A236575 Number of primes between successive numbers that are not squarefree.

2, 0, 1, 1, 1, 1, 1, 0, 0, 0, 2, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 1

Author

Michel Lagneau, Jan 29 2014

Keywords

• nonn

Comments

It seems that a(n) <= 2. [This is true since the maximal gap between nonsquarefree numbers is 4 and 1 in every 3 consecutive numbers is divisible by 3. - Amiram Eldar, Feb 09 2021]

Examples

			A013929(n) = 4, 8, 9, 12, 16, 18, 20, ...
a(1) = 2 because there exists 2 primes between 4 and 8;
a(2) = 0 because there are no prime between 8 and 9;
a(3) = 1 because there exists 1 prime between 9 and 12.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A013929.

Programs

Maple

sqf:={}:t:= n-> product(ithprime(k), k=1..n): for n from 1 to 400 do:if t(n) mod n <>0 then sqf:=sqf union {n} fi od:n1:=nops(sqf):
for m from 1 to n1-1 do :c:=0:i1 :=sqf[m] :i2 :=sqf[m+1] :for p from i1+1 to i2-1 do:if type(p,prime)=true then c:=c+1:else fi:od: printf(`%d, `,c):od:

Mathematica

lst={};aa={};bb={};Do[If[MemberQ[aa,EulerPhi[n]/n],AppendTo[bb,n],AppendTo[aa,EulerPhi[n]/n]],{n,1,1000}];Do[p=0;Do[If[PrimeQ[a],p++],{a,bb[[n]]+1,bb[[n+1]]-1}];AppendTo[lst,p],{n,100}];lst