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Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.
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%I #16 Jun 07 2021 01:15:39

%S 36,81,144,289,484,576,625,676,3600,7396,9801,14400,35344,40000,40804,

%T 44100,45796,56644,59049,71824,112896,121104,172225,226576,231361,

%U 254016,274576,290521,319225,362404,480249,495616,518400,527076,535824

%N Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

%C Corresponding numbers m such that m^2 = a(n) are listed in A136360.

%C Note that some numbers in A136360 are also perfect squares. The corresponding numbers k such that m = k^2 are listed in A136361.

%C Includes all nonzero members of A099764: this occurs when the two pentagonal pyramidal numbers are both equal to i^2*(i+1)/2 where i+1 is a square. - _Robert Israel_, Feb 04 2020

%H Robert Israel, <a href="/A136359/b136359.txt">Table of n, a(n) for n = 1..985</a>

%F a(n) = A136360(n)^2.

%e A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.

%e Thus a(1) = 36, a(2) = 81, a(3) = 144, a(4) = 289 that are the perfect squares in A133459.

%p N:= 200: # for terms up to N^2*(N+1)/2.

%p PP:= [seq(i^2*(i+1)/2, i=1..N)]:

%p PP2:= sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=i..N),i=1..N)},PP[-1]),list)):

%p select(issqr,PP2); # _Robert Israel_, Feb 04 2020

%t Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ]

%Y Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Dec 25 2007