A374674 -id:A374674 - cp's OEIS frontend

A374671 Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

8, 19, 23, 44, 45, 57, 67, 76, 80, 83, 84, 85, 105, 107, 116, 120, 123, 140, 141, 146, 158, 161, 165, 174, 177, 187, 201, 208, 214, 225, 235, 239, 241, 243, 244, 246, 247, 263, 269, 272, 277, 284, 297, 309, 315, 321, 322, 325, 337, 339, 341, 342, 344, 360, 363

Offset: 1

Amiram Eldar, Jul 16 2024

nonn

Positive numbers such that k! and (k+1)! have an equal number of Fermi-Dirac factors (A064547).
Positive numbers k such that A037445(k!) = A037445((k+1)!).
Positive numbers k such that A064547(k!) = A064547((k+1)!).
Positive numbers k such that A177329(k) = A177329(k+1).

			8 is a term since A037445(8!) = A037445(9!) = 64.

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

Cf. A037445, A064547, A177329, A343819, A374672, A374673, A374674.

s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 400], s[#] == s[# + 1] &]

s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}

A374672 Numbers k such that k! has more infinitary divisors than (k+1)!.

Original entry on oeis.org

5, 9, 17, 27, 33, 34, 35, 43, 48, 51, 53, 59, 65, 68, 69, 75, 77, 87, 91, 97, 98, 99, 103, 115, 119, 125, 129, 134, 135, 139, 147, 149, 151, 155, 163, 164, 171, 179, 183, 189, 194, 195, 197, 199, 203, 211, 215, 221, 227, 229, 230, 231, 237, 245, 249, 257, 259

Offset: 1

Amiram Eldar, Jul 16 2024

nonn

Numbers k such that k! has more Fermi-Dirac factors (A064547) than (k+1)!.
Numbers k such that A037445(k!) > A037445((k+1)!).
Numbers k such that A064547(k!) > A064547((k+1)!).
Numbers k such that A177329(k) > A177329(k+1).

			5 is a term since A037445(5!) = 16 > A037445(6!) = 8.

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

Cf. A037445, A064547, A177329, A343819, A374671, A374673, A374674.

s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 300], s[#] > s[# + 1] &]

s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > s2, print1(k-1, ", ")); s1 = s2);}

A374673 a(n) is the start of the least run of exactly n consecutive positive numbers with an equal value of A177329, or -1 if no such run exists.

Original entry on oeis.org

2, 8, 44, 83, 4475, 75093, 164903, 59480, 1342805

Offset: 1

Amiram Eldar, Jul 16 2024

For n > 1, a(n)! is the start of the least run of successive factorials of positive numbers (i.e., ignoring 0!) with an equal number of infinitary divisors (A037445).

a(9) > 320000, if it exists.

			  n |   a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
  --|--------|------------------------------------------------
  1 |      2 | A177329(2) = 1
  2 |      8 | A177329(8) = A177329(9) = 6
  3 |     44 | A177329(44) = A177329(45) = A177329(46) = 21
  4 |     83 | A177329(83) = ... = A177329(86) = 35
  5 |   4475 | A177329(4475) = ... A177329(4479) = 923
  6 |  75093 | A177329(75093) = ... = A177329(75098) = 10857
  7 | 164903 | A177329(164903) = ... = A177329(164909) = 22038
  8 |  59480 | A177329(59480) = ... = A177329(59487) = 8814

Cf. A177329, A318166, A374671, A374672, A374674.

s[n_] := Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; seq[len_] := Module[{v = Table[0, {len}], w = {0}, c = 0, k = 3, m, s1}, While[c < len, s1 = s[k]; m = Length[w]; If[s1 == w[[m]], AppendTo[w, s1], If[m <= len && v[[m]] == 0, v[[m]] = k-m; c++]; w = {s1}]; k++]; v]; seq[5]

s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(len) = {my(v = vector(len), w = [0], c = 0, k = 3, m, s1); while(c < len, s1 = s(k); m = #w; if(s1 == w[m], w = concat(w, s1), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [s1]); k++); v;}

from itertools import count
from collections import Counter
from sympy import factorint
def A374673(n):
    if n==1: return 2
    c, a, l = Counter(), 0, 0
    for m in count(2):
        c += Counter(factorint(m))
        b = sum(map(int.bit_count,c.values()))
        if b==a:
            l += 1
        else:
            if l==n-1:
                return m-n
            l = 0
        a = b # Chai Wah Wu, Jul 18 2024

a(9) from Chai Wah Wu, Jul 18 2024

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A374671 Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

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A374672 Numbers k such that k! has more infinitary divisors than (k+1)!.

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A374673 a(n) is the start of the least run of exactly n consecutive positive numbers with an equal value of A177329, or -1 if no such run exists.

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