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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.
0
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
1,1
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.)
EXAMPLE
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.
MATHEMATICA
Select[Range@ 203, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^2 > #2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved