It is not true that a(n) = A001694(2n-1).
Sum_{n>=1} 1/a(n) = (2/3) * Sum_{n>=1} 1/A001694(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) = (2/3) * A082695 = 1.2957309... - Amiram Eldar, Jun 23 2020

Extensions

Checked by T. D. Noe, May 04 2006

A060859 Powerful numbers of the form k^2 - 1.

Original entry on oeis.org

8, 288, 675, 9800, 235224, 332928, 1825200, 11309768, 384199200, 592192224, 4931691075, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 13325427460800, 15061377048200, 511643454094368
Offset: 1

Views

Author

Labos Elemer, May 04 2001

Keywords

Comments

If k^2-1 is a term, then k-1 is a term of A335851. - Amiram Eldar, Feb 23 2024

Examples

			From _Jon E. Schoenfield_, Sep 06 2017: (Start)
n     k        a(n) =  k^2 - 1          a(n) + 1 = k^2
=   ===   =========================   ==================
1     3        8 = 2^3                  3^2 = 3^2
2    17      288 = 2^5 * 3^2           17^2 = 17^2
3    26      675 = 5^2 * 3^3           26^2 = 2^2 * 13^2
4    99     9800 = 2^3 * 5^2 * 7^2     99^2 = 3^4 * 11^2
5   485   235224 = 2^3 * 3^5 * 11^2   485^2 = 5^2 * 97^2
6   577   332928 = 2^7 * 3^2 * 17^2   577^2 = 577^2
(End)

Links

Crossrefs

Proper subset of A060355.
Cf. A001694, A060860, A335851.

Programs

  • Mathematica
    Select[Range[10^6]^2 - 1, Min[FactorInteger[#][[All, -1]]] > 1 &] (* Michael De Vlieger, Sep 05 2017 *)
seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q, i}, q = Union[p, 2*Select[p, # <= max && OddQ[#] &]]; i = Position[Differences[q], 2] // Flatten; q[[i]]*(q[[i]] + 2)]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)
  • PARI
    isok(n) = issquare(n+1) && ispowerful(n); \\ Michel Marcus, Sep 05 2017

Formula

a(n) = k^2 - 1 and a(n) + 1 = k^2 are consecutive powerful numbers.
a(n) = A060860(n)^2 - 1. - Amiram Eldar, Feb 23 2024

Extensions

Corrected and extended by Jud McCranie, Jul 08 2001
Offset corrected by Donovan Johnson, Nov 15 2011
Name simplified by Jon E. Schoenfield, Nov 30 2023

A365983 Even numbers k such that k^2 - 1 is a powerful number.

Original entry on oeis.org

26, 70226, 130576328, 189750626, 512706121226, 13837575261124, 99612037019890, 1385331749802026
Offset: 1

Views

Author

Jud McCranie, Sep 24 2023

Keywords

Comments

This sequence is a subsequence of A060860 (the even terms) and a supersequence of A094835. All the terms of A094835 are in this sequence, but 130576328 is not in A094835. A094835 also shows that this sequence is infinite.
Terms A076445(n)+1 are terms of this sequence because A076445(n) and A076445(n)+2 are powerful and (A076445(n)+1)^2-1 = A076445(n) * (A076445(n)+2), which is also powerful.
a(n) - 1 is an odd powerful number (A062739). - Amiram Eldar, Feb 23 2024

Examples

			26^2 - 1 = 675 = 3^3 * 5^2 is powerful.
130576328^2 - 1 = 17050177433963583 = 3^2 * 7^3 * 13^2 * 293^2 * 617^2, whose exponents are all greater than 1, so it is powerful.

References

  • Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, entries 70226 and 485.

Links

Crossrefs

Cf. A000466, A001694, A062739, A094835, A060860, A076445.

Programs

  • Mathematica
    seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3), 2}, {i, 1, Sqrt[max/j^3], 2}]]], i}, i = Position[Differences[p], 2] // Flatten; Sqrt[p[[i]]*(p[[i]] + 2) + 1]]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)
  • PARI
    isok(k) = !(k%2) && ispowerful(k^2-1); \\ Michel Marcus, Sep 25 2023

Extensions

a(5)-a(8) from Amiram Eldar, Feb 23 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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