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 1-4 of 4 results.

A232101 a(n) = A000217(A232097(n)) / n!.

Original entry on oeis.org

1, 3, 1, 5, 1, 35, 5, 208, 8073, 36120, 1133197, 456667, 13535367, 2592766, 3409144, 24103053951, 1417826703, 486682892638
Offset: 1

Views

Author

Antti Karttunen, Nov 25 2013

Keywords

Programs

Formula

a(n) = A000217(A232097(n)) / A000142(n).

A232096 a(n) = largest m such that m! divides 1+2+...+n; a(n) = A055881(A000217(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 18 2013

Keywords

Crossrefs

A042963 gives the positions of ones and A014601 the positions of larger terms.

Programs

Formula

a(n) = A055881(A000217(n)).
a(n) = A231719(A226061(n+1)). [Not a practical way to compute this sequence, but follows from the definitions]

A059958 Smallest number m such that m*(m+1) has at least n distinct prime factors.

Original entry on oeis.org

1, 2, 5, 14, 65, 209, 714, 7314, 38570, 254540, 728364, 11243154, 58524465, 812646120, 5163068910, 58720148850, 555409903685, 4339149420605, 69322940121435, 490005293940084, 5819629108725509, 76622240600506314
Offset: 1

Views

Author

Labos Elemer, Mar 02 2001

Keywords

Comments

The original definition left unclear whether "at least" or "exactly" n prime factors are required. Now the "at least" variant was chosen, for the other variant ("exactly"), see A069354: At least up to a(18), both criteria yield the same number, and therefore a(n) = A069354(n) - 1, since m and m+1 are always coprime. - M. F. Hasler, Jan 15 2014
10^13 < a(19) <= 69322940121435. - Giovanni Resta, Mar 24 2020
Terms a(1)-a(10) appear in Erdős and Nicolas (1978-1979). - Amiram Eldar, Jun 24 2023

Examples

			For n = 9, a(9)*(a(9) + 1) = 38570*38571 = (2*5*7*19*29)*(3*13*23*43) with 9 distinct prime factors.
		

Crossrefs

Programs

  • Mathematica
    With[{s = Map[PrimeNu[Times @@ #] &, Partition[Range[10^6], 2, 1]]}, Array[FirstPosition[s, n_/; n>=#][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 02 2017 *)
  • PARI
    a(n) = my(m=1); while(omega(m*(m+1)) < n, m++); m; \\ Michel Marcus, Jul 09 2018

Formula

a(n) = Min_{ m | A001221(m*(m+1)) >= n }.
a(n) <= A002110(n) - 1 because A001221((q-1)*q) >= n+1 for q = A002110(n).
Conjecture: a(n) = A069354(n) - 1. - Robert G. Wilson v, Feb 18 2014

Extensions

More terms from William Rex Marshall, Mar 18 2001
Offset corrected and a(15)-a(16) from Donovan Johnson, Jan 31 2009
a(17) from Donovan Johnson, Sep 15 2010
a(18) from Don Reble, Jan 15 2014
Edited by M. F. Hasler, Jan 15 2014
a(19)-a(20) from Michael S. Branicky, Feb 08 2023
a(21) from Michael S. Branicky, Feb 10 2023
a(22) from Michael S. Branicky, Feb 23 2023

A342930 Least positive number k such that n^n divides k*(k+1)/2.

Original entry on oeis.org

1, 7, 26, 511, 3124, 16767, 823542, 33554431, 387420488, 1787109375, 285311670610, 6737830608896, 302875106592252, 10190301669556224, 12913848876953124, 36893488147419103231, 827240261886336764176, 22831345258932427292672, 1978419655660313589123978, 35357007743740081787109375
Offset: 1

Views

Author

Seiichi Manyama, Mar 29 2021

Keywords

Examples

			  n |     a(n) |         T(a(n)) = n^n * A342931(n).
----+----------+------------------------------------
  1 |        1 |               1 = 1^1 * 1.
  2 |        7 |              28 = 2^2 * 7.
  3 |       26 |             351 = 3^3 * 13.
  4 |      511 |          130816 = 4^4 * 511.
  5 |     3124 |         4881250 = 5^5 * 1562.
  6 |    16767 |       140574528 = 6^6 * 3013.
  7 |   823542 |    339111124653 = 7^7 * 411771.
  8 | 33554431 | 562949936644096 = 8^8 * 33554431.
		

Crossrefs

Programs

  • PARI
    a(n) = my(k=1, m=n^n); while(k*(k+1)/2%m!=0, k++); k;
    
  • PARI
    a(n) = { my(p = 2*n^n, f = factor(p), res = oo); for(i = 2^(#f~-1), 2^#f~-1, b = binary(i); pr = prod(j = 1, #f~, f[j,1]^(b[j]*f[j, 2])); ipr = p/pr; for(j = -1, 0, c = lift(chinese(Mod(-1-j, ipr), Mod(j, pr))); if(c > 0, res = min(res, c)))); res } \\ David A. Corneth, Mar 29 2021

Formula

a(p) = p^p - 1 for odd prime p. - David A. Corneth, Mar 29 2021

Extensions

More terms from David A. Corneth, Mar 29 2021
Showing 1-4 of 4 results.