cp's OEIS Frontend

A294472 Squarefree numbers whose odd prime factors are all consecutive primes.

%I A294472 #35 May 27 2018 01:44:13
%S A294472 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,
%T A294472 46,47,53,58,59,61,62,67,70,71,73,74,77,79,82,83,86,89,94,97,101,103,
%U A294472 105,106,107,109,113,118,122,127,131,134,137,139,142,143,146,149,151,154,157,158,163,166
%N A294472 Squarefree numbers whose odd prime factors are all consecutive primes.
%C A294472 The union of products of any number of consecutive odd primes and twice products of any number of consecutive odd primes.
%C A294472 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]
%C A294472 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.
%C A294472 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]
%C A294472 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
%e A294472 70 is in this sequence because 2*5*7 = 70 is a squarefree number with two consecutive odd prime factors, 5 and 7.
%p A294472 N:= 1000: # to get all terms <= N
%p A294472 R:= 1,2:
%p A294472 Oddprimes:= select(isprime, [seq(i,i=3..N,2)]):
%p A294472 for i from 1 to nops(Oddprimes) do
%p A294472   p:= 1:
%p A294472   for k from i to nops(Oddprimes) do
%p A294472     p:= p*Oddprimes[k];
%p A294472     if p > N then break fi;
%p A294472     if 2*p <= N then R:= R, p, 2*p
%p A294472     else R:= R,p
%p A294472     fi
%p A294472   od;
%p A294472 od:
%p A294472 R:= sort([R]); # _Robert Israel_, Nov 01 2017
%t A294472 Select[Range@ 166, And[Union@ #2 == {1}, Or[# == {1}, # == {}] &@ Union@ Differences@ PrimePi@ DeleteCases[#1, 2]] & @@ Transpose@ FactorInteger[#] &] (* _Michael De Vlieger_, Nov 01 2017 *)
%Y A294472 Cf. A000040, A005117, A073485, A294674.
%K A294472 nonn
%O A294472 1,2
%A A294472 _Juri-Stepan Gerasimov_, Oct 31 2017
%E A294472 Definition corrected by _Michel Marcus_, Nov 01 2017