A061398 -id:A061398 - cp's OEIS frontend

A077381 Number of squarefree numbers between successive squares (exclusive).

2, 3, 5, 5, 7, 8, 8, 11, 11, 14, 14, 14, 17, 19, 18, 20, 22, 20, 24, 26, 28, 26, 28, 30, 31, 32, 33, 36, 34, 37, 40, 36, 43, 42, 44, 46, 47, 46, 49, 48, 48, 51, 50, 56, 55, 57, 58, 60, 63, 59, 63, 63, 63, 69, 70, 67, 71, 71, 73, 71, 74, 78, 76, 78, 81, 79, 84, 83, 87, 85, 84, 87

Offset: 1

Amarnath Murthy, Nov 06 2002

nonn

			a(1) = 2 because there are 2 squarefree integers between 1^2 and 2^2: 2 and 3.
a(3) = 5 = number of squarefree numbers between 3^2 and 4^2: 10, 11, 13, 14 and 15.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Gabriel Mincu and Laurenţiu Panaitopol, On some properties of squarefree and squareful numbers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 49 (97), No. 1 (2006), pp. 63-68; alternative link.

Cf. A005117, A061398.

a:= n-> nops(select(numtheory[issqrfree], [$n^2+1..(n+1)^2-1])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 16 2019

Table[Count[Range[n^2 + 1, (n + 1)^2 - 1], _?(SquareFreeQ[#] &)], {n, 1, 80}]
(* Harvey P. Dale, Jan 25 2014 *)

a(n)=s=0;for(i=n^2+1,(n+1)^2,if(issquarefree(i),s=s+1));return(s); \\ corrected by Hugo Pfoertner, Jul 16 2019

From Amiram Eldar, Feb 16 2021: (Start)
a(n) > n for all n (Mincu and Panaitopol, 2006).
a(n) ~ (12/Pi^2) * n. (End)

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 23 2004

Name clarified by Hugo Pfoertner, Jul 16 2019

A373643 Number of k between consecutive primes such that k is neither squarefree nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 2, 3, 2, 1, 1, 1, 1, 2, 1, 2, 3, 3, 0, 1, 1, 1, 5, 0, 3, 0, 4, 1, 3, 2, 1, 3, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 2, 1, 5, 1, 2, 2, 1, 3, 2, 0, 3, 6, 1, 1, 2, 4, 3, 4, 1, 3, 1, 3, 3, 3, 1, 3, 2, 1, 3, 3, 1, 4, 1, 1, 2, 2, 3

Offset: 1

Michael De Vlieger, Dec 03 2024

			Let S = A126706, the sequence of k neither squarefree nor prime powers.
a(1..4) = 0 since S(1) = 12.
a(5) = 1 since (11, 12, 13) contains S(1) = 12.
a(6) = 0 since (13, 14, 15, 16, 17) contains no number in S.
a(7) = 1 since (17, 18, 19) contains S(2) = 18.
a(8) = 1 since (19, 20, 21, 22, 23) contains S(3) = 20.
a(9) = 2 since (23, 24, 25, 26, 27, 28, 29) contains S(4) = 24 and S(5) = 28, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001223, A061398, A080101, A126706.

Table[Count[Range[Prime[i] + 1, Prime[i + 1] - 1], _?(Nor[SquareFreeQ[#], PrimePowerQ[#]] &)], {i, 120}]

a(n) = A001223(n) - A061398(n) - A080101(n) - 1.

A378111 a(n) is the least prime p such that there are exactly n squarefree numbers strictly between p and the next prime, or -1 if there is no such p.

Original entry on oeis.org

2, 5, 13, 31, 89, 139, 113, 199, 211, 317, 1759, 1381, 1951, 887, 4523, 2179, 2477, 4831, 5351, 4297, 1327, 9973, 14107, 19333, 16141, 20809, 15683, 37907, 28229, 58831, 31907, 19609, 25471, 40289, 114493, 43331, 44293, 34061, 191353, 31397, 107377, 134513, 186481, 448451, 175141, 332317, 188029

Offset: 0

Robert Israel, Nov 29 2024

nonn

a(n) = A000040(k) where k is the least number such that A061398(k) = n.

			a(3) = 31 because there are 3 squarefree numbers between 31 and the next prime 37, namely 33, 34 and 35, and 31 is the least prime that works.

David A. Corneth, Table of n, a(n) for n = 0..253 (first 144 terms from Robert Israel)
David A. Corneth, PARI program

Cf. A000040, A005117, A061398, A377430.

V:= Array(0..100): count:= 0: q:= 2:
for k from 1 while count < 101 do
  p:= q; q:= nextprime(q);
  v:= nops(select(numtheory:-issqrfree,[$p+1 .. q-1]));
  if v <= 100 and V[v] = 0 then
    V[v]:= p; count:= count+1;
  fi
od:

A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

Original entry on oeis.org

0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625

Offset: 1

Gus Wiseman, Feb 06 2025

nonn

A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061399 counts nonsquarefree integers between primes, see A068361, A061398, A068360, A377783, A378086.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers < prime(n), see A373198, A337030.
A112925 gives the greatest squarefree number between primes, least A112926.
Cf. A057627, A065890, A378032 (differences A378034), A378033 (differences A378036).
Cf. A072284, A120327, A224363, A378040, A378082, A378084.

y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]

a(n) = A378086(A068361(n)) = A378086(A068361(n)+1).

cp's OEIS Frontend

A077381 Number of squarefree numbers between successive squares (exclusive).

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A373643 Number of k between consecutive primes such that k is neither squarefree nor prime powers.

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A378111 a(n) is the least prime p such that there are exactly n squarefree numbers strictly between p and the next prime, or -1 if there is no such p.

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A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

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