cp's OEIS Frontend

A380595 a(n) is the first nonsquarefree number k such that the n consecutive nonsquarefree numbers starting with k are in arithmetic progression.

%I A380595 #16 Jan 29 2025 22:13:44
%S A380595 4,4,16,28,28,5050,6348,144946,3348550,221167422,221167422,
%T A380595 47255689915,82462576220,1043460553364,79180770078548,
%U A380595 3215226335143218,23742453640900972,125781000834058568
%N A380595 a(n) is the first nonsquarefree number k such that the n consecutive nonsquarefree numbers starting with k are in arithmetic progression.
%C A380595 Since multiples of 4 are not squarefree, the common difference of the arithmetic progression will be 1, 2 or 4 in each case.
%C A380595 For an arithmetic progression of length 10 or more with initial term even and common difference 2 or 4, there would be an odd multiple of 9 between the first and last term.  Since multiples of 9 are not squarefree, these could not be consecutive nonsquarefree numbers.  Thus for n >= 10, the common difference must be 1, and a(n) = A045882(n).
%e A380595 a(2) = 4 because the 2 nonsquarefree numbers starting with 4 are 4, 6, forming an arithmetic progression with difference 2.
%e A380595 a(3) = 16 because the 3 nonsquarefree numbers starting with 16 are 16, 18, 20, forming an arithmetic progression with difference 2.
%e A380595 a(4) = a(5) = 28 because the 5 nonsquarefree numbers starting with 28 are 28, 32, 36, 40, 44, forming an arithmetic progression with difference 4.
%e A380595 a(6) = 5050 because the 6 nonsquarefree numbers starting with 5050 are 5050, 5052, 5054, 5056, 5058, 5060, forming an arithmetic progression with difference 2.
%e A380595 a(7) = 6348 because the 7 nonsquarefree numbers starting with 6348 are 6348, 6350, 6352, 6354, 6356, 6358, 6360, forming an arithmetic progression with difference 2.
%e A380595 a(8) = 144946, because the 8 nonsquarefree numbers starting with 144946 are 144946, 144948, 144950, 144952, 144954, 144956, 144958, 144960, forming an arithmetic progression with difference 2.
%e A380595 a(9) = 3348550, because the 9 nonsquarefree numbers starting with 3348550 are 3348550, 3348552, 3348554, 3348556, 3348558, 3348560, 3348562, 3348564, 3348566, forming an arithmetic progression with difference 2.
%p A380595 nsf:= remove(numtheory:-issqrfree, [$4..4*10^6]):
%p A380595 S:= nsf[2..-1]-nsf[1..-2]: nS:= nops(S):
%p A380595 R:= NULL:
%p A380595 for m from 1 do
%p A380595   found:= false;
%p A380595   for t from 1 to nS +1-m do
%p A380595     if nops(convert(S[t..t+m-1],set))=1 then R:= R,nsf[t]; found:= true; break fi
%p A380595   od;
%p A380595   if not found then break fi;
%p A380595 od:
%p A380595 R;
%Y A380595 Cf. A013929, A045882, A376267.
%K A380595 nonn,hard,more
%O A380595 1,1
%A A380595 _Robert Israel_, Jan 27 2025
%E A380595 a(1) = 4 prepended by _David A. Corneth_, Jan 28 2025