A038838 Numbers that are divisible by the square of an odd prime.

9, 18, 25, 27, 36, 45, 49, 50, 54, 63, 72, 75, 81, 90, 98, 99, 100, 108, 117, 121, 125, 126, 135, 144, 147, 150, 153, 162, 169, 171, 175, 180, 189, 196, 198, 200, 207, 216, 225, 234, 242, 243, 245, 250, 252, 261, 270, 275, 279, 288, 289, 294, 297, 300, 306

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Condition 1 of Theorem 7.5 (Robinson, 1979) includes: "k is a multiple of a square of an odd prime." - Jonathan Vos Post, Aug 06 2007

If m is a term, every k*m with k > 1 is another term and the primitive terms are the square of odd primes. The subsequence of odd terms is A053850 while the even terms 18, 36, 50, 54, 72, 90, 98, ... are exactly twice the terms of this sequence. - Bernard Schott, Nov 20 2020

The asymptotic density of this sequence is 1 - 8/Pi^2 = 0.189430... - Amiram Eldar, Nov 21 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM ITEM 45.
• Raphael M. Robinson, Multiple tiling of n-dimensional space by unit cubes, Math. Z., Vol. 166 (1979), pp. 225-264.
• Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., Vol. 42, No. 2 (2005), pp. 181-211; Robinson theorem cited on p. 204.

Crossrefs

Cf. A000040, A065091, A122132 (complement).

Cf. A013929 (supersequence of nonsquarefrees).

Subsequences: A001248 \ {2} (primitives), A053850 (odds), A036785 (divisible by the squares of two distinct primes).

Subsequence of A167662. - Reinhard Zumkeller, Nov 08 2009

Programs

PARI

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 22 2006 */

PARI

list(lim)=my(v=List(),n,e,t); forfactored(k=9,lim\1, e=k[2][,2]; t=#e; n=k[1]; if(if(n%2 && t, vecmax(e)>1, t>1, vecmax(e[2..t])>1, 0), listput(v, k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

Formula

{a(n)} = {j such that for some k>1 A001248(k)|j} = {j such that for some k>0 (A065091(k)^2)|j}. - Jonathan Vos Post, Aug 06 2007

A008966(A000265(a(n))) = 0. - Reinhard Zumkeller, Nov 08 2009