A072750 Counting factor 7 in the first n squarefree numbers.

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

Offset: 1

Author

Reinhard Zumkeller, Jul 08 2002

Keywords

• nonn

Examples

			The first 10 squarefree numbers are: 1, 2, 3, 5, 6=2*3, 7, 10=2*5, 11, 13 and 14=2*7: 7 and 14 are divisible by 7, therefore a(10)=2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A008589, A072747, A072748, A072749, A072751.

Programs

Haskell

a072750 n = a072750_list !! (n-1)
a072750_list = scanl1 (+) $ map ((0 ^) . (`mod` 7)) a005117_list
-- Reinhard Zumkeller, Mar 25 2013

Maple

N:= 1000: # to use the squarefree numbers <= N
M:= map(proc(t) if numtheory:-issqrfree(t) then if t mod 7 = 0 then 1 else 0 fi fi end proc, [$1..N]):
ListTools:-PartialSums(M); # Robert Israel, Aug 23 2015

Mathematica

With[{sf=Select[Range[200],SquareFreeQ]},Accumulate[If[Divisible[#,7],1,0]&/@sf]] (* Harvey P. Dale, Mar 21 2013 *)

PARI

n = 94; k = 0; bag = List(); a = vector(n);
until(n == 0, k++; if (issquarefree(k), listput(bag, k); n--));
for (i=2, #bag, a[i] = a[i-1] + (bag[i] % 7 == 0));
print(a); \\ Gheorghe Coserea, Aug 23 2015

Formula

a(n) - a(n-1) = 1 if A005117(n) is in A008589, otherwise 0. - Robert Israel, Aug 23 2015

a(n) ~ n/8. - Amiram Eldar, Feb 24 2021

Extensions

Name clarified by Gheorghe Coserea, Aug 23 2015