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

255, 290, 323, 578, 1484, 2219, 2418, 2491, 4370, 4706, 5243, 6075, 7139, 7930, 9378, 10082, 10554, 10603, 12716, 15872, 16739, 18146, 18938, 22424, 22842, 25227, 25283, 25959, 26910, 28364, 28448, 30255, 33669, 33698, 34316, 34317, 38895, 40179, 41261, 43343, 43999, 47384, 60400, 62695, 64970

Offset: 1

Author

Robert Israel, Jan 01 2023

Keywords

• nonn

Comments

Numbers k such that k and k+1 are both in A051448.

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?

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..750

Crossrefs

Cf. A001414, A051448.

Programs

Maple

A:= select(proc(n) local t; issqr(add(t[1]*t[2], t=ifactors(n)[2])) end proc, {$1..10^5}):
B:= A intersect map(`-`,A,1):
sort(convert(B,list));

Mathematica

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 *)