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A064380 Number of numbers that are infinitarily relatively prime to n; the infinitary EulerPhi. 23
1, 2, 3, 4, 3, 6, 4, 8, 5, 10, 7, 12, 8, 9, 15, 16, 11, 18, 13, 14, 14, 22, 10, 24, 16, 18, 19, 28, 13, 30, 20, 22, 21, 25, 26, 36, 24, 27, 18, 40, 17, 42, 32, 33, 29, 46, 34, 48, 32, 36, 39, 52, 24, 42, 27, 40, 37, 58, 30, 60, 40, 49, 48, 50, 30, 66, 51, 49, 35, 70, 34, 72, 48 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Not the same as A091732.

Let E[n] be the set of different terms of A050376 for which n=Prod{q in E[n]}q. Put Z(n)=n^2/Prod{q in E[n]}(q+1). Then a(n)=Z(n)+o(n^eps), where eps>0 arbitrary small. In fact, in the limits of [2,1000] we have for 636 numbers |a(n)-Z(n)|<=1/2, for 242 numbers 1/2<|a(n)-Z(n)|<=1, for 117 numbers 1<|a(n)-Z(n)|<2 and only for 4 numbers(namely, 308,738,846 and 966) 2<=|a(n)-Z(n)|<3. - Vladimir Shevelev, Apr 17 2010

REFERENCES

V. S. Abramovich(Shevelev), On an analog of the Euler function, Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu) (1981) No. 2, 13-17.

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

LINKS

Wouter Meeussen, Table of n, a(n) for n = 2..2000

S. R. Finch, Unitarism and infinitarism.

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.

FORMULA

a(n) = Sum{t_1>=0} Sum{t_2>=0}... Sum{t_m>=0} (-1)^(t_1+...+t_m} *floor(n/(q_1^t_1*...*q_m^t_m)), where q_i are distinct terms of A050376, such that n=q_1*...*q_m. - Vladimir Shevelev, Apr 17 2010

EXAMPLE

irelprime[6]={1, 4, 5} because iDivisors[6]={1, 2, 3, 6} and iDivisors[4]={1, 4} so 4 is infinitary_relatively_prime to 6 since it lacks common infinitary divisors with 6.

For n = 2 ..8 irelprime[n] gives {1}, {1,2}, {1,2,3}, {1,2,3,4}, {1,4,5}, {1,2,3,4,5,6}, {1,3,5,7}

Let n=10000=16*625 (16 and 625 are terms of A050376). Then a(n) = Sum{t_1>=0} Sum{t_2>=0}(-1)^(t_1+t_2) *floor(16*625/(16^t_1*625^t_2)) =16*625 -16 -625 +1 +floor(625/16) -floor(625/256)=9397. Note that, Z(n)=9396.7 - Vladimir Shevelev, Apr 17 2010

MAPLE

maxpowp := proc(p, n) local f; for f in ifactors(n)[2] do if op(1, f) = p then return op(2, f) ; end if; end do: return 0 ; end proc:

isidiv := proc(d, n) local n2, d2, p, j; if n mod d <> 0 then return false; end if; for p in numtheory[factorset](n) do n2 := maxpowp(p, n) ; n2 := convert(n2, base, 2) ; d2 := maxpowp(p, d) ; d2 := convert(d2, base, 2) ; for j from 1 to nops(d2) do if op(j, n2) = 0 and op(j, d2) <> 0 then return false; end if; end do: end do; return true; end proc:

idivisors := proc(n) local a, d; a := {} ; for d in numtheory[divisors](n) do if isidiv(d, n) then a := a union {d} ; end if; end do: a ; end proc:

isInfrelpr := proc(n, m) idivisors(n) intersect idivisors(m) = {1} ; end proc:

A064380 := proc(n) option remember; local a; a := 0 ; for m from 1 to n-1 do if isInfrelpr(m, n) then a := a+1 ; end if; end do ; a ; end proc: # R. J. Mathar, Feb 19 2011

MATHEMATICA

Table[ Length[ irelprime[ n ] ], {n, 2, 128} ] with irelprime[ n ] defined in A064379.

CROSSREFS

Cf. A037445, A064379.

Sequence in context: A248376 A138796 A186970 * A229110 A229949 A126214

Adjacent sequences:  A064377 A064378 A064379 * A064381 A064382 A064383

KEYWORD

nonn,nice

AUTHOR

Wouter Meeussen, Sep 27 2001

STATUS

approved

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.