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A241148 Number of factorials k!, 0<=k<=n, relatively prime to n! in Fermi-Dirac arithmetic. 3
1, 2, 2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 4, 4, 4, 2, 2, 2, 5, 5, 7, 4, 3, 3, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 4, 4, 3, 2, 4, 4, 3, 3, 2, 4, 5, 5, 4, 4, 2, 2, 2, 2, 6, 5, 2, 2, 3, 3, 7, 7, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 2, 2, 2, 2, 6, 6, 4, 4, 2, 2, 2, 3, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, equivalently, the number of factorials k!, 0<=k<=n,  for which k! and n! have no common A050376-factors in their factorizations over distinct terms of A050376.

Note that 1 (=0!=1!) corresponds to an empty subset of A050376.

REFERENCES

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (Russian; MR 2000f: 11097, pp. 3912-3913).

LINKS

Amiram Eldar, Table of n, a(n) for n = 0..500

S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

EXAMPLE

0!=1, 1!=1; further we have the following factorizations of k! over distinct terms of A050376 for k = 2,3,4,5,6:

2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16. Thus, in the sense of the factorizations being considered, 6! is relatively prime to 0!,1!,2!,3!, and 4!. So a(6)=5.

MATHEMATICA

b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], _?(# == 1 &)]) // Flatten; infp[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}]; (Prime /@ Range[np])^(b /@ v) // Flatten]; infCoprimeQ[x_, y_] := Intersection[infp[x], infp[y]] == {}; a[n_] := Length @ Select[Range[0, n], infCoprimeQ[n, #] & ]; Array[a, 87, 0] (* Amiram Eldar, Sep 17 2019 *)

CROSSREFS

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139.

Sequence in context: A162487 A215924 A115101 * A172414 A266995 A051887

Adjacent sequences:  A241145 A241146 A241147 * A241149 A241150 A241151

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Apr 16 2014

EXTENSIONS

More terms from Peter J. C. Moses, Apr 18 2014

STATUS

approved

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Last modified October 19 02:41 EDT 2019. Contains 328211 sequences. (Running on oeis4.)