Oleg P. Kirillov - cp's OEIS frontend

A235933 Numbers coprime to 35.

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 96, 97, 99

Offset: 1

Oleg P. Kirillov, Jan 17 2014

The asymptotic density of this sequence is 24/35. - Amiram Eldar, Oct 23 2020

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1).

Cf. A160547 (numbers coprime to 31), A229968 (numbers coprime to 33), A204458 (numbers coprime to 34), A007310 (numbers coprime to 36).

Cf. A045572 (numbers not divisible by 5 or 2), A229829 (numbers not divisible by 5 or 3), A047201 (numbers not divisible by 5), A236207 (numbers not divisible by 5 or 11).

a235933 n = a235933_list !! (n-1)
a235933_list = filter ((== 1) . gcd 35) [1..]
-- Reinhard Zumkeller, Mar 27 2014

[n: n in [1..100] | GCD(n,35) eq 1]; // Bruno Berselli, Mar 27 2014

Select[Range[100], GCD[#, 35] == 1 &] (* Bruno Berselli, Mar 27 2014 *)

[i for i in range(100) if gcd(i, 35) == 1] # Bruno Berselli, Mar 27 2014

Signature corrected from Georg Fischer, Feb 07 2021

Erroneous recurrence removed from Bruno Berselli, Feb 08 2021

A236204 Numbers not divisible by 2, 3 or 11.

Original entry on oeis.org

1, 5, 7, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 133, 137, 139, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175, 179, 181, 185, 191, 193, 197, 199, 203, 205

Offset: 1

Oleg P. Kirillov, Jan 20 2014

All primes except 2, 3 and 11 are in this sequence. Any product of terms is also a term in the sequence. - Alonso del Arte, Feb 04 2014
Also integers n such that least prime factor of 32^n-1 is 31. - Giovanni Resta, Mar 22 2014
Numbers coprime to 66. The asymptotic density of this sequence is 10/33. - Amiram Eldar, Oct 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A235583.

Select[2Range[125] - 1, GCD[#, 66] == 1 &] (* Alonso del Arte, Feb 04 2014 *)

is(n)=gcd(n,66)==1 \\ Charles R Greathouse IV, Mar 26 2014

a(n) = a(n-1) + a(n-20) - a(n-21). - Charles R Greathouse IV, Mar 26 2014
For n > 20, a(n) = a(n - 20) + 66. - Zak Seidov, Mar 27 2014
G.f.: x*(x^20 + 4*x^19 + 2*x^18 + 6*x^17 + 4*x^16 + 2*x^15 + 4*x^14 + 2*x^13 + 4*x^12 + 2*x^11 + 4*x^10 + 2*x^9 + 4*x^8 + 2*x^7 + 4*x^6 + 2*x^5 + 4*x^4 + 6*x^3 + 2*x^2 + 4*x + 1)/(x^21 - x^20 - x + 1). - Chai Wah Wu, Aug 03 2020

A236207 Numbers not divisible by 5 or 11.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 84, 86, 87, 89, 91, 92, 93, 94, 96, 97

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1).

Intersection of A047201 and A160542.

Select[Range[100], Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[100], Or @@ Divisible[#, {5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)

A236217 Numbers not divisible by 3, 5 or 11.

Original entry on oeis.org

1, 2, 4, 7, 8, 13, 14, 16, 17, 19, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119, 122, 124, 127, 128

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 165. The asymptotic density of this sequence is 16/33. - Amiram Eldar, Oct 23 2020

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Intersection of: A160542 and A229829; A047201 and A229968; A001651, A047201 and A160542.

Cf. A236206, A236208.

Select[Range[200], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[200], Or @@ Divisible[#, {3, 5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)
Select[Range[130], CoprimeQ[165, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-80) - a(n-81) for n > 81. - Bruno Berselli, Mar 25 2014

A236208 Numbers not divisible by 2, 5 or 11.

Original entry on oeis.org

1, 3, 7, 9, 13, 17, 19, 21, 23, 27, 29, 31, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 87, 89, 91, 93, 97, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 110. The asymptotic density of this sequence is 4/11. - Amiram Eldar, Oct 23 2020

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1).

Cf. A007775, A235583, A236204.

Select[Range[200], CoprimeQ[110, #] &] (* Amiram Eldar, Oct 23 2020 *)

G.f.: (x^36 +x^35 +2*x^34 -2*x^33 +2*x^32 +x^31 -x^30 +2*x^29 -2*x^28 +4*x^27 -x^26 -x^25 +6*x^24 -6*x^23 +4*x^22 -x^21 +x^20 +4*x^19 -6*x^18 +4*x^17 +x^16 -x^15 +4*x^14 -6*x^13 +6*x^12 -x^11 -x^10 +4*x^9 -2*x^8 +2*x^7 -x^6 +x^5 +2*x^4 -2*x^3 +2*x^2 +x +1)*x / ((x+1) *(x^2+1) *(x^4-x^3+x^2-x+1) *(x^4+1) *(x^8-x^6+x^4-x^2+1) *(x^16-x^12+x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Feb 13 2014

A236206 Numbers not divisible by 3, 5 or 7.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 17, 19, 22, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 118, 121, 122, 124, 127, 128, 131, 134, 136

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers whose odd part is 11-rough: products of terms of A008364 and powers of 2 (terms of A000079). - Peter Munn, Aug 03 2020

Numbers coprime to 105. The asymptotic density of this sequence is 16/35. - Amiram Eldar, Oct 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Subsequences: A000079, A008364.
Intersection of any 2 of A160545, A229829, A235933.
Other sequences with similar definitions: A007775, A236217.

Select[Range[300], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 7] > 0 &] (* T. D. Noe, Feb 05 2014 *)
Select[Range[300],Or@@Divisible[#,{3,5,7}]==False&] (* Harvey P. Dale, Mar 13 2014 *)
Select[Range[150], CoprimeQ[105, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-48) - a(n-49). - Amiram Eldar, Oct 23 2020

A235583 Numbers not divisible by 2, 5 or 7.

Original entry on oeis.org

1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 127, 129, 131, 137, 139, 141, 143, 149, 151, 153, 157, 159, 163, 167, 169, 171, 173, 177, 179, 181, 183

Offset: 1

Oleg P. Kirillov, Jan 12 2014

All primes, except 2, 5 and 7, are in this sequence. Any product of terms is also a term in the sequence. For example, a(2)a(4) = 3 * 11 = 33 = a(12). - Alonso del Arte, Jan 12 2014
In other words, numbers equivalent 1,3,9,...,69 modulo 70. This means the first differences of the sequence are 24-periodic. - Ralf Stephan, Jan 14 2014
Numbers coprime to 70. The asymptotic density of this sequence is 12/35. - Amiram Eldar, Oct 23 2020

			51 = 3 * 17, and gcd(51, 70) = 1, so it is in the sequence.
53 is prime, so it is in the sequence.
55 = 5 * 11, and gcd(55, 70) = 5, so it is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1).

Cf. A007775, A008364 (subsequence).

Select[Range[210], GCD[#, 70] == 1 &] (* Alonso del Arte, Jan 12 2014 *)
Select[Range[300], Mod[#, 2]>0 &&Mod[#, 5]>0 &&Mod[#, 7]>0&] (* Vincenzo Librandi, Feb 08 2014 *)

G.f.: x*(x^22 +3*x^21 +8*x^20 +7*x^19 +x^18-2*x^17 -x^16 +5*x^15 +10*x^14 +7*x^13 -x^12 -6*x^11 -x^10 +7*x^9 +10*x^8 +5*x^7 -x^6 -2*x^5 +x^4 +7*x^3 +8*x^2 +3*x +1) / ((x+1) *(x^2+1) *(x^2+x+1) *(x^4-x^2+1) *(x^4+1) *(x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Jan 12 2014

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User: Oleg P. Kirillov

A235933 Numbers coprime to 35.

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A236204 Numbers not divisible by 2, 3 or 11.

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A236207 Numbers not divisible by 5 or 11.

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A236217 Numbers not divisible by 3, 5 or 11.

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A236208 Numbers not divisible by 2, 5 or 11.

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A236206 Numbers not divisible by 3, 5 or 7.

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A235583 Numbers not divisible by 2, 5 or 7.

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