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

Author

Amiram Eldar, Jul 16 2024

Keywords

• nonn

Comments

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).

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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] &]

PARI

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);}