A325276 -id:A325276 - cp's OEIS frontend

A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

1, 2, 1, 4, 1, 5, 2, 1, 7, 3, 1, 9, 3, 1, 1, 12, 3, 1, 1, 14, 5, 1, 1, 16, 6, 2, 1, 17, 7, 3, 1, 1, 20, 8, 3, 1, 1, 22, 9, 3, 1, 1, 1, 25, 9, 3, 2, 1, 1, 27, 11, 4, 2, 1, 1, 31, 11, 4, 2, 1, 1, 33, 11, 4, 3, 1, 1, 1, 36, 13, 4, 3, 1, 1, 1, 39, 13, 4, 3, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 12 2019

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
Row n is the sequence of nonzero exponents in the q-factorization of n!.
Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n!.

			We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).
Triangle begins:
  {}
   1
   2  1
   4  1
   5  2  1
   7  3  1
   9  3  1  1
  12  3  1  1
  14  5  1  1
  16  6  2  1
  17  7  3  1  1
  20  8  3  1  1
  22  9  3  1  1  1
  25  9  3  2  1  1
  27 11  4  2  1  1
  31 11  4  2  1  1
  33 11  4  3  1  1  1
  36 13  4  3  1  1  1
  39 13  4  3  1  1  1  1
  42 14  5  3  1  1  1  1

Row lengths are A000720.
Row sums are A325544(n) - 1.
Column k = 1 is A325543.
Cf. A056239, A067255, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.
q-factorization: A324922, A324923, A324924, A325614, A325615, A325660.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length/@Split[difac[n!]],{n,20}]

A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.

Original entry on oeis.org

3, 4, 5, 7, 26, 65, 942, 24147

Offset: 3

Gus Wiseman, Apr 24 2019

If infinite terms were allowed, we would have a(0) = 1, a(1) = 2, a(2) = infinity. It is possible this sequence is finite, or that there are additional gaps.

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

			Column n is the sequence of images under A181819 starting with a(n)!:
  6  24  120  5040  403291461126605635584000000
  4  10  20   84    11264760
  3  4   6    12    240
     3   4    6     28
         3    4     6
              3     4
                    3

a(n) is the first position of n in A325272.
Cf. A000142, A022559, A181819, A181821, A323023, A325238, A325273, A325274, A325275, A325276, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
dat=Table[fdadj[n!],{n,1000}];
Table[Position[dat,k][[1,1]],{k,3,Max@@dat}]

A325510 Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 7, 16, 98, 269, 1397, 7582, 70520, 259906, 1677259, 5229112, 44726100, 666355170, 4917007185, 18459879921

Offset: 0

Gus Wiseman, May 08 2019

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{12}}    {{1222}}        {{12333}}
         {{1}{2}}  {{1}{222}}      {{1}{2333}}
                   {{12}{22}}      {{12}{333}}
                   {{2}{122}}      {{13}{233}}
                   {{1}{2}{22}}    {{3}{1233}}
                   {{2}{2}{12}}    {{33}{123}}
                   {{1}{2}{2}{2}}  {{1}{2}{333}}
                                   {{1}{23}{33}}
                                   {{1}{3}{233}}
                                   {{3}{12}{33}}
                                   {{3}{13}{23}}
                                   {{3}{3}{123}}
                                   {{1}{1}{1}{23}}
                                   {{1}{2}{3}{33}}
                                   {{1}{3}{3}{23}}
                                   {{1}{2}{3}{3}{3}}

Cf. A000142, A001055, A007716, A011371, A022559, A076716, A115627, A317791, A318285, A322583, A325272, A325276, A325508, A325509, A325511.

\\ Requires C(sig) from A318285.
a(n)={if(n<2, 1, my(f=factor(n!)[,2], sig=vector(vecmax(f))); for(i=1, #f, sig[f[i]]++); C(sig))} \\ Andrew Howroyd, Jan 17 2023

a(n) = A317791(n!).

a(n) = A318285(A181819(n!)) = A318285(A325508(n)). - Andrew Howroyd, Jan 17 2023

a(9)-a(17) from Andrew Howroyd, Jan 17 2023

A325511 Numbers whose prime signature is that of a factorial number.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 103, 104, 106

Offset: 1

Gus Wiseman, May 08 2019

nonn

A181819(a(n)) belongs to A325508.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}
  26: {1,6}
  29: {10}
  31: {11}

Cf. A000142, A011371, A022559, A076934, A115627, A181819, A325272, A325276, A325508, A325509, A325510.

Select[Range[30],MemberQ[Table[Sort[Last/@FactorInteger[k!]],{k,#}],Sort[Last/@FactorInteger[#]]]&]

A307734 Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.

Original entry on oeis.org

1, 2, 0, 3, 4, 5, 7, 26, 65, 942, 24147

Offset: 0

Gus Wiseman, Apr 25 2019

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

Conjecture: this sequence has infinitely many nonzero terms.

			Column n is the sequence of images under A181819 starting with a(n)!:
  -  2  -  6  24  120  5040  403291461126605635584000000
           4  10  20   84    11264760
           3  4   6    12    240
              3   4    6     28
                  3    4     6
                       3     4
                             3

Essentially the same as A325410.
a(n) is zero or the first position of n in A325272.
Cf. A000142, A323023, A325238, A325273, A325274, A325275, A325276, A325277, A325416.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

cp's OEIS Frontend

A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

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A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.

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A325510 Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.

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A325511 Numbers whose prime signature is that of a factorial number.

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Mathematica

A307734 Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.

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