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.

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

Original entry on oeis.org

1, 2, 33, 19940, 204323, 380480345, 440738966073
Offset: 1

Views

Author

Dean Hickerson, Oct 01 1998

Keywords

Comments

a(8) > 10^13. - Donovan Johnson, Oct 20 2009
Don Reble has shown that a(8) < 1.9*10^42, cf. link.
From David Wasserman, Jan 05 2019: (Start)
a(8) <= 108111092880293127811946663766147737122,
a(9) <= 6850672946809600696044301071559918192380244,
a(10) <= 96037988156124494415303285590850571857698741869620,
a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)

Examples

			a(4) = 19940 because 19940, ..., 19943 all have the form p^2 q r.

Links

Crossrefs

Cf. A034174.
Cf. A006558, A083790.
Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - M. F. Hasler, Oct 28 2012

Programs

  • PARI
    A034173(n)={my(f);for(k=1,oo,f=0;for(i=1,n, f==(f=vecsort(factor(k+n-i)[,2])) || i==1 || [k+=n-i; next(2)]);return(k))} \\ M. F. Hasler, Oct 23 2012

Formula

a(n) = A034174(n) - n + 1. - Max Alekseyev, Nov 10 2009
a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - M. F. Hasler, Oct 28 2012

Extensions

a(7) from Donovan Johnson, Oct 20 2009
Don Reble link repaired by N. J. A. Sloane, Oct 24 2024

A113458 Least k such that k, k+n and k+2n have the same prime signature.

Original entry on oeis.org

33, 3, 155, 3, 77, 5, 51, 3, 77, 3, 35, 5, 50, 3, 187, 6, 21, 5, 39, 3, 145, 33, 39, 5, 69, 39, 91, 3, 33, 7, 15, 12, 221, 3, 28, 7, 21, 3, 55, 3, 33, 5, 91, 66, 209, 69, 35, 5, 50, 3, 115, 39, 141, 5, 51, 6, 145, 85, 15, 7, 21, 93, 95, 3, 57, 5, 51, 3, 65, 15, 35, 7, 69, 55, 287, 6
Offset: 1

Views

Author

David Wasserman, Jan 08 2006

Keywords

Comments

Third row of A113456.

Examples

			a(4) = 3 because 3, 7 and 11 have the same prime signature.

Links

Crossrefs

Cf. A052214, A113456, A113467.

Programs

  • Maple
    s:= n-> sort(map(i-> i[2], ifactors(n)[2])):
a:= proc(n) option remember; local k; for k
      while s(k)<>s(k+n) or s(k)<>s(k+2*n) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018
  • Mathematica
    s[n_] := FactorInteger[n][[All, 2]] // Sort;
a[n_] := Module[{k}, For[k = 2, True, k++, If[s[k] == s[k+n] == s[k+2n], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Nov 05 2020 *)

A113465 Rectangular array read by antidiagonals: a(n, d) is the smallest number that starts an arithmetic progression with common difference d of n numbers with the same number of divisors.

Original entry on oeis.org

1, 2, 1, 33, 3, 1, 242, 3, 2, 1, 11605, 213, 119, 3, 1, 28374, 213, 3445, 3, 2, 1, 171893, 1383, 15026, 111, 77, 5, 1, 1043710445721, 3091, 74783, 201, 8718, 5, 8, 1
Offset: 1

Views

Author

David Wasserman, Jan 08 2006

Keywords

Comments

First two columns are A006558 and A113466. First four rows are A000012, A065559, A113467 and A113468. a(9, 1) is unknown; the rest of the 9th antidiagonal is 8129,88015,201,8718,5,8,3,1.

Examples

			a(4, 3) = 3445 because 3445, 3448, 3451 and 3454 each have 8 divisors.

Crossrefs

Cf. A006558, A065559, A113456, A113466, A113467, A113468.

A113457 Least number that begins an n-term arithmetic progression with common difference 2 in which all terms have the same prime signature.

Original entry on oeis.org

1, 3, 3, 213, 213, 1383, 3091, 8129
Offset: 1

Views

Author

David Wasserman, Jan 08 2006

Keywords

Comments

Second column of A113456. a(9) >= 37887000000.

Crossrefs

Cf. A092209 A113456 A113466.
Showing 1-4 of 4 results.

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

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