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

4, 4, 16, 28, 28, 5050, 6348, 144946, 3348550, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Author

Robert Israel, Jan 27 2025

Keywords

• nonn
• hard
• more

Comments

Since multiples of 4 are not squarefree, the common difference of the arithmetic progression will be 1, 2 or 4 in each case.

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).

Examples

			a(2) = 4 because the 2 nonsquarefree numbers starting with 4 are 4, 6, forming an arithmetic progression with difference 2.
a(3) = 16 because the 3 nonsquarefree numbers starting with 16 are 16, 18, 20, forming an arithmetic progression with difference 2.
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.
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.
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.
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.
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.
		

Crossrefs

Cf. A013929, A045882, A376267.

Programs

Maple

nsf:= remove(numtheory:-issqrfree, [$4..4*10^6]):
S:= nsf[2..-1]-nsf[1..-2]: nS:= nops(S):
R:= NULL:
for m from 1 do
  found:= false;
  for t from 1 to nS +1-m do
    if nops(convert(S[t..t+m-1],set))=1 then R:= R,nsf[t]; found:= true; break fi
  od;
  if not found then break fi;
od:
R;

Extensions

a(1) = 4 prepended by David A. Corneth, Jan 28 2025