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.

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A336941 Number of strict chains of divisors starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 8, 604, 691968, 16359233536, 10083474928244288, 195661337707783118840768, 139988400203593571474134024847360, 4231553868972506381329450624389969130848256, 6090860257621637852755610879241895108657182173073604608, 464479854191019594417264488167571483344961210693790188774166838214656
Offset: 0

Views

Author

Gus Wiseman, Aug 13 2020

Keywords

Examples

			The a(2) = 8 chains:
  12/1
  12/2/1
  12/3/1
  12/4/1
  12/6/1
  12/4/2/1
  12/6/2/1
  12/6/3/1

Links

Crossrefs

A022915 is the maximal case.
A076954 can be used instead of A006939.
A336571 is the case with distinct prime multiplicities.
A336942 is the case using members of A130091.
A337070 is the version ending with any divisor of A006939(n).
A000005 counts divisors.
A074206 counts chains of divisors from n to 1.
A006939 lists superprimorials or Chernoff numbers.
A067824 counts divisor chains starting with n.
A181818 gives products of superprimorials, with complement A336426.
A253249 counts chains of divisors.
A317829 counts factorizations of superprimorials.
A336423 counts chains using A130091, with maximal case A336569.
Cf. A000142, A001055, A002033, A008480, A022559, A027423, A124010, A167865, A181796, A336417, A336420, A337069.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chns[n_]:=If[n==1,1,Sum[chns[d],{d,Most[Divisors[n]]}]];
Table[chns[chern[n]],{n,0,3}]
  • PARI
    a(n)={my(sig=vector(n,i,i), m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k)))} \\ Andrew Howroyd, Aug 30 2020

Formula

a(n) = A337070(n)/2 for n > 0.
a(n) = A074206(A006939(n)).

Extensions

Terms a(8) and beyond from Andrew Howroyd, Aug 30 2020

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2024

Keywords

Comments

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Links

Crossrefs

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

Programs

  • Maple
    N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024
  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 2, 10, 141, 6769, 1298995, 1148840085, 5307091649182, 143026276277298216, 24801104674619158730662, 30190572492693121799801655311, 278937095127086600900558327826721594
Offset: 0

Views

Author

Gus Wiseman, Aug 15 2020

Keywords

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1), which has n! divisors.
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of the superprimorial A006939(n).

Examples

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6    2*6*30
       2*2*3  6*6*10
              2*5*6*6
              2*2*3*30
              2*2*6*15
              2*3*6*10
              2*2*3*5*6
              2*2*2*3*15
              2*2*3*3*10
              2*2*2*3*3*5
The a(1) = 1 through a(3) = 10 set multipartitions:
     {1}  {1}{12}    {1}{12}{123}
          {1}{1}{2}  {12}{12}{13}
                     {1}{1}{12}{23}
                     {1}{1}{2}{123}
                     {1}{2}{12}{13}
                     {1}{3}{12}{12}
                     {1}{1}{1}{2}{23}
                     {1}{1}{2}{2}{13}
                     {1}{1}{2}{3}{12}
                     {1}{1}{1}{2}{2}{3}

Crossrefs

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103774 is the version for factorials instead of superprimorials.
A337073 is the strict case (strict factorizations into squarefree numbers).
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[facsqf[chern[n]]],{n,0,3}]
  • PARI
    \\ See A318360 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

Formula

a(n) = A050320(A006939(n)).
a(n) = A318360(A002110(n)). - Andrew Howroyd, Aug 31 2020

Extensions

a(7)-a(12) from Andrew Howroyd, Aug 31 2020

A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 2, 14, 422, 59433, 43181280, 178025660042, 4550598470020490, 782250333882971717562, 974196106965358319940100513, 9412280190038329162111356578977100, 751537739224674099813783040471383322758327
Offset: 0

Views

Author

Gus Wiseman, Aug 15 2020

Keywords

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). It has n! divisors.
Also the number of strict set multipartitions (sets of sets) of the multiset of prime factors of the superprimorial A006939(n).

Examples

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6  2*6*30    2*6*30*210
            2*3*6*10  6*10*30*42
                      2*3*6*30*70
                      2*5*6*30*42
                      2*3*10*30*42
                      2*3*6*10*210
                      2*6*10*15*42
                      2*6*10*21*30
                      2*6*14*15*30
                      3*6*10*14*30
                      2*3*5*6*10*42
                      2*3*5*6*14*30
                      2*3*6*7*10*30
                      2*3*6*10*14*15
The a(1) = 1 through a(3) = 14 set multipartitions:
    {1}  {1}{12}  {1}{12}{123}    {1}{12}{123}{1234}
                  {1}{2}{12}{13}  {12}{13}{123}{124}
                                  {1}{12}{13}{23}{124}
                                  {1}{12}{13}{24}{123}
                                  {1}{12}{14}{23}{123}
                                  {1}{2}{12}{123}{134}
                                  {1}{2}{12}{13}{1234}
                                  {1}{2}{13}{123}{124}
                                  {1}{3}{12}{123}{124}
                                  {2}{12}{13}{14}{123}
                                  {1}{2}{12}{13}{14}{23}
                                  {1}{2}{12}{4}{13}{123}
                                  {1}{2}{3}{12}{13}{124}
                                  {1}{2}{3}{12}{14}{123}

Crossrefs

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103775 is the version for factorials instead of superprimorials.
A337072 is the non-strict version.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts strict set multipartitions of integer partitions.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A283877 counts non-isomorphic strict set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A022559, A103774, A124010, A181818, A318361, A336417, A337070.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
ystfac[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[ystfac[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[ystfac[chern[n]]],{n,0,4}]
  • PARI
    \\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

Formula

a(n) = A050326(A006939(n)).
a(n) = A318361(A002110(n)). - Andrew Howroyd, Sep 01 2020

Extensions

a(7)-a(13) from Andrew Howroyd, Sep 01 2020
Previous Showing 11-14 of 14 results.

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