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).

2, 6, 15, 29, 60, 87, 137, 176, 247, 360, 422, 568, 689, 776, 923, 1136, 1369, 1494, 1764, 1978, 2128, 2451, 2710, 3074, 3562, 3870, 4077, 4411, 4638, 4995, 6026, 6426, 6987, 7271, 8180, 8493, 9134, 9802, 10319, 11030, 11767, 12139, 13314, 13712, 14329, 14742

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..4000

Formula

a(n) = A184677(n) - 1.

Example

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.

Maple

A[1]:= 2: p:= 2: P:= 1:
f:= proc(n) local x, y;
  x:= n;
  do
    y:= igcd(x, P);
    x:= x/y;
    if x = 1 then return true fi;
    if y = 1 then return false fi
  od;
end proc:
for nn from 2 to 100 do
  q:= p; p:= nextprime(p); P:= P*q;
  A[nn]:= A[nn-1] + p + numboccur(true, map(f, [$q^2+1 .. p^2-1]))
od:
seq(A[i], i=1..100); # Robert Israel, Apr 06 2021

Mathematica

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 *)

Prog

(PARI) forprime(n = 2, prime(35), i = 0; for(k = 2, n^2, v = factor(k)~[1, ]; if(vecmax(v) <= n, i++)); print1(i", "))
(PARI) a(n) = my(p=prime(n)); sum(k=2, p^2, vecmax(factor(k)[, 1]) <= p); \\ Michel Marcus, Mar 03 2021

Crossrefs

Cf. A000040, A001248, A184677.

Sequence in context: A033286 A182724 A374218 * A098651 A087427 A370440

Adjacent sequences: A342160 A342161 A342162 * A342164 A342165 A342166

Keyword

nonn

Author

Dimitris Valianatos, Mar 03 2021

Extensions

Definition clarified by Robert Israel, Apr 06 2021

Status

approved