A338473 Numbers that can be written as the sum of two brilliant numbers (A078972).

8, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 39, 40, 41, 42, 44, 45, 46, 49, 50, 53, 55, 56, 58, 59, 60, 63, 64, 70, 74, 84, 98, 125, 127, 130, 131, 135, 136, 142, 146, 147, 149, 152, 153, 156, 157, 158, 164, 168, 170

Offset

1,1

Comments

The sequence is infinite.

There are an infinite number of term pairs (a(k), a(k + 1)) that are consecutive numbers. Indeed, if p is a prime number, then 9 + p^2 and 10 + p^2 are terms. Also, numbers 14 + p^2 and 15 + p^2 are terms.

There are also larger sequences of consecutive numbers that are terms. For example, the 21 consecutive numbers 780, 781, ..., 800 or 4184, 4185, ..., 4204 are terms.

Links

Table of n, a(n) for n=1..61.

Example

8 = 4 + 4 = A078972(1) + A078972(1), so 8 is a term.
10 = 4 + 6 = A078972(1) + A078972(2), so 10 is a term.
15 = 6 + 9 = A078972(2) + A078972(3), so 15 is a term.

Mathematica

m = 200; brils = Select[Range[m], (f = FactorInteger[#])[[;; , 2]] == {2} || f[[;; , 2]] == {1, 1} && Equal @@ IntegerLength@f[[;; , 1]] &]; Select[Range[m], Length[IntegerPartitions[#, {2}, brils]] > 0 &] (* Amiram Eldar, Dec 06 2020 *)

Prog

(Magma) f:=Factorization; br:=func<n|#Divisors(n) eq 3 or &+[d[2]:d in f(n)] eq 2 and #Intseq(f(n)[1][1]) eq #Intseq(f(n)[2][1])>; [k:k in [4..200]|exists(i){m:m in [4..k-4]|br(m) and br(k-m)}];

Crossrefs

Cf. A078972, A338474.

Sequence in context: A046031 A102758 A176815 * A336143 A347267 A220571

Adjacent sequences: A338470 A338471 A338472 * A338474 A338475 A338476

Keyword

nonn,base

Author

Marius A. Burtea, Dec 06 2020

Status

approved