A378900 Squares of numbers divisible by the squares of two distinct primes.

1296, 5184, 10000, 11664, 20736, 32400, 38416, 40000, 46656, 50625, 63504, 82944, 90000, 104976, 129600, 153664, 156816, 160000, 186624, 194481, 202500, 219024, 234256, 250000, 254016, 291600, 331776, 345744, 360000, 374544, 419904, 455625, 456976, 467856, 490000

Offset: 1

Author

Michael De Vlieger, Dec 12 2024

Keywords

• nonn
• easy

Comments

Also, the squares in A376936.

Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.

Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:

Type A: d = k/d, which pertain to square k (in A000290).

Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.

Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.

Since divisors d, k/d may either divide or not divide the other, there are no other cases.

In addition the following kinds of divisor pairs are also seen:

Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).

Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.

Examples

			Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
   n    a(n)   Factors of a(n)    b(n)   Type B       Type C
  -------------------------------------------------------------
   1    1296   2^4  * 3^4          36    6 * 216      24 * 54
   2    5184   2^6  * 3^4          72    6 * 864      48 * 108
   3   10000   2^4  * 5^4         100   10 * 1000     40 * 250
   4   11664   2^4  * 3^6         108    6 * 1944     24 * 486
   5   20736   2^8  * 3^4         144    6 * 3456     54 * 384
   6   32400   2^4  * 3^4 * 5^2   180   30 * 1080    120 * 270
   7   38416   2^4  * 7^4         196   14 * 2744     56 * 686
   8   40000   2^6  * 5^4         200   10 * 4000     80 * 500
   9   46656   2^6  * 3^6         216    6 * 7776     48 * 972
  10   50625   3^4  * 5^4         225   15 * 3375    135 * 375
  11   63504   2^4  * 3^4 * 7^2   252   42 * 1512    168 * 378
  12   82944   2^10 * 3^4         288    6 * 13824    54 * 1536
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.

Crossrefs

Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.

Cf. A013661, A082020.

Programs

Mathematica

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}],   IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Formula

a(n) = A036785(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - Amiram Eldar, Dec 21 2024