A117204 -id:A117204 - cp's OEIS frontend

A117344 Duplicate of A117204.

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 23, 26, 29, 30, 33, 34, 35, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 59, 61, 65, 66, 69, 70, 71, 74, 77, 78, 79, 82, 83, 86, 89, 91

Offset: 1

dead

A117203 Odd squarefree positive integers k such that (k-1)/2 is also squarefree.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 21, 23, 29, 31, 35, 39, 43, 47, 53, 59, 61, 67, 69, 71, 77, 79, 83, 85, 87, 93, 95, 103, 107, 111, 115, 119, 123, 131, 133, 139, 141, 143, 149, 155, 157, 159, 165, 167, 173, 179, 183, 187, 191, 195, 203, 205, 211, 213, 215, 219, 221, 223, 227

Offset: 1

Leroy Quet, Mar 02 2006

nonn

The asymptotic density of this sequence is (3/4)*A065474 = 0.2419755742... (Erdős and Ivić, 1987). - Amiram Eldar, Feb 17 2021

			21 and (21-1)/2 = 10 are both squarefree, so 21 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A065474, A117204, A117205.

with(numtheory): a:=proc(n) if n mod 2 =1 and abs(mobius(n))>0 and abs(mobius((n-1)/2))>0 then n else fi end: seq(a(n),n=2..300); # Emeric Deutsch, Mar 07 2006

fQ[n_] := Max @@ Last /@ FactorInteger@n < 2; Select[ 2Range@113 + 1, fQ@# && fQ[(# - 1)/2] &] (* Robert G. Wilson v, Apr 18 2006 *)

for (i=1,100,if(issquarefree(2*i+1) && issquarefree(i),print1(2*i+1,","))) \\ Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 23 2006

a(n) = 2*A117204(n) + 1.

More terms from Emeric Deutsch and Reinhard Zumkeller, Mar 07 2006
More terms from Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 23 2006
More terms from Robert G. Wilson v, Apr 18 2006

A117206 Squarefree positive integers k such that 2*k-1 is also squarefree.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 15, 17, 19, 21, 22, 26, 29, 30, 31, 33, 34, 35, 37, 39, 42, 43, 46, 47, 51, 53, 55, 57, 58, 62, 65, 66, 67, 69, 70, 71, 73, 78, 79, 82, 83, 87, 89, 91, 93, 94, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119, 127, 129, 130, 133

Offset: 1

Leroy Quet, Mar 02 2006

nonn

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Erdős and Ivić, 1987). - Amiram Eldar, Feb 17 2021

			10 and 2*10 - 1 = 19 are both squarefree, so 10 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A065474, A117204, A117205.

with(numtheory): a:=proc(n) if abs(mobius(n))>0 and abs(mobius(2*n-1))>0 then n else fi end: seq(a(n),n=1..160); # Emeric Deutsch, Mar 08 2006

Select[Range[200],And@@SquareFreeQ/@{#,2#-1}&] (* Harvey P. Dale, Dec 17 2012 *)

a(n) = (A117205(n) + 1)/2.

More terms from Emeric Deutsch, Mar 08 2006

A115170 Non-cubefree numbers k such that 2k+1 is also non-cubefree (A046099).

Original entry on oeis.org

40, 256, 312, 472, 688, 904, 1120, 1200, 1312, 1336, 1552, 1768, 1984, 2187, 2200, 2312, 2416, 2456, 2632, 2848, 2875, 3064, 3280, 3312, 3429, 3496, 3712, 3928, 3944, 4144, 4312, 4360, 4576, 4792, 5008, 5224, 5312, 5440, 5562, 5656, 5872, 6088, 6250, 6304, 6312, 6345, 6520, 6655, 6688, 6736, 6952

Offset: 1

Jonathan Vos Post, Mar 03 2006

The probability that a random integer is cubefree is 1/zeta(3) (see A088453).

			312 is there because 2^3 divides 312 and 5^3 divides 312*2+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..700 from R. J. Mathar)

Cf. A005384 (n and 2n+1 prime), A046099, A088453, A111153 (n and 2n+1 semiprime), A117204 (n and 2n+1 squarefree), A115228.

noncfQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 2 &]; Select[Range[7000], noncfQ[#] && noncfQ[2*# + 1] &] (* Amiram Eldar, May 25 2025 *)

Edited by Don Reble, Mar 05 2006

2875 inserted by R. J. Mathar, Dec 08 2015

A062837 Numbers k such that k, 2k+1, 3k+1 are all squarefree.

Original entry on oeis.org

2, 3, 6, 7, 10, 11, 14, 15, 19, 23, 26, 30, 34, 35, 38, 39, 42, 43, 46, 47, 51, 55, 59, 66, 70, 71, 74, 78, 79, 82, 86, 91, 95, 102, 106, 107, 110, 111, 115, 118, 119, 123, 127, 131, 134, 138, 142, 143, 146, 151, 154, 155, 159, 167, 170, 174, 178, 179, 182, 186

Offset: 1

Jason Earls, Jul 21 2001

nonn

The asymptotic density of this sequence is (7/3) * Product_{p prime} (1 - 3/p^2) = (7/3) * A206256 = 0.292802955446... (Tsang, 1985). - Amiram Eldar, Feb 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Kai-Man Tsang, The distribution of r-tuples of square-free numbers, Mathematika, Vol. 32, No. 2 (1985), pp. 265-275.

Subsequence of A005117 and A117204.

Cf. A206256.

Select[Range[200],AllTrue[{#,2#+1,3#+1},SquareFreeQ]&] (* Harvey P. Dale, Oct 20 2014 *)

j=[]; for(n=1,200, if(issquarefree(n) && issquarefree(2*n+1) && issquarefree(3*n+1),j=concat(j,n))); j

cp's OEIS Frontend

A117344 Duplicate of A117204.

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A117203 Odd squarefree positive integers k such that (k-1)/2 is also squarefree.

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A117206 Squarefree positive integers k such that 2*k-1 is also squarefree.

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A115170 Non-cubefree numbers k such that 2k+1 is also non-cubefree (A046099).

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A062837 Numbers k such that k, 2*k+1, 3*k+1 are all squarefree.

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PARI

A062837 Numbers k such that k, 2k+1, 3k+1 are all squarefree.