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A342163 a(n) is the number of numbers greater than 1 and up to prime(n)^2 whose prime factors are all less than or equal to prime(n).

%I A342163 #25 Jun 25 2022 11:13:38
%S A342163 2,6,15,29,60,87,137,176,247,360,422,568,689,776,923,1136,1369,1494,
%T A342163 1764,1978,2128,2451,2710,3074,3562,3870,4077,4411,4638,4995,6026,
%U A342163 6426,6987,7271,8180,8493,9134,9802,10319,11030,11767,12139,13314,13712,14329,14742
%N A342163 a(n) is the number of numbers greater than 1 and up to prime(n)^2 whose prime factors are all less than or equal to prime(n).
%H A342163 Robert Israel, <a href="/A342163/b342163.txt">Table of n, a(n) for n = 1..4000</a>
%F A342163 a(n) = A184677(n) - 1.
%e A342163 For n=3, prime(3) = 5. Then the numbers up to 5^2 = 25 that have prime factors <= 5 are 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25. So a(3) = 15.
%p A342163 A[1]:= 2: p:= 2: P:= 1:
%p A342163 f:= proc(n) local x,y;
%p A342163   x:= n;
%p A342163   do
%p A342163     y:= igcd(x,P);
%p A342163     x:= x/y;
%p A342163     if x = 1 then return true fi;
%p A342163     if y = 1 then return false fi
%p A342163   od;
%p A342163 end proc:
%p A342163 for nn from 2 to 100 do
%p A342163   q:= p; p:= nextprime(p); P:= P*q;
%p A342163   A[nn]:= A[nn-1] + p + numboccur(true,map(f, [$q^2+1 .. p^2-1]))
%p A342163 od:
%p A342163 seq(A[i],i=1..100); # _Robert Israel_, Apr 06 2021
%t A342163 Block[{nn = 46, w}, w = Array[FactorInteger[#][[All, 1]] &, Prime[nn]^2]; Table[-1 + Count[w[[1 ;; p^2]], _?(AllTrue[#, # <= p &] &)], {p, Prime@ Range@ nn}]] (* _Michael De Vlieger_, Mar 13 2021 *)
%o A342163 (PARI) forprime(n = 2, prime(35), i = 0; for(k = 2, n^2, v = factor(k)~[1,]; if(vecmax(v) <= n, i++)); print1(i", "))
%o A342163 (PARI) a(n) = my(p=prime(n)); sum(k=2, p^2, vecmax(factor(k)[,1]) <= p); \\ _Michel Marcus_, Mar 03 2021
%Y A342163 Cf. A000040, A001248, A184677.
%K A342163 nonn
%O A342163 1,1
%A A342163 _Dimitris Valianatos_, Mar 03 2021
%E A342163 Definition clarified by _Robert Israel_, Apr 06 2021