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A342163 a(n) is the number of numbers greater than 1 and up to prime(n)^2 such that their prime factors are all smaller or equal to prime(n). 1
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 (list; graph; refs; listen; history; text; internal format)
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 until 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: A331773 A033286 A182724 * A098651 A087427 A031126

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

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Last modified September 26 03:19 EDT 2021. Contains 347664 sequences. (Running on oeis4.)