A112929 -id:A112929 - cp's OEIS frontend

A061400 Primes p such that there is no squarefree number between p and the next prime.

2, 3, 11, 17, 59, 71, 97, 107, 149, 179, 191, 197, 227, 239, 269, 311, 347, 349, 419, 431, 521, 599, 659, 809, 827, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1277, 1319, 1427, 1447, 1451, 1487, 1607, 1619, 1663, 1667, 1787, 1871, 1931, 1949, 1997, 2027, 2087

Offset: 1

Labos Elemer, Jun 07 2001

nonn

Primes in sequence A112925. - Leroy Quet, Oct 06 2005

			Between 71 and 73, the only composite is 72 = 2*2*2*3*3, not squarefree. Each of the integers between 97 and 101 has at least one squared divisor.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005117, A013929.

Cf. A112925, A112926, A112928, A112929, A112930.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: A:=[seq(a(m),m=1..400)]: b:=proc(k) if isprime(A[k])=true then A[k] else fi end: seq(b(i),i=1..400); # Emeric Deutsch, Oct 14 2005

Select[Prime@ Range@ 310, Count[Range[# + 1, NextPrime@ # - 1], k_ /; SquareFreeQ@ k] == 0 &] (* Michael De Vlieger, Feb 19 2017 *)

{ n=0; p=2; forprime (q=3, 109621, c=0; for (i=p+1, q-1, c+=issquarefree(i); if (c, break)); if (c==0, write("b061400.txt", n++, " ", p)); p=q ) } \\ Harry J. Smith, Jul 22 2009

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A112928 Primes in sequence A112926.

Original entry on oeis.org

3, 5, 13, 19, 61, 73, 101, 109, 151, 181, 193, 199, 229, 241, 271, 313, 349, 353, 421, 433, 523, 601, 661, 811, 829, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1279, 1321, 1429, 1451, 1453, 1489, 1609, 1621, 1667, 1669, 1789, 1873, 1933, 1951, 1999, 2029

Offset: 1

Leroy Quet, Oct 06 2005

nonn

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

Cf. A061400, A112925, A112926, A112929, A112930.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: A:=[seq(a(m),m=1..400)]: b:=proc(k) if isprime(A[k])=true then A[k] else fi end: seq(b(i),i=1..400); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Select[Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 300}], PrimeQ]] (* Michael De Vlieger, Sep 11 2017 *)

More terms from Emeric Deutsch, Oct 14 2005

A112930 a(n) = order of n-th term of A112926 among squarefree integers.

Original entry on oeis.org

3, 4, 5, 7, 9, 10, 13, 14, 17, 19, 21, 25, 28, 30, 32, 34, 38, 39, 43, 46, 47, 51, 53, 57, 62, 63, 65, 68, 69, 72, 79, 82, 85, 87, 93, 94, 97, 101, 104, 106, 110, 111, 118, 119, 122, 123, 131, 140, 142, 143, 146, 150, 151, 155, 159, 163, 167, 168, 171, 173

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			The 5th term of A112926 is 13 and 13 is the 9th squarefree integer (with 1 counted as the first squarefree integer). So a(5) = 9.

Michael De Vlieger and Diana Mecum, Table of n, a(n) for n = 1..10000 (first 450 terms from Diana Mecum)

Cf. A061400, A112925, A112926, A112928, A112929.

With[{k = 120, s = Select[Range[10^4], SquareFreeQ]}, Flatten@ Array[Position[s, SelectFirst[Range[Prime@ # + 1, Prime@ # + k], SquareFreeQ]] &, 60]] (* Michael De Vlieger, Aug 16 2017 *)

More terms from Diana L. Mecum, Jun 13 2007

A169646 Number of squarefree numbers of form k*n, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 0, 3, 7, 0, 8, 5, 6, 0, 11, 0, 12, 0, 8, 9, 15, 0, 0, 10, 0, 0, 17, 8, 19, 0, 13, 13, 15, 0, 23, 15, 17, 0, 26, 11, 28, 0, 0, 18, 30, 0, 0, 0, 21, 0, 32, 0, 25, 0, 23, 23, 36, 0, 37, 25, 0, 0, 30, 18, 41, 0, 29, 22, 44, 0, 45, 30, 0, 0, 36, 22, 49, 0, 0, 32, 51, 0, 41, 34

Offset: 1

Reinhard Zumkeller, Apr 05 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000040, A008966, A065473, A073311, A112929, A118259.

[&+[MoebiusMu(k*n)^2: k in [1..n]]: n in [1..80]]; // Vincenzo Librandi, Jul 25 2019

with(numtheory): seq(add(mobius(n*i)^2, i = 1 .. n), n = 1 .. 90); # Ridouane Oudra, Jul 24 2019

Count[#,?SquareFreeQ]&/@Table[k*n,{n,90},{k,n}] (* _Harvey P. Dale, Sep 05 2012 *)
Table[Sum[MoebiusMu[i n]^2, {i, n}], {n, 100}] (* Vincenzo Librandi, Jul 25 2019 *)

a(n) = sum(i = 1, n, issquarefree(i*n)); \\ Amiram Eldar, May 12 2025

a(n) = A008966(n)*A073311(n).
a(A000040(n)) = A112929(n).
a(n) = Sum_{i=1..n} A008966(n*i). - Ridouane Oudra, Jul 24 2019
a(n) = (A118259(n) - A118259(n-1))/2, for n>1. - Ridouane Oudra, May 04 2025
Sum_{k=1..n} a(k) ~ c * n / 2, where c = Product_{p prime} (1 - (3*p-2)/(p^3)) (A065473). - Amiram Eldar, May 12 2025

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A061400 Primes p such that there is no squarefree number between p and the next prime.

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A112928 Primes in sequence A112926.

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A112930 a(n) = order of n-th term of A112926 among squarefree integers.

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A169646 Number of squarefree numbers of form k*n, 1 <= k <= n.

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