A338474 -id:A338474 - cp's OEIS frontend

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

Marius A. Burtea, Dec 06 2020

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.

			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.

Cf. A078972, A338474.

f:=Factorization; br:=func; [k:k in [4..200]|exists(i){m:m in [4..k-4]|br(m) and br(k-m)}];

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 *)

cp's OEIS Frontend

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

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