A229829 -id:A229829 - cp's OEIS frontend

A270297 Numbers which are representable as a sum of seven but no fewer consecutive nonnegative integers.

28, 56, 112, 196, 224, 308, 364, 392, 448, 476, 532, 616, 644, 728, 784, 812, 868, 896, 952, 1036, 1064, 1148, 1204, 1232, 1288, 1316, 1372, 1456, 1484, 1568, 1624, 1652, 1708, 1736, 1792, 1876, 1904, 1988, 2044, 2072, 2128, 2156, 2212, 2296, 2324, 2408, 2464

Offset: 1

Martin Renner, Mar 14 2016

nonn

			28 = 1 + 2 + 3 + 4 + 5 + 6 + 7;
35 = 2 + 3 + 4 + 5 + 6 + 7 + 8 = 17 + 18 (not in sequence);
56 = 5 + 6 + 7 + 8 + 9 + 10 + 11;
112 = 13 + 14 + 15 + 16 + 17 + 18 + 19.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A138591, A229829, A270296, A270298, A270299, A270300, A270301, A270302, A270303.

A163169(a(n)) = 7. - Ray Chandler, Mar 22 2016

a(n) = 28*A229829(n). - Hugo Pfoertner, Feb 04 2021

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

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A235933 Numbers coprime to 35.

Original entry on oeis.org

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

A281746 Nonnegative numbers k such that k == 0 (mod 3) or k == 0 (mod 5).

Original entry on oeis.org

0, 3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 125, 126, 129, 130, 132, 135

Offset: 1

Seiichi Manyama, Jan 29 2017

In the game "FizzBuzz", players replace any number divisible by three with the word "Fizz", and any number divisible by five with the word "Buzz". But multiples of both three and five are replaced by "FizzBuzz". For example, 1, 2, Fizz, 4, Buzz, Fizz, 7, 8, Fizz, Buzz, 11, Fizz, 13, 14, FizzBuzz, 16, 17, Fizz, 19, Buzz, Fizz, 22, 23, Fizz, Buzz, 26, Fizz, 28, 29, FizzBuzz, ...
The asymptotic density of this sequence is 7/15. - Amiram Eldar, Mar 25 2021
For a neat way to supplement the set to achieve equal density with its complement, see A080307. - Peter Munn, Oct 12 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Project Euler, Problem 1: Multiples of 3 or 5.
Rosetta Code, FizzBuzz.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Complement of A229829.
Union of A008585 and A008587.
Subsequence of {0} U A080307.
Cf. A281787.

Select[Range[0, 200], MemberQ[Mod[#, {3, 5}], 0]&] (* or *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 3, 5, 6, 9, 10, 12, 15}, 80] (* Harvey P. Dale, Apr 01 2018 *)
Union[3Range[0, 33], 5Range[20]] (* Alonso del Arte, Sep 03 2018 *)
CoefficientList[Series[-(3*x^7 + 2*x^6 + x^5 + 3*x^4 + x^3 + 2*x^2 + 3*x) / (-x^8 + x^7 + x - 1) , {x, 0, 80}], x] (* Stefano Spezia, Sep 16 2018 *)

concat(0, Vec(x^2*(3 + 2*x + x^2 + 3*x^3 + x^4 + 2*x^5 + 3*x^6) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^100))) \\ Colin Barker, Feb 07 2017

G.f.: -(3*x^8 + 2*x^7 + x^6 + 3*x^5 + x^4 + 2*x^3 + 3*x^2) / (-x^8 + x^7 + x - 1).
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8. - Colin Barker, Feb 07 2017
a(n) = 15n/7 + O(1). - Charles R Greathouse IV, Jan 13 2025

A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 24, 30, 33, 35, 36, 39, 40, 42, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 78, 80, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 126, 129, 130, 132, 138, 140, 141, 144, 145, 147, 150

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003593 and A229829.
Analogous to A081062 and A105115 that apply to A120944(1)=6 and A120944(2)=10, respectively.
This sequence applies to term A120944(4)=15.

			6 is in the sequence since gcd(6, 15) = 3 and 6 does not divide 15^e with integer e >= 0.
2 and 4 are not in the sequence since they are coprime to 15.
3 and 5 are not in the sequence since they are divisors of 15.
9 is not in the sequence since 9 | 15^2.

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

Cf. A003593, A081062, A105115, A120944, A229829, A316991.

With[{nn = 150, k = 15}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A175550 Period of the decimal expansion of 1/F as F runs through the Fibonacci numbers greater than 1 and not divisible by 2 or 5.

Original entry on oeis.org

1, 6, 6, 44, 232, 84, 138, 133, 336, 396, 28656, 3016, 84, 514228, 335824, 152214, 67830, 4440, 261744, 504628, 108373609, 47124, 3295440, 2971215072, 49349664, 45240, 4438362040, 203028, 3599596, 10841042784, 104340657248, 252736776688

Offset: 1

Michel Lagneau, Jun 26 2010

The Fibonacci numbers contributing to this sequence are {3, 13, 21, 89, 233, ...}, i.e., Fibonacci(k) for k = 4, 7, 8, 11, 13, ... (A229829, starting with A229829(3)).

			For n = 1, the 1st Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(4) = 3, and period(1/3) = 1, so a(1) = 1.
For n = 2, the 2nd Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(7) = 13, and period (1/13) = 6, so a(2) = 6.

Cf. A000045, A175561, A178505, A045572, A002329, A229829.

with(combinat, fibonacci):nn:= 50:for q from 1 to nn do:n:=fibonacci(q):indic:=0:for p from 1 to n do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic=0 then printf(`%d, `, p):indic:=1:else fi:od:od:

Table[MultiplicativeOrder[10, n/Times @@ ({2, 5}^IntegerExponent[n, {2, 5}])], {n, Select[Fibonacci[Range[3, 70]], CoprimeQ[#, 10] &]}] (* Amiram Eldar, May 27 2024 *)

a(15) onwards from Robert G. Wilson v, Jun 29 2010

A229973 Numbers coprime to 39.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 56, 58, 59, 61, 62, 64, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 100, 101, 103, 106

Offset: 1

Gary Detlefs, Oct 04 2013

Numbers not divisible by 3 or 13.
For n from 1 to 24, a(n) mod 39-n - floor(11*n/25)-2*floor(n/8) has a period of 24, consisting of all zeros except a -2 at indices 8, 16, and 24.
The asymptotic density of this sequence is 8/13. - 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 (2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1).

Cf. A160545, A229829, A229968.

for n from 1 to 50 do if n mod 3<>0 and n mod 13<>0 then print(n) fi od

CoefficientList[Series[(x^22 + x^20 + x^18 + x^16 + 2 x^14 - x^12 + 3 x^11 - x^10 + 2 x^8 + x^6 + x^4 + x^2 + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + 1) (x^4 - x^2 + 1) (x^4 + 1) (x^8 - x^4 + 1)), {x, 0, 80}], x] (* Vincenzo Librandi, Oct 08 2013 *)
Select[Range[100], CoprimeQ[39, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n+24) = a(n) + 39.
a(n) = 39*floor((n-1)/24) + f(n) + floor(11*f(n)/25) + 2*floor(f(n)/8) - 2*floor(((n-1)mod 8)/7) + 40*floor(f(n-1)/23), where f(n) = n mod 24.
G.f.: x*(x^22+x^20+x^18+x^16+2*x^14-x^12+3*x^11-x^10+2*x^8+x^6+x^4+x^2+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)*(x^4+1)*(x^8-x^4+1)). - Colin Barker, Oct 07 2013

More terms from Colin Barker, Oct 07 2013

a(34) corrected by Vincenzo Librandi, Oct 08 2013

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

A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

Original entry on oeis.org

1, 49, 121, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 5929, 6241, 6859, 6889, 7921, 8281, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14161, 14641, 16129, 16807, 17161, 17689, 18769

Offset: 1

Amiram Eldar, Jul 02 2024

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(25) = 7^2*11^2 = 5929.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to powerful numbers.

Intersection of A007775 and A001694.
Intersection of A229829 and A062739.
Intersection of A047201 and A374289.
Cf. A000961, A082695.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]

is(k) = gcd(k, 30) == 1 && ispowerful(k);

Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .

In general, the sum of reciprocals of the p-rough powerful numbers is (zeta(2)*zeta(3)/zeta(6)) * Product_{prime q < p} ((q-1)*q/(q^2-q+1)).

cp's OEIS Frontend

A270297 Numbers which are representable as a sum of seven but no fewer consecutive nonnegative integers.

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A236206 Numbers not divisible by 3, 5 or 7.

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A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

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A235933 Numbers coprime to 35.

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A281746 Nonnegative numbers k such that k == 0 (mod 3) or k == 0 (mod 5).

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A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

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A175550 Period of the decimal expansion of 1/F as F runs through the Fibonacci numbers greater than 1 and not divisible by 2 or 5.

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Maple

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A229973 Numbers coprime to 39.

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