A081402 a(n) = A008475(n!).

0, 2, 5, 11, 16, 30, 37, 149, 221, 369, 380, 1310, 1323, 2389, 2975, 33695, 33712, 72312, 72331, 269439, 282855, 545109, 545132, 4254514, 4269514, 8463974, 9999248, 35167130, 35167159, 71972737, 71972768, 2152347552, 2161914700

Offset

1,2

Links

Table of n, a(n) for n=1..33.

Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Formula

From Amiram Eldar, Dec 10 2024: (Start)

a(n) = 2^(n-s_2(n)) + O(sqrt(3)^n), where s_2(n) = A000120(n).

Sum_{k=1..n} a(k) = 2^(n+O(log(n))).

Both formulas from De Koninck and Verreault (2024, pp. 51-52, eq. (3.10) and (3.16)). (End)

Example

a(1) = 0 since 1! = 1 has no prime factor.
a(8) = 2^7 + 3^2 + 5 + 7 = 149 since 8! = 2^7*3^2*5*7.

Mathematica

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}]; supo[x_] := Apply[Plus, ba[x]^ep[x]]; Table[supo[w], {w, 1, 25}]

Prog

(PARI) a(n) = my(f=factor(n!)); sum(k=1, #f~, f[k, 1]^f[k, 2]); \\ Michel Marcus, Jul 09 2018

Crossrefs

Cf. A008475, A000120, A000142, A081403, A081404.

Sequence in context: A082083 A287558 A024917 * A281023 A359399 A118660

Adjacent sequences: A081399 A081400 A081401 * A081403 A081404 A081405

Keyword

nonn

Author

Labos Elemer, Mar 31 2003

Status

approved