A076954 -id:A076954 - cp's OEIS frontend

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

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

Offset: 0

Gus Wiseman, Aug 15 2020

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).

			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}

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.

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}]

\\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A050326(A006939(n)).

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

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750

Offset: 1

Jon Awbrey, Jul 09 2005

nonn

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

Suggested by Franklin T. Adams-Watters

J. Awbrey, Riffs and Rotes
Rémy Sigrist, PARI program for A109299

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

PARI
```
\\ See Links section.
```

is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

A076955 Product_{i=1..n} i^prime(i).

Original entry on oeis.org

1, 8, 1944, 31850496, 1555200000000000, 20311991333683200000000000, 4725188984058417024032990822400000000000, 680971499131541103151460312870118086801149132800000000000

Offset: 1

Amarnath Murthy, Oct 20 2002

nonn

n-th partial product of A062481.

Cf. A076954.

seq(product(i^ithprime(i),i=1..n),n=1..13);

a(1) =1, a(n+1) = a(n)*(n+1)^prime(n+1).

More terms from Sascha Kurz, Jan 22 2003

A135505 a(0) = 1; a(n) = [product_(i = 1..n) prime(i)^i] - 1, where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 17, 2249, 5402249, 870037764749, 4199506113235182749, 1723219765760312626547490749, 29266411525287522788837599332989370749, 52713275010243038997421106186697438702252144407249, 22176856087751973465466098269669474342964368337745368642450857249

Offset: 0

Ctibor O. Zizka, Feb 19 2008

Gheorghe Coserea, Table of n, a(n) for n = 0..50
C. K. Caldwell and Y. Gallot, On the primality of n!-1 and 2*3*..*p -1, Math. Comp., Volume 71, Number 237, Pages 441-448.
Leo Corry, Number crunching vs. number theory: computers and FLT, from Kummer to SWAC (1850-1960) and beyond, Archive for History of Exact Sciences, Vol. 62, No. 4 (July 2008), pp. 393-455.
Roland Queme, Some applications of Kummer and Stickelberger relations, arXiv:math/0601136 [math.NT], 2006.

Cf. A057588, A002110, A057705.

a(n) = { if (n <= 0, return(1)); prod(i = 1, n, prime(i)^i) - 1;}
vector(11, i, a(i-1))  \\ Gheorghe Coserea, Aug 24 2015

a(n) = A076954(n)-1, n>0. - R. J. Mathar, Nov 01 2009

Converted references to links - R. J. Mathar, Oct 30 2009

A344530 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+e_k)^k (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 2, 3, 18, 5, 50, 75, 2250, 7, 98, 147, 6174, 245, 17150, 25725, 5402250, 11, 242, 363, 23958, 605, 66550, 99825, 32942250, 847, 130438, 195657, 90393534, 326095, 251093150, 376639725, 870037764750, 13, 338, 507, 39546, 845, 109850, 164775, 64262250, 1183

Offset: 0

Rémy Sigrist, May 22 2021

All terms are distinct and belong to A304678.

			For n = 42:
- 42 = 2^1 + 2^3 + 2^5,
- a(42) = prime(1+1) * prime(1+3)^2 * prime(1+5)^3,
- a(42) = 3 * 7^2 * 13^3 = 322959.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A003961, A045967, A076954, A304678.

a(n) = { my (v=1, e); for (k=1, oo, if (n==0, return (v), n-=2^e=valuation(n, 2); v*=prime(1+e)^k)) }

a(0) = 1.
a(2*n) = A003961(a(n)).
a(2*n + 1) = A045967(a(n)).
a(2^n) = prime(n+1) for any n >= 0.
a(2^n - 1) = A076954(n) for any n >= 0.

A302156 a(n) = Product_{k=1..n} prime(k+1)^(n-k+1).

Original entry on oeis.org

1, 3, 45, 4725, 5457375, 81942485625, 20916229168209375, 101440469450294396296875, 11315322731906749607393607890625, 36603333436941101463129791457625571484375, 3670591247252362378693685549273035871463800818359375, 13619248222892703567716797493618519282116254094632750020888671875

Offset: 0

Ilya Gutkovskiy, Apr 02 2018

a(n) is the smallest odd number with n distinct exponents in its prime factorization.

			+---+-------------------------------+
| n |  prime factorization of a(n)  |
+---+-------------------------------+
| 1 | 3                             |
| 2 | 3^2*5                         |
| 3 | 3^3*5^2*7                     |
| 4 | 3^4*5^3*7^2*11                |
| 5 | 3^5*5^4*7^3*11^2*13           |
| 6 | 3^6*5^5*7^4*11^3*13^2*17      |
| 7 | 3^7*5^6*7^5*11^4*13^3*17^2*19 |
+---+-------------------------------+

Cf. A000079, A002110, A006939, A070826, A076954, A087315.

Table[Product[Prime[k + 1]^(n - k + 1), {k, 1, n}], {n, 0, 11}]

a(n) = prod(k=1, n, prime(k+1)^(n-k+1)); \\ Altug Alkan, Apr 02 2018

a(0) = 1; a(n) = A002110(n+1)*a(n-1)/2.

a(n) = A006939(n+1)/A000079(n+1).

A318956 For any number n > 0 with prime factorization Product_{k=1..w} p_k ^ x_k (where p_1 < p_2 < ... < p_w and x_k > 0 for k=1..w), let (o_1, ..., o_w) be the ordinal transform of (x_1, ..., x_w); a(n) = Product_{k=1..w} p_k ^ o_k.

Original entry on oeis.org

1, 2, 3, 2, 5, 18, 7, 2, 3, 50, 11, 6, 13, 98, 75, 2, 17, 6, 19, 10, 147, 242, 23, 6, 5, 338, 3, 14, 29, 2250, 31, 2, 363, 578, 245, 18, 37, 722, 507, 10, 41, 6174, 43, 22, 15, 1058, 47, 6, 7, 10, 867, 26, 53, 6, 605, 14, 1083, 1682, 59, 150, 61, 1922, 21, 2

Offset: 1

Rémy Sigrist, Sep 20 2018

nonn

The ordinal transform of a sequence b(n) is the sequence t(n) = number of values in b(1), ..., b(n) which are equal to b(n).

Cf. A002110, A005117, A076954, A130091.

a(n) = if (n==1, 1, my (f=factor(n), o=vector(vecmax(f[,2]))); for (i=1, #f~, f[i,2] = o[f[i,2]]++); factorback(f))

a(p) = p iff p = 1 or p is prime.
a(n^k) = a(n) for any n > 0 and k > 0.
A007947(a(n)) = A007947(n) for any n > 0.
a(a(a(n))) = a(n) for any n > 0.
a(n) belongs to A005117 iff n belongs to A130091 and vice versa.
a(A002110(n)) = A076954(n) for any n >= 0.
a(A076954(n)) = A002110(n) for any n >= 0.

cp's OEIS Frontend

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

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A109299 Primal codes of canonical finite permutations on positive integers.

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A076955 Product_{i=1..n} i^prime(i).

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A135505 a(0) = 1; a(n) = [product_(i = 1..n) prime(i)^i] - 1, where prime(i) is i-th prime.

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A344530 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+e_k)^k (where prime(k) denotes the k-th prime number).

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A302156 a(n) = Product_{k=1..n} prime(k+1)^(n-k+1).

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A318956 For any number n > 0 with prime factorization Product_{k=1..w} p_k ^ x_k (where p_1 < p_2 < ... < p_w and x_k > 0 for k=1..w), let (o_1, ..., o_w) be the ordinal transform of (x_1, ..., x_w); a(n) = Product_{k=1..w} p_k ^ o_k.

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