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 1-5 of 5 results.

A074191 Duplicate of A090630.

Original entry on oeis.org

1, 1, 1, 8, 8, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400
Offset: 1

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Keywords

A336416 Number of perfect-power divisors of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 7, 7, 11, 18, 36, 36, 47, 47, 84, 122, 166, 166, 221, 221, 346, 416, 717, 717, 1001, 1360, 2513, 2942, 4652, 4652, 5675, 5675, 6507, 6980, 13892, 17212, 20408, 20408, 39869, 45329, 51018, 51018, 68758, 68758, 105573, 138617, 284718, 284718, 338126, 421126
Offset: 0

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Author

Gus Wiseman, Jul 22 2020

Keywords

Comments

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

Examples

			The a(1) = 0 through a(9) = 18 divisors:
       1: 1
       2: 1
       6: 1
      24: 1,4,8
     120: 1,4,8
     720: 1,4,8,9,16,36,144
    5040: 1,4,8,9,16,36,144
   40320: 1,4,8,9,16,32,36,64,128,144,576
  362880: 1,4,8,9,16,27,32,36,64,81,128,144,216,324,576,1296,1728,5184

Links

Crossrefs

The maximum among these divisors is A090630, with quotient A251753.
The version for distinct prime exponents is A336414.
The uniform version is A336415.
Replacing factorials with Chernoff numbers (A006939) gives A336417.
Prime powers are A000961.
Perfect powers are A001597, with complement A007916.
Prime power divisors are counted by A022559.
Cf. A000005, A001221, A001222, A011371, A032741, A118914, A124010, A130091, A327527.
Factorial numbers: A000142, A007489, A027423, A048656, A071626, A181796, A325272, A325273, A325617.

Programs

  • Mathematica
    perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[n!],perpouQ]],{n,0,15}]
  • PARI
    a(n) = sumdiv(n!, d, (d==1) || ispower(d)); \\ Michel Marcus, Aug 19 2020
  • PARI
    addhelp(val, "exponent of prime p in n!")
val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = {if(n<=3, return(1)); my(pr = primes(primepi(n\2)), v = vector(#pr, i, val(n, pr[i])), res = 1, cv); for(i = 2, v[1], if(issquarefree(i), cv = v\i; res-=(prod(i = 1, #cv, cv[i]+1)-1)*(-1)^omega(i) ) ); res } \\ David A. Corneth, Aug 19 2020

Formula

a(p) = a(p-1) for prime p. - David A. Corneth, Aug 19 2020

Extensions

a(26)-a(34) from Jinyuan Wang, Aug 19 2020
a(35)-a(49) from David A. Corneth, Aug 19 2020

A336417 Number of perfect-power divisors of superprimorials A006939.

Original entry on oeis.org

1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522
Offset: 0

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Author

Gus Wiseman, Jul 24 2020

Keywords

Comments

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.
The n-th superprimorial number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Examples

			The a(0) = 1 through a(4) = 15 divisors:
  1  2  12  360  75600
-------------------------
  1  1   1    1      1
         4    4      4
              8      8
              9      9
             36     16
                    25
                    27
                    36
                   100
                   144
                   216
                   225
                   400
                   900
                  3600

Links

Crossrefs

A000325 is the uniform version.
A076954 can be used instead of A006939.
A336416 gives the same for factorials instead of superprimorials.
A000217 counts prime power divisors of superprimorials.
A000961 gives prime powers.
A001597 gives perfect powers, with complement A007916.
A006939 gives superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A091050 counts perfect power divisors.
A181818 gives products of superprimorials.
A294068 counts factorizations using perfect powers.
A317829 counts factorizations of superprimorials.
Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]
  • PARI
    a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020

Formula

a(n) = A091050(A006939(n)).
a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 30 2020

A251753 n!/pp, where pp is the largest perfect power (A001597) which divides n!.

Original entry on oeis.org

1, 1, 2, 6, 3, 15, 5, 35, 70, 70, 7, 77, 231, 3003, 858, 1430, 1430, 24310, 12155, 230945, 46189, 230945, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353, 3357800061, 353452638, 1531628098, 3829070245, 156991880045
Offset: 0

Views

Author

Robert G. Wilson v, Dec 07 2014

Keywords

Links

Crossrefs

Cf. A000142, A001597, A090630.

Programs

  • Mathematica
    perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; f[n_] := Block[{d = Divisors[n!], k = 1},  While[ ! perfectPowerQ[ d[[-k]]], k++]; n!/d[[-k]]]; Array[f, 41, 0] (* or *)
f[n_] := Block[{fi = FactorInteger[n!]}, n!/Times @@ (#1[[1]] ^ (2 Quotient[#1[[2]],2])&) /@ fi]; f[4] = 3; f[5] = 15; f[21] = 230945; Array[f, 40]

Formula

If p is prime, then a(p) = p*a(p-1).
a(n) = n! / A090630(n). - Joerg Arndt, Dec 08 2014

A336418 Numbers with a factorial number of divisors.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163
Offset: 1

Views

Author

Gus Wiseman, Jul 27 2020

Keywords

Comments

Positive integers n such that A000005(n) belongs to A000142.

Crossrefs

A000005 counts divisors.
A048656 counts squarefree divisors of n!.
A325617 counts permutations of the multiset of prime factors of n!.
A336414 counts divisors of n! with distinct prime exponents.
Cf. A000142, A001221, A001222, A002982, A007489, A022559, A027423, A032741, A054991, A090630, A108731, A181796, A251753, A336416.

Programs

  • Mathematica
    nn=5;Select[Range[nn!],MemberQ[Array[Factorial,nn],Length[Divisors[#]]]&]
Showing 1-5 of 5 results.

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