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A064380
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Number of numbers that are infinitarily relatively prime to n; the infinitary EulerPhi.
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23
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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; internal format)
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OFFSET
| 2,2
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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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
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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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
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LINKS
| S. R. Finch, Unitarism and infinitarism.
Wouter Meeussen, Table of n, a(n) for n = 2..2000
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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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
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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 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2010]
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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
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MATHEMATICA
| Table[ Length[ irelprime[ n ] ], {n, 2, 128} ] with irelprime[ n ] defined in A064379.
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CROSSREFS
| Cf. A037445, A064379.
Sequence in context: A088043 A138796 A186970 * A126214 A126801 A076945
Adjacent sequences: A064377 A064378 A064379 * A064381 A064382 A064383
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KEYWORD
| nonn,nice
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 27 2001
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