The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A177329

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A177329

Number of factors in the representation of n! as a product of distinct terms of A050376.
(history; published version)

#33 by Michel Marcus at Sat Aug 24 06:00:15 EDT 2024

STATUS

reviewed

approved

#32 by Joerg Arndt at Sat Aug 24 05:44:15 EDT 2024

STATUS

proposed

reviewed

#31 by Amiram Eldar at Sat Aug 24 04:20:42 EDT 2024

STATUS

editing

proposed

#30 by Amiram Eldar at Sat Aug 24 04:03:17 EDT 2024

FORMULA

a(n) = sum Sum_{i} A000120(e_i), where n! = product Product_{i} p_i^e_i is the prime factorization of n!.

CROSSREFS

Cf. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.

#29 by Amiram Eldar at Sat Aug 24 04:02:32 EDT 2024

FORMULA

a(n) = A064547(n!). [_- _R. J. Mathar_, May 28 2010]

CROSSREFS

Cf. A050376, A176525, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A064380, A050292A176525.

#28 by Amiram Eldar at Sat Aug 24 04:00:58 EDT 2024

REFERENCES

V. Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

#27 by Amiram Eldar at Sat Aug 24 04:00:26 EDT 2024

PROG

(PARI) a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[, 2])); \\ Amiram Eldar, Aug 24 2024

#26 by Amiram Eldar at Sat Aug 24 03:59:28 EDT 2024

MATHEMATICA

b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], _?(# == 1 &)]) // Flatten; a[n_] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; Length[(b /@ v) // Flatten]]; Array[a, 77, 2] (* Amiram Eldar, Sep 17 2019 *)

f[p_, e_] := DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n!]; Array[a, 100, 2] (* Amiram Eldar, Aug 24 2024 *)

STATUS

approved

editing

Discussion

Sat Aug 24

04:00

Amiram Eldar: I replaced my (correct) code with a faster and simpler version.

#25 by OEIS Server at Thu Aug 22 12:44:00 EDT 2024

LINKS

Chai Wah Wu, <a href="/A177329/b177329_1.txt">Table of n, a(n) for n = 2..10000</a> (terms 2..1000 from Amiram Eldar)

#24 by Michel Marcus at Thu Aug 22 12:44:00 EDT 2024

STATUS

reviewed

approved

Discussion

Thu Aug 22

12:44

OEIS Server: Installed new b-file as b177329.txt.  Old b-file is now b177329_1.txt.