A375635 -id:A375635 - cp's OEIS frontend

A375636 The number of infinitary divisors of n! that are factorials.

1, 2, 3, 4, 5, 2, 3, 5, 3, 2, 3, 4, 5, 5, 3, 5, 6, 2, 3, 2, 3, 5, 6, 2, 3, 3, 5, 4, 5, 2, 3, 6, 12, 2, 2, 2, 3, 5, 3, 4, 5, 7, 8, 4, 4, 2, 3, 2, 2, 3, 6, 4, 5, 2, 3, 5, 7, 4, 5, 4, 5, 5, 3, 4, 12, 2, 3, 2, 2, 3, 4, 2, 3, 3, 6, 4, 4, 2, 3, 4, 2, 3, 4, 4, 4, 2, 2

Offset: 1

Amiram Eldar, Aug 22 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A000720, A037445, A077609, A090971, A375635, A375637, A375638.

expQ[e1_, e2_] := Module[{m = Length[e2], ans = 1}, Do[If[BitAnd[e1[[i]], e2[[i]]] < e2[[i]], ans = 0; Break[]], {i, 1, m}]; ans];
e[n_] := e[n] = FactorInteger[n!][[;; , 2]]; a[n_] := 1 + Sum[expQ[e[n], e[m]], {m, 2, n}]; Array[a, 100]

isexp(e1, e2) = {my(m = #e2, ans = 1); for(i=1,m,if(bitand(e1[i], e2[i]) < e2[i], ans = 0; break)); ans;}
e(n) = factor(n!)[,2];
a(n) = 1 + sum(m = 2, n, isexp(e(n), e(m)));

a(n) >= 2 for n >= 2.
a(n) <= 2 if and only if n is in A375637.
a(A375638(n)) = n or -1.
a(p) = a(p-1) + 1 for a prime p.
a(n) = 1 + Sum_{k=2..n} [Sum_{p prime <= A007917(k)} A090971(v_p(n!), v_p(k!)) = primepi(k)], where v_p(n) is the p-adic valuation of n, primepi(k) = A000720(k), and [] is the Iverson bracket.

A375637 Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.

Original entry on oeis.org

1, 2, 6, 10, 18, 20, 24, 30, 34, 35, 36, 46, 48, 49, 54, 66, 68, 69, 72, 78, 81, 86, 87, 90, 91, 92, 96, 102, 108, 114, 116, 117, 120, 121, 126, 130, 142, 143, 150, 155, 156, 161, 166, 171, 172, 180, 184, 190, 192, 198, 204, 205, 212, 216, 222, 228, 232, 238, 240

Offset: 1

Amiram Eldar, Aug 22 2024

nonn

The trivial infinitary divisors of a number m are 1 and m itself. Therefore if k >=2 then k! has at least 2 infinitary divisors that are factorials, 1 and k!.

Numbers k such that A375636(k) <= 2, or equivalently, 1 and numbers k such that A375636(k) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A037445, A077609, A375635, A375636.

expQ[e1_, e2_] := Module[{m = Length[e2], ans = 1}, Do[If[BitAnd[e1[[i]], e2[[i]]] < e2[[i]], ans = 0; Break[]], {i, 1, m}]; ans];
e[n_] := e[n] = FactorInteger[n!][[;; , 2]]; q[n_] := Sum[expQ[e[n], e[m]], {m, 2, n}] <= 1; Select[Range[240], q]

isexp(e1, e2) = {my(m = #e2, ans = 1); for(i=1,m,if(bitand(e1[i], e2[i]) < e2[i], ans = 0; break)); ans;}
e(n) = factor(n!)[,2];
is(n) = sum(m = 2, n, isexp(e(n), e(m))) <= 1;

A375638 a(n) is the least positive integer k such that k! has exactly n infinitary divisors that are factorials, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 17, 42, 43, 138, 125, 220, 33, 387, 1766, 3269, 7014, 1398, 1399, 1958, 19143, 30759

Offset: 1

Amiram Eldar, Aug 22 2024

a(n) is the least k > 0 such that A375636(k) = n.

			The n values m(i) such that m(i)! is an infinitary divisor of a(n)! for n = 1..12 are:
   n | a(n) | m(i), i = 1..n
  ---+------+-----------------------------------------
   1 |   1  | 1
   2 |   2  | 1, 2
   3 |   3  | 1, 2, 3
   4 |   4  | 1, 2, 3, 4
   5 |   5  | 1, 2, 3, 4, 5
   6 |  17  | 1, 2, 6, 15, 16, 17
   7 |  42  | 1, 2, 3, 4, 5, 6, 42
   8 |  43  | 1, 2, 3, 4, 5, 6, 42, 43
   9 | 138  | 1, 2, 3, 4, 5, 6, 7, 8, 138
  10 | 125  | 1, 2, 3, 4, 5, 6, 7, 8, 124, 125
  11 | 220  | 1, 2, 3, 4, 5, 6, 7, 8, 218, 219, 220
  12 |  33  | 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33

Cf. A000142, A037445, A077609, A375635, A375636.

cp's OEIS Frontend

A375636 The number of infinitary divisors of n! that are factorials.

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A375637 Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.

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A375638 a(n) is the least positive integer k such that k! has exactly n infinitary divisors that are factorials, or -1 if no such number exists.

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