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A294472
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Squarefree numbers whose odd prime factors are all consecutive primes.
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2
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1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 34, 35, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 70, 71, 73, 74, 77, 79, 82, 83, 86, 89, 94, 97, 101, 103, 105, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 143, 146, 149, 151, 154, 157, 158, 163, 166
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OFFSET
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1,2
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COMMENTS
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The union of products of any number of consecutive odd primes and twice products of any number of consecutive odd primes.
A073485 lists the squarefree numbers with no gaps in their prime factors >= prime(1), and {a(n)} lists the squarefree numbers with no gaps in their prime factors >= prime(2). If we let {b(n)} be the squarefree numbers with no gaps in their prime factors >= prime(3), ..., and let {x(n)} be the squarefree numbers with no gaps in their prime factors >= prime(y), ..., then A073485(n) >= a(n) >= b(n) >= ... >= x(n) >= ... >= A005117(n). [edited by Jon E. Schoenfield, May 26 2018]
Conjecture: if z(n) is the smallest y such that n*k - k^2 is a squarefree number with no gaps in their prime factors >= prime(y) for some k < n, then z(n) >= 1 for all n > 1.
The terms a(n) for which a(n-1) + 1 = a(n) = a(n+1) - 1 begin 2, 6, 14, 30, 106, ... [corrected by Jon E. Schoenfield, May 26 2018]
Squarefree numbers for which any two neighboring odd prime factors in the ordered list of prime factors are consecutive primes. - Felix Fröhlich, Nov 01 2017
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LINKS
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EXAMPLE
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70 is in this sequence because 2*5*7 = 70 is a squarefree number with two consecutive odd prime factors, 5 and 7.
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MAPLE
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N:= 1000: # to get all terms <= N
R:= 1, 2:
Oddprimes:= select(isprime, [seq(i, i=3..N, 2)]):
for i from 1 to nops(Oddprimes) do
p:= 1:
for k from i to nops(Oddprimes) do
p:= p*Oddprimes[k];
if p > N then break fi;
if 2*p <= N then R:= R, p, 2*p
else R:= R, p
fi
od;
od:
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MATHEMATICA
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Select[Range@ 166, And[Union@ #2 == {1}, Or[# == {1}, # == {}] &@ Union@ Differences@ PrimePi@ DeleteCases[#1, 2]] & @@ Transpose@ FactorInteger[#] &] (* Michael De Vlieger, Nov 01 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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