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A359445 Numbers k such that the sums (with multiplicity) of prime factors of k and k+1 are both squares.

%I A359445 #15 Apr 12 2024 09:25:09
%S A359445 255,290,323,578,1484,2219,2418,2491,4370,4706,5243,6075,7139,7930,
%T A359445 9378,10082,10554,10603,12716,15872,16739,18146,18938,22424,22842,
%U A359445 25227,25283,25959,26910,28364,28448,30255,33669,33698,34316,34317,38895,40179,41261,43343,43999,47384,60400,62695,64970
%N A359445 Numbers k such that the sums (with multiplicity) of prime factors of k and k+1 are both squares.
%C A359445 Numbers k such that k and k+1 are both in A051448.
%C A359445 Numbers k such that k, k+1 and k+2 are all in A051448 include 34316, 594044, and 869123.  Are there numbers k for which k, k+1, k+2 and k+3 are all in A051448?
%H A359445 Robert Israel, <a href="/A359445/b359445.txt">Table of n, a(n) for n = 1..750</a>
%e A359445 a(3) = 323 is a term because 323 = 17*19 with 17+19 = 36 = 6^2 and 324 = 2^2*3^4 with 2*2 + 4*3 = 16 = 4^2.
%p A359445 A:= select(proc(n) local t; issqr(add(t[1]*t[2], t=ifactors(n)[2])) end proc, {$1..10^5}):
%p A359445 B:= A intersect map(`-`,A,1):
%p A359445 sort(convert(B,list));
%t A359445 Module[{nn=65000,sq},sq=Table[If[IntegerQ[Sqrt[Total[Times@@@FactorInteger[n]]]],1,0],{n,nn}];SequencePosition[sq,{1,1}]][[;;,1]] (* _Harvey P. Dale_, Apr 12 2024 *)
%Y A359445 Cf. A001414, A051448.
%K A359445 nonn
%O A359445 1,1
%A A359445 _Robert Israel_, Jan 01 2023