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A206852 Numbers N such that N/2 is a square, N/3 is a cube, and N/5 is a fifth power.

%I A206852 #59 Dec 16 2022 09:02:12
%S A206852 30233088000000,32462531054272512000000,6224724715037147546112000000,
%T A206852 34856377305871210027941888000000,28156757354736328125000000000000000,
%U A206852 6683747269421867033919422988288000000,681433858470444619689081338982912000000
%N A206852 Numbers N such that N/2 is a square, N/3 is a cube, and N/5 is a fifth power.
%C A206852 The terms must be of the form N = 2^a*3^b*5^c*m^(2*3*5) where gcd(m, 2*3*5) = 1 and a-1, b-1 and c-1 must be a multiple of 2, 3 and 5, respectively, and a, b, c must be a multiple of the two other prime factors, respectively. This gives (a, b, c) == (3*5, 2*5, 2*3) [mod 2*3*5], whence N = 2^15*3^10*5^6*n^30. - _M. F. Hasler_, Jul 22 2022
%H A206852 Georg Fischer, <a href="/A206852/b206852.txt">Table of n, a(n) for n = 1..1000</a>
%H A206852 Shyam Sunder Gupta, <a href="http://www.shyamsundergupta.com/doyouknow.htm">Do you know</a>, as of Feb 15 2012.
%H A206852 Michael Penn, <a href="https://www.youtube.com/watch?v=SzJnsX8L-wE">a sunny number puzzle!</a>, YouTube video, 2021.
%H A206852 <a href="/index/Rec#order_31">Index entries for linear recurrences with constant coefficients</a>, signature (31, -465, 4495, -31465, 169911, -736281, 2629575, -7888725, 20160075, -44352165, 84672315, -141120525, 206253075, -265182525, 300540195, -300540195, 265182525, -206253075, 141120525, -84672315, 44352165, -20160075, 7888725, -2629575, 736281, -169911, 31465, -4495, 465, -31, 1).
%F A206852 a(n) = 30233088000000 * n^30 = 2^15 * 3^10 * 5^6 * n^30. - _Charles R Greathouse IV_, Apr 25 2012
%t A206852 Table[30233088000000 * n^30, {n,1,1000}] (* _Georg Fischer_, Feb 07 2021 *)
%o A206852 (PARI) {is_A206852(n)=(n=divrem(n,3^10*5^6<<15))[2]==0 && ispower(n[1],30)} \\ replacing obsolete PARI code from 2012. - _M. F. Hasler_, Jul 22 2022
%o A206852 (PARI) a(n)=30233088000000*n^30 \\ _Charles R Greathouse IV_, Apr 25 2012
%o A206852 (Python) def A206852(n): return 30233088000000*n**30 # _M. F. Hasler_, Jul 24 2022
%o A206852 (Python)
%o A206852 def is_A206852(n):
%o A206852     for p in (2, 3, 5):
%o A206852         for e in range(n):
%o A206852             if n % p: break
%o A206852             n //= p
%o A206852         if e % 30 != 30//p: return False
%o A206852     return is_A122971(n) # _M. F. Hasler_, Jul 24 2022
%Y A206852 Cf. A000290 (squares), A000578 (cubes), A000584 (5th powers), A122971 (30th powers).
%K A206852 nonn,easy
%O A206852 1,1
%A A206852 _M. F. Hasler_, Feb 15 2012