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.

A177687 Number of distinct transpositions of digits (zeros and units) in n-th prime written in base 2.

Original entry on oeis.org

2, 1, 3, 1, 4, 4, 10, 10, 5, 5, 1, 20, 20, 15, 6, 15, 6, 6, 35, 35, 35, 21, 35, 35, 35, 35, 21, 21, 21, 35, 1, 56, 56, 70, 70, 56, 56, 70, 56, 56, 56, 56, 8, 56, 70, 56, 56, 8, 56, 56, 56, 8, 56, 8, 36, 126, 126, 126, 126, 126, 126, 126, 126, 84, 126, 84, 126, 126, 84, 84, 126, 84
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 12 2010, May 14 2010

Keywords

Examples

			a(1)=2 because 2 =  10_2 and |{10, 01}| = 2,
a(2)=1 because 3 =  11_2 and |{11}| = 1,
a(3)=3 because 5 = 101_2 and |{101, 011, 110}| = 3.

Links

Crossrefs

Cf. A004676, A168324, A178244.

Programs

  • Maple
    A177687 := proc(n) p := ithprime(n) ; combinat[numbperm](convert(p,base,2)) ; end proc: seq(A177687(n),n=1..120) ; # R. J. Mathar, May 15 2010
  • Mathematica
    Array[Length@ Permutations@ IntegerDigits[Prime@ #, 2] &, 72] (* Michael De Vlieger, Feb 15 2019 *)

Formula

a(n) = binomial(A000120(p) + A023416(p), A000120(p)), where p=prime(n). - R. J. Mathar, May 15 2010
a(n) = A178244(A000040(n)). - Robert Israel, Oct 25 2023

Extensions

Terms corrected by D. S. McNeil and R. J. Mathar, May 14 2010

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 2, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 0, 2, 2, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 2, 0, 0, 2, 2, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2025

Keywords

Comments

Are the rows all unimodal?
Counts permutations of prime factors by "inseparability". For "separability" we have A374252.

Examples

			The prime indices of 12 are {1,1,2}, and we have:
- 1 permutation (1,2,1) with 0 adjacent equal parts
- 2 permutations (1,1,2), (2,1,1) with 1 adjacent equal part
- 0 permutations with 2 adjacent equal parts
so row 12 is (1,2,0).
Row 48 counts the following permutations:
  .  .  (1,1,1,2,1)  (1,1,1,1,2)  .
        (1,1,2,1,1)  (2,1,1,1,1)
        (1,2,1,1,1)
Row 144 counts the following permutations:
  .  (1,1,2,1,2,1)  (1,1,1,2,1,2)  (1,1,1,2,2,1)  (1,1,1,1,2,2)  .
     (1,2,1,1,2,1)  (1,1,2,1,1,2)  (1,1,2,2,1,1)  (2,2,1,1,1,1)
     (1,2,1,2,1,1)  (1,2,1,1,1,2)  (1,2,2,1,1,1)
                    (2,1,1,1,2,1)  (2,1,1,1,1,2)
                    (2,1,1,2,1,1)
                    (2,1,2,1,1,1)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 0  1
   6: 1
   6: 2  0
   7: 1
   8: 0  0  1
   9: 0  1
  10: 2  0
  11: 1
  12: 1  2  0
  13: 1
  14: 2  0
  15: 2  0
  16: 0  0  0  1
  17: 1
  18: 1  2  0
  19: 1
  20: 1  2  0
  21: 2  0
  22: 2  0
  23: 1
  24: 0  2  2  0

Crossrefs

Row lengths are A001222.
The minima of each row are A010051.
Sorted positions of first appearances appear to be A025487.
Column k = last is A069513.
Row sums are A168324 or A008480.
The number of trailing zeros in each row is A297155 = A001221-1.
Column k = 1 is A335452.
The number of leading zeros in each row is A374246.
For separability instead of inseparability we have A374252.
For a multiset with prescribed multiplicities we have A386578, separability A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A051903, A051904, A106351, A238130, A336102, A373957, A382525, A386581, A386632.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==k]]],{n,30},{k,0,PrimeOmega[n]-1}]

A160810 Numbers k such that the number of partitions of k into prime divisors of k exceeds the number of distinct transpositions of prime factors of k.

Original entry on oeis.org

18, 24, 30, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 136, 138, 140, 144, 147, 150, 152, 153, 154, 156, 160, 162, 165, 168, 170, 171, 174, 175, 176
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 23 2009

Keywords

Comments

Numbers k such that A066882(k) > A168324(k).

Crossrefs

Cf. A066882, A168324.

Programs

  • Maple
    A066882 := proc(n) gf := 1 ; for d in numtheory[divisors](n) do if isprime(d) then gf := gf/(1-x^d) ; gf := taylor(gf,x=0,n+2) ; end if; end do: coeftayl(gf,x=0,n) ; end proc:
A168324 := proc(n) if n = 1 then 0; else multn := numtheory[bigomega](n) ; multn := factorial(multn) ; for p in ifactors(n)[2] do multn := multn/factorial(op(2,p)) ; end do: multn ; end if; end proc:
for n from 1 to 300 do if A066882(n) > A168324(n) then printf("%d,\n",n) ; end if; end do: # R. J. Mathar, May 21 2010

Extensions

More terms from R. J. Mathar, May 21 2010
Showing 1-3 of 3 results.

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