A184677 Number of numbers <= p^2 with largest prime factor <= p, where p is the n-th prime; a(0) = 1.

1, 3, 7, 16, 30, 61, 88, 138, 177, 248, 361, 423, 569, 690, 777, 924, 1137, 1370, 1495, 1765, 1979, 2129, 2452, 2711, 3075, 3563, 3871, 4078, 4412, 4639, 4996, 6027, 6427, 6988, 7272, 8181, 8494, 9135, 9803, 10320, 11031, 11768, 12140, 13315, 13713, 14330

Offset

0,2

Comments

a(n) = #{m: m<=A001248(n) and A006530(m)<=A000040(n)} for n > 0.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Example

a(1) = #{1,2,4} = 3 = number of binary powers <= 4;
a(2) = #{1,2,3,4,6,8,9} = 7 = number of 3-smooth numbers <= 9;
a(3) = #{1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25} = 16 = number of 5-smooth numbers <= 25.

Mathematica

Block[{nn = 45, w}, w = Array[FactorInteger[#][[All, 1]] &, Prime[nn]^2]; {1}~Join~Table[Count[w[[1 ;; p^2]], _?(AllTrue[#, # <= p &] &)], {p, Prime@ Range@ nn}]] (* Michael De Vlieger, Mar 13 2021 *)

Prog

(PARI) a(n)=if(n==0, return(1)); my(p=prime(n), s=p); forfactored(k=p+1, p^2, if(vecmax(k[2][, 1])<=p, s++)); s \\ Charles R Greathouse IV, Nov 27 2017

Crossrefs

Smooth numbers: A000079, A003586, A051037, A002473, A051038, A080197, A080681, A080682, A080683.

Cf. A027424.

Sequence in context: A211379 A213180 A110585 * A224340 A240739 A301117

Adjacent sequences: A184674 A184675 A184676 * A184678 A184679 A184680

Keyword

nonn

Author

Reinhard Zumkeller, Jun 27 2011

Status

approved