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A038111 Denominator of density of integers with smallest prime factor prime(n).

%I A038111 #75 Feb 05 2025 01:52:39
%S A038111 2,6,15,105,385,1001,17017,323323,7436429,19605131,86822723,
%T A038111 3212440751,131710070791,5663533044013,266186053068611,
%U A038111 613385252723321,2783825377744303,5855632691117327,392327390304860909,27855244711645124539,2033432863950094091347,160641196252057433216413
%N A038111 Denominator of density of integers with smallest prime factor prime(n).
%C A038111 Denominator of (Product_{k=1..n-1} (1 - 1/prime(k)))/prime(n). - _Vladimir Shevelev_, Jan 09 2015
%C A038111 a(n)/a(n-1) = prime(n)/q(n) where q(n) is 1 or a prime for all n < 1000. What are the first indices for which q(n) is composite? - _M. F. Hasler_, Dec 04 2018
%H A038111 Robert Israel, <a href="/A038111/b038111.txt">Table of n, a(n) for n = 1..277</a>
%H A038111 Fred Kline and Gerry Myerson, <a href="http://math.stackexchange.com/q/867135/28555">Identity for frequency of integers with smallest prime(n) divisor</a>, Mathematics Stack Exchange, Jul 2014.
%H A038111 Vladimir Shevelev, <a href="https://doi.org/10.1155/2008/908045">Generalized Newman phenomena and digit conjectures on primes</a>, Int'l J. Math. and Math. Sci. (2008) Art. ID 908045, 1-12. See Eq. (5.8).
%F A038111 a(n) = denominator of phi(e^(psi(p_n-1)))/e^(psi(p_n)), where psi(.) is the second Chebyshev function and phi(.) is Euler's totient function. - _Fred Daniel Kline_, Jul 17 2014
%F A038111 a(n) = prime(n)*A060753(n). - _Vladimir Shevelev_, Jan 10 2015
%F A038111 a(n) = a(n-1)*prime(n)/q(n), where q(n) = 1 except for q({3, 5, 6, 10, 11, 16, 17, 18, ...}) = (2, 3, 5, 11, 7, 23, 13, 29, ...), cf. A112037. - _M. F. Hasler_, Dec 03 2018
%e A038111 From _M. F. Hasler_, Dec 03 2018: (Start)
%e A038111 The density of the even numbers is 1/2, thus a(1) = 2.
%e A038111 The density of the numbers divisible by 3 but not by 2 is 1/6, thus a(2) = 6.
%e A038111 The density of multiples of 5 not divisible by 2 or 3 is 2/30, thus a(3) = 15. (End)
%p A038111 N:= 100: # for the first N terms
%p A038111 Q:= 1: p:= 1:
%p A038111 for n from 1 to N do
%p A038111   p:= nextprime(p);
%p A038111   A[n]:= denom(Q/p);
%p A038111   Q:= Q * (1 - 1/p);
%p A038111 end:
%p A038111 seq(A[n],n=1..N); # _Robert Israel_, Jul 14 2014
%t A038111 Denominator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 1, 64} ]
%t A038111 (* _Wouter Meeussen_ *)
%t A038111 Denominator@
%t A038111 Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/
%t A038111 Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 1, 21}]
%t A038111 (* _Fred Daniel Kline_, Jul 14 2014 *)
%o A038111 (PARI) apply( A038111(n)=denominator(prod(k=1,n-1,1-1/prime(k)))*prime(n), [1..30]) \\ _M. F. Hasler_, Dec 03 2018
%Y A038111 Cf. A038110, A060753, A112037.
%K A038111 nonn,frac
%O A038111 1,1
%A A038111 _Wouter Meeussen_
%E A038111 Name edited by _M. F. Hasler_, Dec 03 2018