A185077 Numbers such that the largest prime factor equals the sum of the squares of the other prime factors.

78, 156, 234, 290, 312, 468, 580, 624, 702, 742, 936, 1014, 1160, 1248, 1404, 1450, 1484, 1872, 2028, 2106, 2320, 2496, 2808, 2900, 2968, 3042, 3744, 4056, 4212, 4498, 4640, 4992, 5194, 5616, 5800, 5936, 6084, 6318, 7250, 7488, 8112, 8410, 8424, 8715, 8996, 9126, 9280, 9962

Offset: 1

Author

Michel Lagneau, Feb 18 2011

Keywords

• nonn

Comments

Observation : it seems that the prime divisors of a majority of numbers n are of the form {2, p, q} with q = 2^2 + p^2, but there exists more rarely numbers with more prime divisors (examples : 8715 = 3*5*7*83; 153230 = 2*5*7*11*199).

Terms which are odd: 8715, 26145, 41349, 43575, 61005, 61971, 78435, ..., . - Robert G. Wilson v, Jul 02 2014

Examples

			8996 is in the sequence because the prime divisors are {2, 13, 173} and 173 = 13^2 + 2^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..725 from Robert Israel)

Crossrefs

Cf. A071140.

See also the related sequences A048261, A121518.

Programs

Maple

filter:= proc(n)
local F,f,x;
F:= numtheory:-factorset(n);
f:= max(F);
evalb(f = add(x^2,x=F minus {f}));
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 02 2014

Mathematica

Reap[Do[p = First /@ FactorInteger[n]; If[p[[-1]] == Plus@@(Most[p]^2), Sow[n]], {n, 9962}]][[2, 1]]
lpfQ[n_]:=With[{f=FactorInteger[n][[;;,1]]},Total[Most[f]^2]==Last[f]]; Select[Range[10000],lpfQ] (* Harvey P. Dale, Jul 28 2024 *)

PARI

isok(n) = {my(f = factor(n)); f[#f~, 1] == sum(i=1, #f~ - 1, f[i, 1]^2);} \\ Michel Marcus, Jul 02 2014

Extensions

Corrected by T. D. Noe, Feb 18 2011