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.

A243289 n minus the index of the greatest prime dividing n-th squarefree number: a(n) = n - A243290(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 4, 3, 3, 6, 8, 5, 5, 10, 10, 7, 11, 8, 16, 9, 16, 15, 19, 12, 17, 20, 14, 24, 15, 21, 16, 25, 17, 29, 27, 26, 20, 20, 28, 34, 36, 23, 34, 40, 25, 25, 35, 43, 43, 28, 38, 29, 46, 40, 45, 32, 51, 47, 44, 52, 36, 36, 56, 37, 61, 50, 39, 39, 64, 58
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2014

Keywords

Comments

If A005117(n) <= 2n, or equally, if A243351 is always positive, then this sequence is certainly positive as well.

Links

Crossrefs

Cf. A005117, A006530, A049084, A243290, A243291, A243348, A243349, A243351.

Programs

  • Mathematica
    With[{t = Table[PrimePi[FactorInteger[k][[-1, 1]]], {k, Select[Range[120], SquareFreeQ]}]}, Range[Length[t]] - t] (* Amiram Eldar, Mar 04 2024 *)

Formula

a(n) = n - A243290(n).

A243291 Difference between n and the index of its largest prime factor: a(n) = n - A061395(n).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 3, 7, 7, 7, 6, 10, 7, 10, 12, 15, 10, 16, 11, 17, 17, 17, 14, 22, 22, 20, 25, 24, 19, 27, 20, 31, 28, 27, 31, 34, 25, 30, 33, 37, 28, 38, 29, 39, 42, 37, 32, 46, 45, 47, 44, 46, 37, 52, 50, 52, 49, 48, 42, 57, 43, 51, 59, 63, 59, 61, 48, 61, 60
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2014

Keywords

Comments

All terms are strictly positive because A006530(n) <= n, A049084(n) < n and thus A061395(n) < n. [As A061395(n) = A049084(A006530(n))].
Please see also the comments at A243289.

Links

Crossrefs

Cf. A006530, A049084, A061395, A243289, A243290.

Programs

  • Mathematica
    a[n_] := n - PrimePi[FactorInteger[n][[-1, 1]]]; Array[a, 100] (* Amiram Eldar, Mar 04 2024 *)
  • Scheme
    (define (A243291 n) (- n (A061395 n)))

Formula

a(n) = n - A061395(n).
For n > 1, a(2^n) = (2^n)-1.

A243349 Difference between the n-th squarefree number and the index of its largest prime factor.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 7, 6, 7, 10, 12, 10, 11, 17, 17, 14, 20, 19, 27, 20, 28, 27, 31, 25, 30, 33, 28, 38, 29, 37, 32, 44, 37, 50, 49, 48, 42, 43, 51, 59, 61, 48, 60, 66, 51, 52, 62, 72, 72, 57, 69, 60, 78, 72, 77, 65, 85, 82, 79, 87, 72, 75, 95, 76, 101, 90, 79, 80, 105
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2014

Keywords

Comments

Different from A243289. A243348 gives the difference a(n) - A243289(n).

Links

Crossrefs

Cf. A005117, A243289, A243290, A243291, A243348.

Programs

  • Mathematica
    With[{s = Select[Range[120], SquareFreeQ]}, s - Table[PrimePi[FactorInteger[k][[-1, 1]]], {k, s}]] (* Amiram Eldar, Mar 04 2024 *)
  • Scheme
    (define (A243349 n) (A243291 (A005117 n)))

Formula

a(n) = A243291(A005117(n)).
a(n) = A005117(n) - A061395(A005117(n)).

A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 1, 1, 4, 1, 2, 5, 6, 1, 3, 2, 1, 7, 8, 2, 2, 1, 4, 9, 1, 5, 10, 1, 1, 1, 11, 2, 3, 1, 6, 3, 1, 12, 1, 7, 2, 4, 13, 1, 1, 2, 14, 1, 8, 15, 2, 5, 16, 3, 2, 2, 6, 1, 9, 17, 18, 1, 10, 3, 3, 1, 1, 3, 19, 2, 7, 1, 2, 1, 20, 21, 1, 11, 4, 1, 1, 1, 4, 22, 1, 12, 23, 3, 4
Offset: 1

Views

Author

Gus Wiseman, May 23 2025

Keywords

Comments

All rows are different.

Examples

			The 28-th squarefree number is 42, with 0-prepended prime indices (0,1,2,4), with differences (1,1,2), so row 28 is (1,1,2).
The squarefree numbers and corresponding rows begin:
    1: ()        23: (9)        47: (15)
    2: (1)       26: (1,5)      51: (2,5)
    3: (2)       29: (10)       53: (16)
    5: (3)       30: (1,1,1)    55: (3,2)
    6: (1,1)     31: (11)       57: (2,6)
    7: (4)       33: (2,3)      58: (1,9)
   10: (1,2)     34: (1,6)      59: (17)
   11: (5)       35: (3,1)      61: (18)
   13: (6)       37: (12)       62: (1,10)
   14: (1,3)     38: (1,7)      65: (3,3)
   15: (2,1)     39: (2,4)      66: (1,1,3)
   17: (7)       41: (13)       67: (19)
   19: (8)       42: (1,1,2)    69: (2,7)
   21: (2,2)     43: (14)       70: (1,2,1)
   22: (1,4)     46: (1,8)      71: (20)

Crossrefs

Row-lengths are A072047, sums A243290.
This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.

Programs

  • Mathematica
    sql=Select[Range[100],SquareFreeQ];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]
Showing 1-4 of 4 results.

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