

A104477


Number of successive squarefree intervals between primes.


2



1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 3, 0, 4, 0, 4, 0, 3, 0, 5, 0, 6, 0, 4, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 0, 5, 0, 8, 0, 7, 0, 6, 0, 7, 0, 8, 0, 7, 0, 7, 0, 9, 0, 8, 0, 9, 0, 8, 0, 9, 0, 8, 0, 8, 0, 11, 0, 10, 0, 11, 0, 10, 0, 8, 0, 11, 0, 10, 0, 12, 0, 9, 0, 12, 0, 14, 0, 9, 0, 10, 0
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OFFSET

1,7


COMMENTS

Find the number (the "run length") of successive intervals [p, p'=nextprime(p)] (followed by [p', p''], then [p'', p'''] etc.) which do not contain a square. When a square (n+1)^2 is found in such an interval, this will result in a term a(2n) = 0, preceded by a(2n1) = the number of intervals of primes counted before reaching that square, i.e., between n^2 and (n+1)^2.  M. F. Hasler, Oct 01 2018


LINKS



FORMULA

a(2n) = 0: this is the interval from the greatest prime less than the (n+1)th square, through that square and up to the least prime greater than that square.  Robert G. Wilson v, Apr 23 2005
a(2n1) = the difference between the indices of the greatest prime less than (n+1)^2 and the least prime greater than n^2.  Robert G. Wilson v, Apr 23 2005


EXAMPLE

a(1)=1 because the first interval between primes (2 to 3) is free of squares.
a(2)=0 because there is a square between 3 and 5.
a(7)=2 because there are two successive squarefree intervals: 17 to 19; and 19 to 23.
a(8)=0 because between 23 and 29 there is a square: 25.


MATHEMATICA

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; PrevPrim[n_] := Block[{k = n  1}, While[ !PrimeQ[k], k ]; k]; f[n_] := If[ EvenQ[n], 0, PrimePi[ PrevPrim[(n + 3)^2/4]]  PrimePi[ NextPrim[(n + 1)^2/4]]]; Table[ f[n], {n, 100}] (* Robert G. Wilson v, Apr 23 2005 *)


PROG

(PARI) p=2; c=0; forprime(np=p+1, 1e4, if( sqrtint(p) < sqrtint(np), print1(c", ", c=0, ", "), c++); p=np) \\ For illustrative purpose. Better:
A104477(n)=if(bittest(n, 0), primepi((1+n\/=2)^2)primepi(n^2)1, 0) \\ M. F. Hasler, Oct 01 2018


CROSSREFS

Equals A014085  1 without the initial term, interleaved with 0's.


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



