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A064380
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Number of numbers less than n that are infinitarily relatively prime to n; the infinitary Euler phi function.
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27
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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
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OFFSET
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2,2
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COMMENTS
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Let E[n] be the set of different terms of A050376 for which n = Product_{q in E[n]}q. Put Z(n) = n^2/Product_{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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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(10000) = 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
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MAPLE
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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
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Table[ Length[ irelprime[ n ] ], {n, 2, 128} ] (* with irelprime[ n ] defined in A064379 *)
infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]]; a[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]; Array[a, 100, 2] (* Amiram Eldar, Mar 26 2023 *)
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PROG
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(PARI) isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
a(n) = sum(j = 1, n-1, isinfcoprime(j, n)); \\ Amiram Eldar, Mar 26 2023
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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