cp's OEIS Frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

%I A376936 #29 Nov 08 2024 04:08:28
%S A376936 216,432,648,864,1000,1296,1728,1944,2000,2592,2744,3375,3456,3888,
%T A376936 4000,5000,5184,5400,5488,5832,6912,7776,8000,9000,9261,10000,10125,
%U A376936 10368,10584,10648,10800,10976,11664,13500,13824,15552,16000,16200,16875,17496,17576,18000
%N A376936 Powerful numbers divisible by cubes of 2 distinct primes.
%C A376936 Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
%C A376936 Proper subset of A320966.
%C A376936 Contains A372695, A177493, and A162142. Does not contain A085986.
%H A376936 Michael De Vlieger, <a href="/A376936/b376936.txt">Table of n, a(n) for n = 1..10000</a>
%H A376936 Michael De Vlieger, <a href="/A376936/a376936.txt">Notes on this sequence</a>
%F A376936 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - _Amiram Eldar_, Nov 08 2024
%e A376936 216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
%e A376936 432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
%e A376936 Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
%e A376936    n |   a(n)   divisor pairs d X a(n)/d
%e A376936   ---------------------------------------------------------------------------
%e A376936    1 |   216:   12 X 18;
%e A376936    2 |   432:   18 X 24;
%e A376936    3 |   648:   12 X 54;
%e A376936    4 |   864:   24 X 36, 18 X 48;
%e A376936    5 |  1000:   20 X 50;
%e A376936    6 |  1296:   24 X 54;
%e A376936    7 |  1728:   18 X 96, 36 X 48;
%e A376936    8 |  1944:   12 X 162, 36 X 54;
%e A376936    9 |  2000:   40 X 50;
%e A376936   10 |  2592:   24 X 108, 48 X 54;
%e A376936   11 |  2744:   28 X 98;
%e A376936   12 |  3375:   45 X 75;
%e A376936   13 |  3456:   18 X 192, 36 X 96, 48 X 72;
%e A376936   22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
%e A376936   58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192
%t A376936 Union@ Select[
%t A376936   Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
%t A376936   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]
%Y A376936 Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.
%Y A376936 Cf. A082020, A082695.
%K A376936 nonn,easy
%O A376936 1,1
%A A376936 _Michael De Vlieger_, Oct 16 2024