

A241139


Number of nonprimes in factorization of n! over distinct terms of A050376.


4



0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7, 7, 4, 4, 5, 5, 6, 6, 8, 9, 10, 10, 9, 9, 11, 11, 12, 12, 10, 9, 8, 8, 9, 10, 11, 11, 12, 12, 11, 12, 14, 14, 16, 15, 15, 15, 13, 13, 14, 14, 14, 14, 16, 16, 16, 16, 17, 19, 21, 21, 18, 18, 19, 16, 14, 14, 16, 16, 17
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OFFSET

2,5


REFERENCES

V. S. Shevelev, Multiplicative functions in the FermiDirac arithmetic, Izvestia Vuzov of the NorthCaucasus region, Nature sciences 4 (1996), 2843 [Russian].


LINKS

Amiram Eldar, Table of n, a(n) for n = 2..1000
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, 136.


FORMULA

a(n) = A177329(n)  A055460(n).


EXAMPLE

Factorization of 4! over distinct terms of A050376 is 4! = 2*3*4. This factorization contains only one A050376nonprime. So a(4)=1.


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[Select[(b /@ v) // Flatten, # > 1 &]]]; Array[a, 73, 2] (* Amiram Eldar, Sep 17 2019 *)


PROG

(PARI) a(n)={my(f=factor(n!)[, 2]); sum(i=1, #f~, hammingweight(f[i]>>1))} \\ Andrew Howroyd, Sep 17 2019


CROSSREFS

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124.
Sequence in context: A342678 A108356 A239499 * A294242 A241092 A055656
Adjacent sequences: A241136 A241137 A241138 * A241140 A241141 A241142


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Apr 16 2014


EXTENSIONS

More terms from Peter J. C. Moses, Apr 17 2014


STATUS

approved



