A072749 -id:A072749 - cp's OEIS frontend

A072747 Counting factor 2 in the first n squarefree numbers.

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Reinhard Zumkeller, Jul 08 2002

nonn

			The first 10 squarefree numbers are: 1, 2, 3, 5, 6=2*3, 7, 10=2*5, 11, 13 and 14=2*7: 2, 6, 10 and 14 are divisible by 2, therefore a(10)=4.

Gheorghe Coserea, Table of n, a(n) for n = 1..10000

Cf. A005117, A072748, A072749, A072750, A072751.

Accumulate[If[Divisible[#, 2], 1, 0]&/@Select[Range[100], SquareFreeQ]] (* Vincenzo Librandi, Aug 23 2015 *)

n = 77; 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] % 2 == 0)); print(a); \\ Gheorghe Coserea, Aug 22 2015

a(n) ~ n/3. - Amiram Eldar, Feb 24 2021

Name clarified by Gheorghe Coserea, Aug 22 2015

A072748 Counting factor 3 in the first n squarefree numbers.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 19, 20, 20, 21, 21

Offset: 1

Reinhard Zumkeller, Jul 08 2002

nonn

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

Gheorghe Coserea, Table of n, a(n) for n = 1..10000

Cf. A005117, A072747, A072749, A072750, A072751.

Accumulate[If[Divisible[#, 3], 1, 0]&/@Select[Range[100], SquareFreeQ]] (* Vincenzo Librandi, Aug 23 2015 *)

n = 80; 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] % 3 == 0));
print(a); \\ Gheorghe Coserea, Aug 22 2015

a(n) ~ n/4. - Amiram Eldar, Feb 24 2021

Name clarified by Gheorghe Coserea, Aug 22 2015

A072750 Counting factor 7 in the first n squarefree numbers.

Original entry on oeis.org

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

Reinhard Zumkeller, Jul 08 2002

nonn

			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.

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

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

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

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

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

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

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

Name clarified by Gheorghe Coserea, Aug 23 2015

A072751 Greatest of the most frequent prime factors of squarefree numbers <= n; a(1)=1.

Original entry on oeis.org

1, 2, 3, 5, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 08 2002

a(n)=2 for n>14.

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.

Cf. A005117, A072747, A072748, A072749, A072750.

max = 200; primeFactors = FactorInteger[#][[All, 1]]& /@ Select[Range[max], SquareFreeQ]; a[n_] := Sort[ Tally[ Take[ primeFactors, n] // Flatten], Which[#1[[2]] > #2[[2]], True, #1[[2]] == #2[[2]], #1[[1]] > #2[[1]], True, False]& ][[1, 1]]; Table[a[n], {n, 1, primeFactors // Length}] (* Jean-François Alcover, Oct 14 2013 *)

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A072747 Counting factor 2 in the first n squarefree numbers.

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A072748 Counting factor 3 in the first n squarefree numbers.

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A072750 Counting factor 7 in the first n squarefree numbers.

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A072751 Greatest of the most frequent prime factors of squarefree numbers <= n; a(1)=1.

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