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A256447 Number of integers in range (prime(n)^2)+1 .. (prime(n)*prime(n+1)) whose smallest prime factor is at least prime(n): a(n) = A250477(n) - A250474(n). 7

%I

%S 2,3,3,7,5,9,6,13,23,9,28,22,12,24,39,37,17,44,32,16,53,37,53,76,46,

%T 23,43,20,49,161,48,82,23,142,27,91,90,66,103,97,41,181,41,74,39,228,

%U 228,86,45,86,130,44,217,134,141,138,46,148,106,47,261,355,116,53,109,387,166,284,65,119,181,243,198,195,122,190,268,125,265,330,78

%N Number of integers in range (prime(n)^2)+1 .. (prime(n)*prime(n+1)) whose smallest prime factor is at least prime(n): a(n) = A250477(n) - A250474(n).

%C a(n) = number of integers in range [(prime(n)^2)+1, (prime(n) * prime(n+1))] whose smallest prime factor is at least prime(n).

%C All the terms are strictly positive, because at least for the last number in the range we have A020639(prime(n)*prime(n+1)) = prime(n).

%C See the conjectures in A256448.

%H Antti Karttunen, <a href="/A256447/b256447.txt">Table of n, a(n) for n = 1..564</a>

%H A. Karttunen, <a href="https://oeis.org/plot2a?name1=A256447&amp;name2=A256448&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio a(n)/A256448(n) plotted with OEIS Plot2-script</a>

%H A. Karttunen, <a href="https://oeis.org/plot2a?name1=A256447&amp;name2=A251723&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio a(n)/A251723(n) plotted with OEIS Plot2-script</a>

%F a(n) = A250477(n) - A250474(n).

%F a(n) = A251723(n) - A256448(n).

%F a(n) = A256448(n) + A256449(n).

%F a(n) = A256468(n) + 1.

%F Other identities. For all n >= 1:

%F a(n+1) = A256446(n) - A256448(n).

%e For n=1, we have in range [(prime(1)^2)+1, (prime(1) * prime(2))], that is, in range [5,6], two numbers, 5 and 6, whose smallest prime factor (A020639) is at least 2, thus a(1) = 2.

%e For n=2, we have in range [10, 15] three numbers, {11, 13, 15}, whose smallest prime factor is at least 3, thus a(2) = 3.

%e For n=3, we have in range [26, 35] three numbers, {29, 31, 35}, whose smallest prime factor is at least prime(3) = 5, thus a(3) = 3.

%t f[n_] := Count[Range[Prime[n]^2 + 1, Prime[n] Prime[n + 1]],

%t x_ /; Min[First /@ FactorInteger[x]] >=

%t Prime@n]; Array[f, 81] (* _Michael De Vlieger_, Mar 30 2015 *)

%o (Scheme) (define (A256447 n) (- (A250477 n) (A250474 n)))

%Y One more than A256468.

%Y Cf. A020639, A250474, A250477, A251723, A256446, A256448, A256449.

%K nonn

%O 1,1

%A _Antti Karttunen_, Mar 29 2015

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Last modified March 21 11:49 EDT 2019. Contains 321368 sequences. (Running on oeis4.)