A115627 -id:A115627 - 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}]

A226460 Let m! have prime factorization Product (p_j^e_j); a(n) = number of distinct prime factors p_j such that e_j = n has no solution for any m!.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 3, 2, 2, 1, 1, 1, 3, 0, 1, 2, 1, 0, 4, 2, 1, 0, 2, 1, 4, 2, 2, 0, 2, 0, 2, 2, 3, 1, 3, 1, 2, 1, 3, 1, 1, 0, 5, 2, 0, 0, 2, 2, 2, 1, 3, 2, 0, 1, 5, 3, 3, 1, 1, 2, 2, 0, 2, 1, 3, 0, 4, 2, 3, 0, 2, 1, 2, 1, 4, 2, 0, 0, 6, 1, 1, 0

Offset: 0

Naohiro Nomoto, Jun 08 2013

If n belongs to A048247 then a(n) is equal to zero.

For a given prime p and n satisfying p^k + p^(k-1) + ... + 1 <= n < p^(k+1) + ... + 1 for some k, let r_k = n mod (p^k + p^(k-1) + ... + 1), r_(k-1) = r_k mod (p^(k-1) + ... + 1), and so on down to r_1 = r_2 mod (p + 1). Then, p^n appears in a factorial m! iff none of the r_i is congruent to -1. - Charlie Neder, Nov 03 2018

			For n = 11, there are three distinct prime factors (3, 5, 11) in factorization of m!.
3^10 divides 26! ( 26! is not divisible by 3^11).
3^13 divides 27!.
5^10 divides 49! ( 49! is not divisible by 5^11).
5^12 divides 50!.
11^10 divides 120! ( 120! is not divisible by 11^11).
11^12 divides 121!.
The exponent of three distinct prime factors never becomes equal to 11. (It searches for all the exponent of prime factorization of factorials [A000142].)
Therefore a(11)=3.

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A000142, A048247, A115627.

is(k, p) = my(c, s); while(sk;
a(n) = sum(p=2, n, isprime(p)&&is(n, p)); \\ Jinyuan Wang, Aug 22 2021

A266620 a(n) = least non-divisor of n!.

Original entry on oeis.org

2, 3, 4, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71

Offset: 1

Jeffrey Shallit, Jan 01 2016

It appears that a(n) = A151800(n) with the exception of n = 3. - Robert Israel, Jan 13 2016

			For n = 4 the least non-divisor of 4! = 24 = 2^3 * 3 is 5.
For n = 5 the least non-divisor of 5! = 120 = 2^3 * 3 * 5 is 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007918, A007978, A066169, A115627, A151800.

N:= 100: # to get a(1)..a(N)
m:= 1 + numtheory:-pi(N):
Primes:= [seq(ithprime(i),i=1..m)]:
for i from 1 to m do pindex[Primes[i]]:= i od:
V:= Vector(m):
k:= 0:
for n from 1 to N do
  for f in ifactors(n)[2] do
    q:= pindex[f[1]];
    V[q]:= V[q] + f[2];
    k:= max(k, q);
  od:
  a[n]:= min(seq(Primes[i]^(1+V[i]),i=1..k),Primes[k+1]);
od:
seq(a[n],n=1..N); # Robert Israel, Jan 13 2016

Table[Complement[Range[2n], Divisors[n!]][[1]], {n, 30}] (* Alonso del Arte, Sep 23 2017 *)
Table[Block[{m = n!, k = n + 1}, While[Divisible[m, k], k++]; k], {n, 67}] (* Michael De Vlieger, Sep 23 2017 *)

from sympy import nextprime
def A266620(n): return 4 if n == 3 else nextprime(n) # Chai Wah Wu, Feb 22 2023

a(n) = min_{k >= 1} prime(k)^(1 + v(n!, prime(k))) where v(m, p) is the p-adic order of m. - Robert Israel, Jan 13 2016

a(n) = prime(pi(n) + 1) except for n = 3, in which case the least non-divisor of 3! is 4, not 5. - Alonso del Arte, Sep 23 2017

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[#]]]&]

A115628 Factorization of n! into primorials.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 1, 0, 1, 5, 1, 0, 1, 3, 3, 0, 1, 4, 2, 1, 1, 4, 2, 1, 0, 1, 5, 3, 1, 0, 1, 5, 3, 1, 0, 0, 1, 6, 3, 0, 1, 0, 1, 5, 3, 1, 1, 0, 1, 9, 3, 1, 1, 0, 1, 9, 3, 1, 1, 0, 0, 1, 8, 5, 1, 1, 0, 0, 1, 8, 5, 1, 1, 0, 0, 0, 1, 10, 4, 2, 1, 0, 0, 0, 1, 9, 5, 1, 2, 0, 0, 0, 1, 10, 5, 1, 1, 1

Offset: 2

Franklin T. Adams-Watters, Jan 26 2006

Factorial of n has a unique representation as a product of primorials: n! = 2#^T(n,1)*3#^T(n,2)*...*P_{Pi(n)}#^T(n,Pi(n)).

			Rows start: 1; 0,1; 2,1; 2,0,1; 2,1,1; 2,1,0,1; 5,1,0,1; 3,3,0,1.

Cf. A115627, A000142, A002110. Row lengths are A000720.

T(n, k) = A115727(n, k)-A115727(n, k+1).

A176383 Triangle of the powers of the prime factorization of n! in row n.

Original entry on oeis.org

2, 2, 3, 8, 3, 8, 3, 5, 16, 9, 5, 16, 9, 5, 7, 128, 9, 5, 7, 128, 81, 5, 7, 256, 81, 25, 7, 256, 81, 25, 7, 11, 1024, 243, 25, 7, 11, 1024, 243, 25, 7, 11, 13, 2048, 243, 25, 49, 11, 13, 2048, 729, 125, 49, 11, 13, 32768, 729, 125, 49, 11, 13, 32768, 729, 125, 49, 11, 13, 17, 65536

Offset: 2

Vladimir Shevelev, Apr 16 2010

Row n contains pi(n) = A000720(n) terms. The exponents are in A115627.

The first column contains the maximum power of 2 dividing n!, the second column the maximum power of 3 dividing n! etc.

			The irregular tables starts with n=2:
2; # =2! = 2
2*3; # =3! = 6
8*3; # =4! = 24
8*3*5; # =5! = 120
16*9*5; # =6!
16*9*5*7; # =7!
128*9*5*7; # =8!
128*81*5*7;
256*81*25*7;

Alois P. Heinz, Rows n = 2..300, flattened

Cf. A000142, A000720.

with(numtheory):
b:= proc(n) option remember; `if`(n=1, 1, b(n-1)+
      add(i[2]*x^pi(i[1]), i=ifactors(n)[2]))
    end:
T:= n->(p->seq(ithprime(i)^coeff(p, x, i), i=1..pi(n)))(b(n)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 22 2014

T[n_] := List @@ Power @@@ FactorInteger[n!];
Array[T, 20, 2] // Flatten (* Jean-François Alcover, Mar 27 2017 *)
Rest[Flatten[Table[#[[1]]^#[[2]]&/@FactorInteger[n!],{n,20}]]] (* Harvey P. Dale, Jan 04 2019 *)

a(n)=my(i=2);while(n-primepi(i)>1,n-=primepi(i);i++);p=prime(n-1);p^sum(j=1,log(i)\log(p),i\=p) \\ David A. Corneth, Jun 21 2014

Arbitrarily defined first 2 terms removed by R. J. Mathar, Apr 23 2010

A321362 a(n) is the least k such that in the prime power factorization of k! the exponents of primes p_1, ..., p_n are odd, while the exponent of p_(n+1) is even.

Original entry on oeis.org

2, 3, 5, 119, 57, 220, 1131, 2986, 1505, 3211, 21300, 26795, 11820, 14575, 67385, 221051, 33782, 132512, 819236, 1478432, 1630903, 26736550, 1095752, 41815849, 24813938, 31982450, 142574286, 860986855, 602660826, 2638930495, 2664421881, 1309662955, 33767540563

Offset: 1

Michel Marcus, Nov 07 2018

nonn

Analog to A240537 where odd and even are swapped.

			For k=2, 2!=2 with factorization 2 where the first exponent is odd.
For k=3, 3!=6 with factorization 2*3 where the first 2 exponents are odd.
For k=5, 5!=120 with factorization 2*3*5 where the first 3 exponents are odd.
In each case, there are no lesser numbers with the same property.

Giovanni Resta, Table of n, a(n) for n = 1..37 (terms < 10^12)
Rémy Sigrist, C program for A321362

Cf. A115627, A240537.

C
```
See Links section.
```

aQ[k_,n_] := Module[{e=FactorInteger[k!][[;;,2]]}, Length[e]>=n && AllTrue[ e[[1;;n]], OddQ ] && If[Length[e]>n, EvenQ[e[[n+1]]], True]]; a[n_] := Module[ {k=2}, While[!aQ[k,n], k++]; k ]; Array[a,10] (* Amiram Eldar, Nov 07 2018 *)

isok(v, n) = {if (#v < n, return (0)); for (i=1, n, if (!(v[i] % 2), return(0));); (#v == n) || !(v[n+1] % 2);}
newv(v, i) = {if (isprime(i), return(concat(v, 1))); f = factor(i); for (k=1, #f~, v[primepi(f[k,1])] += f[k,2];); return (v);}
a(n) = {my(v =[1], i = 2); while (!isok(v, n), i++; v = newv(v, i)); i;}

a(16)-a(32) from Rémy Sigrist, Nov 08 2018

a(33) from Giovanni Resta, Nov 09 2018

A330706 Numbers m such that the prime factorization of m! contains no composite exponents.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 14

Offset: 1

Devansh Singh, Mar 29 2020

This sequence is finite and a(7) = 14 is the last term. Nagura (see references) proves that for n >= 25, there is always a prime between n and 1.2*n. Hence, for any prime p > 25, there is always a number m between 4p and 4.8*p, and so floor(m/p) = 4. Since by assumption p > 4, floor(floor(m/p)/p) = 0 and so m! is divisible by p^4 but not p^5. It remains to check the primes up to 25 individually. - Charles R Greathouse IV, Apr 14 2020

			4 is a term since 4! = (2^3)*(3^1) and the multiplicity of 2 is 3 which is prime and the multiplicity of 3 is 1.

J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.

Cf. A000142, A115627.

Select[Range[100], !AnyTrue[FactorInteger[#!][[;;,2]], CompositeQ] &] (* Amiram Eldar, Mar 29 2020 *)

ok(n)={my(f=factor(n!)[,2]); for(i=1, #f, if(f[i]<>1 && !isprime(f[i]), return(0))); 1}
{select(ok, [1..100])} \\ Andrew Howroyd, Mar 29 2020

f(m,p)=my(s); while(m\=p, s+=m); s;
is(n)=forprime(p=2,n\4+1, if(!isprime(f(n,p)), return(0))); 1;
select(is,[1..25]) \\ Charles R Greathouse IV, Apr 14 2020

A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 4, 4, 2, 4, 5, 3, 5, 6, 6, 1, 6, 7, 7, 2, 5, 8, 8, 2, 6, 8, 9, 3, 7, 9, 10, 10, 2, 7, 10, 11, 11, 3, 8, 11, 12, 12, 12, 3, 9, 12, 12, 13, 13, 4, 9, 12, 13, 14, 14, 1, 10, 13, 14, 15, 15, 2, 11, 14, 15, 16, 16, 16, 2, 10, 15, 16, 17, 17, 17, 3, 11, 16, 17, 18, 18, 18, 18

Offset: 2

Michel Marcus, Apr 21 2022

This is the "s" in the American Mathematical Monthly problem.

			First few rows are:
  1;
  2, 2;
  1, 3;
  2, 4, 4;
  2, 4, 5;
  3, 5, 6, 6;
  1, 6, 7, 7;
  2, 5, 8, 8;
  2, 6, 8, 9;
  3, 7, 9, 10, 10;
  2, 7, 10, 11, 11;
  ...

Michel Marcus, Table of n, a(n) for n = 2..10388 (Rows n=2..300).
R. D. Carmichael, Diophantine Analysis: 141, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), p. 107.

Cf. A115627, A000120 (column 1), A089792 (column 2).

T[n_, k_] := n - IntegerExponent[n!, Prime[k]]; Table[T[n, k], {n, 2, 19}, {k, 1, PrimePi[n]}] // Flatten (* Amiram Eldar, Apr 21 2022 *)

row(n) = vector(primepi(n), k, n-valuation(n!, prime(k)));

T(n,k) = n - A115627(n, k).

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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A226460 Let m! have prime factorization Product (p_j^e_j); a(n) = number of distinct prime factors p_j such that e_j = n has no solution for any m!.

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A266620 a(n) = least non-divisor of n!.

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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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A115628 Factorization of n! into primorials.

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A176383 Triangle of the powers of the prime factorization of n! in row n.

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A321362 a(n) is the least k such that in the prime power factorization of k! the exponents of primes p_1, ..., p_n are odd, while the exponent of p_(n+1) is even.

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