A290965 Let n = p1^e1*...*pj^ej*...*pm^em be the prime factorization of n > 1, with the primes in increasing order and ej > 0. If for some j < m the sum p1^2 + ... + pj^2 > p(j+1), then n is in the sequence.

6, 12, 15, 18, 21, 24, 30, 35, 36, 42, 45, 48, 54, 55, 60, 63, 65, 66, 70, 72, 75, 77, 78, 84, 85, 90, 91, 95, 96, 102, 105, 108, 110, 114, 115, 119, 120, 126, 130, 132, 133, 135, 138, 140, 143, 144, 147, 150, 154, 156, 161, 162, 165, 168, 170, 174, 175, 180, 182, 186, 187, 189, 190, 192, 195, 198, 203

Offset: 1

Author

Richard Locke Peterson, Aug 15 2017

Keywords

• nonn

Comments

Sequence is a semigroup, since it is closed under multiplication, an associative operation--in fact, it is provably superclosed, i.e., a product of a term in sequence and an arbitrary number is a term in the sequence since the preexisting primes will still be in the new number.

Density: There are 28 terms in the sequence less than 100. Using WolframAlpha, 72% of numbers from 10^20 + 1 through 10^20 + 50 were found to be in the sequence.

Other facts: No primes or prime powers are in the sequence.

Related sequences: Some other sequences that are superclosed semigroups are the counting numbers, the numbers that are not squarefree, and the numbers with initial product in factorization greater than a later prime in the factorization. (See crossrefs.)

Examples

			6 = 2*3 is a term since 2^2 > 3.
1095 = 3*5*73 is a term because 3^2 > 5.
10, 20, and 100 are not terms since 2^2 < 5.
66 = 2*3*11 and 78 = 2*3*13 are terms since 2^2 + 3^2 > 11 and 2^2 + 3^2 = 13.
975560 = 2^3*5*29^3 is a term since 2^2 + 5^2 = 29.
		

Crossrefs

Cf. A000027, A013929, A289484.

Programs

Mathematica

Select[Range@ 203, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^2 > #2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)