A045572 -id:A045572 - cp's OEIS frontend

A047201 Numbers not divisible by 5.

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87

Offset: 1

N. J. A. Sloane

Original name was: Numbers that are congruent to {1, 2, 3, 4} mod 5.
More generally the sequence of numbers not divisible by some fixed integer m>=2 is given by a(n,m) = n-1+floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009
Complement of A008587. - Reinhard Zumkeller, Nov 30 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A019692, A045572, A179290.

a047201 n = a047201_list !! (n-1)
a047201_list = [x | x <- [1..], mod x 5 > 0]
-- Reinhard Zumkeller, Dec 17 2011

[Floor((15*n-1)/12): n in [1..70]]; // Vincenzo Librandi, Apr 06 2015

seq(floor((15*n-1)/12), n=1..56); # Gary Detlefs, Mar 07 2010

Select[Table[n,{n,200}],Mod[#,5]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)

a(n)= n+(n-1)\4 \\ corrected by Michel Marcus, Sep 02 2022

a(n)=n-1+floor((n+3)/4) \\ Benoit Cloitre, Jul 11 2009

def A047201(n): return n+(n-1>>2) # Chai Wah Wu, Feb 24 2025

[i for i in range(72) if gcd(5,i) == 1] # Zerinvary Lajos, Apr 21 2009

G.f.: (x+2*x^2+3*x^3+4*x^4+4*x^5+3*x^6+2*x^7+x^8)/(1-x^4)^2 (not reduced). - Len Smiley
a(n) = 5+a(n-4).
G.f.: x*(1+x+x^2+x^3+x^4)/((1-x)*(1-x^4)).
a(n) = n-1+floor((n+3)/4). - Benoit Cloitre, Jul 11 2009
A011558(a(n))=1; A079998(a(n))=0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor((15*n-1)/12). - Gary Detlefs, Mar 07 2010
a(n) = A225496(n) for n <= 42. - Reinhard Zumkeller, May 09 2013
From Wesley Ivan Hurt, Jun 22 2015: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5), n>5.
a(n) = (10*n-5-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))/8. (End)
E.g.f.: 1 + (1/4)*(-cos(x) + (-3 + 5*x)*cosh(x) + sin(x) + (-2 + 5*x)*sinh(x)). - Stefano Spezia, Dec 01 2019
a(n) = floor((5*n-1)/4). - Wolfdieter Lang, Sep 30 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2-2/sqrt(5))*Pi/5 = A179290 * A019692 / 10. - Amiram Eldar, Dec 07 2021

Comment from Lekraj Beedassy, Dec 17 2006 is now the current name. - Wesley Ivan Hurt, Jun 25 2015

A002329 Periods of reciprocals of integers prime to 10.

Original entry on oeis.org

1, 1, 6, 1, 2, 6, 16, 18, 6, 22, 3, 28, 15, 2, 3, 6, 5, 21, 46, 42, 16, 13, 18, 58, 60, 6, 33, 22, 35, 8, 6, 13, 9, 41, 28, 44, 6, 15, 96, 2, 4, 34, 53, 108, 3, 112, 6, 48, 22, 5, 42, 21, 130, 18, 8, 46, 46, 6, 42, 148, 75, 16, 78, 13, 66, 81, 166, 78, 18, 43

Offset: 1

N. J. A. Sloane

Nonzero terms of A084680.

J. W. L. Glaisher, On circulating decimals, Proc. Camb. Phil. Soc., 3 (1878), 185-206.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Decimal Expansion.
Eric Weisstein's World of Mathematics, Exponent.
Eric Weisstein's World of Mathematics, Multiplicative Order
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A084680, A045572.

Reap[ Do[ If[ CoprimeQ[n, 10], Sow[ MultiplicativeOrder[10, n]]], {n, 1, 150}]][[2, 1]] (* Jean-François Alcover, Nov 07 2012 *)

do(lim)=my(v=List()); for(n=1, lim, if(gcd(n, 10)==1, listput(v, znorder(Mod(10,n))))); Vec(v) \\ Charles R Greathouse IV, Jun 21 2017

a(n) = A084680(A045572(n)). - Jianing Song, Jun 15 2021

a(1) = 1 inserted by Jinyuan Wang, Jun 14 2021

A017509 a(n) = 11*n + 10.

Original entry on oeis.org

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, 131, 142, 153, 164, 175, 186, 197, 208, 219, 230, 241, 252, 263, 274, 285, 296, 307, 318, 329, 340, 351, 362, 373, 384, 395, 406, 417, 428, 439, 450, 461, 472, 483, 494, 505, 516, 527, 538, 549, 560, 571, 582

Offset: 0

N. J. A. Sloane

If k is any member of A045572, the sequence lists the numbers n such that (n^k+1)/11 is a nonnegative integer. See also A267541. - Bruno Berselli, Jan 16 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 989
Leo Tavares, Illustration: Triangular Lines
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A017401, A017413, A045572, A267541.
Cf. A211013 (partial sums), A254322 (partial products).
Powers of the form (11*n+10)^m: this sequence (m=1), A017510 (m=2), A017511 (m=3), A017512 (m=4), A017513 (m=5), A017514 (m=6), A017515 (m=7), A017516 (m=8), A017517 (m=9), A017518 (m=10), A017519 (m=11), A017520 (m=12).
Cf. A008591, A005408.

List([0..60], n-> (11*n+10)); # G. C. Greubel, Oct 29 2019

[11*n+10: n in [0..60]]; // Vincenzo Librandi, Sep 18 2011

seq((11*n+10), n=0..60); # G. C. Greubel, Oct 29 2019

Range[10, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)
(11*Range[60] -1) (* G. C. Greubel, Oct 29 2019 *)

a(n)=11*n+10 \\ Charles R Greathouse IV, Jul 10 2016

def a(n): return 11*n + 10
print([a(n) for n in range(53)]) # Michael S. Branicky, Oct 21 2021

[(11*n+10) for n in (0..60)] # G. C. Greubel, Oct 29 2019

From G. C. Greubel, Oct 29 2019: (Start)
G.f.: (10 + x)/(1-x)^2.
E.g.f.: (10 + 11*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)
a(n) = A008591(n+1) + A005408(n). - Leo Tavares, Oct 25 2022

A045798 Oddish numbers (prime to 10 and 10's digit is odd).

Original entry on oeis.org

11, 13, 17, 19, 31, 33, 37, 39, 51, 53, 57, 59, 71, 73, 77, 79, 91, 93, 97, 99, 111, 113, 117, 119, 131, 133, 137, 139, 151, 153, 157, 159, 171, 173, 177, 179, 191, 193, 197, 199, 211, 213, 217, 219, 231, 233, 237, 239, 251, 253, 257, 259

Offset: 1

J. H. Conway

From Jianing Song, Apr 27 2019: (Start)
Numbers congruent to {11, 13, 17, 19} mod 20.
Numbers k such that Kronecker(-20,k) = A289741(k) = -1. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1).

Complement of A045797 with respect to A045572.

a045798 n = a045798_list !! (n-1)
a045798_list = filter (odd . (`mod` 10) . (`div` 10)) a045572_list
-- Reinhard Zumkeller, Dec 10 2011

seq(seq(20*j + k, k = [11, 13, 17, 19]),j=0..100); # Robert Israel, Mar 27 2017

Table[10n+{1,3,7,9},{n,1,31,2}]//Flatten (* Harvey P. Dale, Oct 01 2019 *)

is(n)=gcd(n,10)==1 && n\10%2 \\ Charles R Greathouse IV, Feb 07 2017

Conjecture a(n) = a(n-1)+a(n-4)-a(n-5). G.f.: x*(11+2*x+4*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2)). - Colin Barker, Apr 14 2012
a(n) = 5n + O(1). - Charles R Greathouse IV, Feb 07 2017
a(n+4) = a(n) + 20. This confirms Barker's conjecture. - Robert Israel, Mar 27 2017

More terms from Erich Friedman.

A045797 Evenish numbers (prime to 10 and 10's digit is even).

Original entry on oeis.org

1, 3, 7, 9, 21, 23, 27, 29, 41, 43, 47, 49, 61, 63, 67, 69, 81, 83, 87, 89, 101, 103, 107, 109, 121, 123, 127, 129, 141, 143, 147, 149, 161, 163, 167, 169, 181, 183, 187, 189, 201, 203, 207, 209, 221, 223, 227, 229, 241, 243, 247, 249, 261, 263, 267, 269, 281

Offset: 1

J. H. Conway

From Jianing Song, Apr 27 2019: (Start)
Numbers congruent to {1, 3, 7, 9} mod 20.
Numbers k such that Kronecker(-20,k) = A289741(k) = +1. (End)
First 20 terms are congruences of 3^k mod 100. - Dario Vuksan, Jan 09 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Complement of A045798 with respect to A045572.

a045797 n = a045797_list !! (n-1)
a045797_list = filter (even . (`mod` 10) . (`div` 10)) a045572_list
-- Reinhard Zumkeller, Dec 10 2011

Flatten[Table[10n+{1,3,7,9},{n,0,30,2}]] (* Harvey P. Dale, Dec 05 2012 *)

is(n)=gcd(n,10)==1 && n\10%2==0 \\ Charles R Greathouse IV, Sep 24 2015

Conjecture a(n) = a(n-1)+a(n-4)-a(n-5). G.f.: x*(1+2*x+4*x^2+2*x^3+11*x^4) / ((1-x)^2*(1+x)*(1+x^2)). - Colin Barker, Apr 14 2012
The conjecture above is correct. - Jianing Song, Apr 27 2019
a(n) = 5n + O(1). - Charles R Greathouse IV, Jan 09 2023

More terms from Erich Friedman

Offset changed by Reinhard Zumkeller, Dec 10 2011

A017401 a(n) = 11n + 1.

Original entry on oeis.org

1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 243, 254, 265, 276, 287, 298, 309, 320, 331, 342, 353, 364, 375, 386, 397, 408, 419, 430, 441, 452, 463, 474, 485, 496, 507, 518, 529, 540, 551, 562, 573, 584, 595, 606, 617, 628, 639, 650, 661

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=11, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=-charpoly(A,x^(n-1)). - Milan Janjic, Feb 21 2010

Sequence lists all nonnegative solutions to x^k == 1 (mod 11), where k is a member of A045572. - Bruno Berselli, Jan 18 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A017281, A017413.

[11*n+1: n in [0..60]]; // Vincenzo Librandi, Jul 30 2011

Range[1, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=11*n+1 \\ Charles R Greathouse IV, Jan 18 2016

G.f.: (1+10*x)/(1-x)^2.

E.g.f.: exp(x)*(1 + 11*x). - Stefano Spezia, Oct 08 2022

A229829 Numbers coprime to 15.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119

Offset: 1

Gary Detlefs, Oct 01 2013

A001651 INTERSECT A047201.
a(n) - 15*floor((n-1)/8) - 2*((n-1) mod 8) has period 8, repeating [1,0,0,1,0,1,1,0].
Numbers whose odd part is 7-rough: products of terms of A007775 and powers of 2 (terms of A000079). - Peter Munn, Aug 04 2020
The asymptotic density of this sequence is 8/15. - Amiram Eldar, Oct 18 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Lists of numbers coprime to other semiprimes: A007310 (6), A045572 (10), A162699 (14), A160545 (21), A235933 (35).
Cf. A008676, A078588, A113763, A160542, A160543, A160544.
Subsequence of: A001651, A047201.
Subsequences: A000079, A007775.

[n: n in [1..120] | IsOne(GCD(n,15))]; // Bruno Berselli, Oct 01 2013

for n from 1 to 500 do if n mod 3<>0 and n mod 5<>0 then print(n) fi od

Select[Range[120], GCD[#, 15] == 1 &] (* or *) t = 70; CoefficientList[Series[(1 + x + 2 x^2 + 3 x^3 + x^4 + 3 x^5 + 2 x^6 + x^7 + x^8)/((1 - x)^2 (1 + x) (1 + x^2) (1 + x^4)) , {x, 0, t}], x] (* Bruno Berselli, Oct 01 2013 *)
Select[Range[120],CoprimeQ[#,15]&] (* Harvey P. Dale, Oct 31 2013 *)

[i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013

a(n+8) = a(n) + 15.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor(2*phi*(f(n+1)+2)) -2*floor(phi*(f(n+1)+2)), where f(n) = (n-1) mod 8 and phi=(1+sqrt(5))/2.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor((2*f(n)+5)/5) -floor((f(n)+2)/3), where f(n) = (n-1) mod 8.
From Bruno Berselli, Oct 01 2013: (Start)
G.f.: x*(1 +x +2*x^2 +3*x^3 +x^4 +3*x^5 +2*x^6 +x^7 +x^8) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). -
a(n) = a(n-1) +a(n-8) -a(n-9) for n>9. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(7 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 2021

A045800 0-ish numbers (end in 01, 07, 43, 49).

Original entry on oeis.org

1, 7, 43, 49, 101, 107, 143, 149, 201, 207, 243, 249, 301, 307, 343, 349, 401, 407, 443, 449, 501, 507, 543, 549, 601, 607, 643, 649, 701, 707, 743, 749, 801, 807, 843, 849, 901, 907, 943, 949, 1001, 1007, 1043, 1049, 1101, 1107, 1143, 1149, 1201, 1207

Offset: 1

J. H. Conway

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

Cf. A045801-A045809, A045572, A045797, A045798.

import Data.List (findIndices)
a045800 n = a045800_list !! (n-1)
a045800_list = findIndices (`elem` [1,7,43,49]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

LinearRecurrence[{1,0,0,1,-1},{1,7,43,49,101},60] (* Harvey P. Dale, Jul 26 2015 *)

G.f.: x*(1+6*x+36*x^2+6*x^3+51*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

a(n) = (50*n-8*i^(n*(n+1))-19*(-1)^n-75)/2, where i=sqrt(-1). - Bruno Berselli, Feb 22 2012

More terms from Erich Friedman.

A045809 9-ish numbers (end in 13, 37, 59, 91).

Original entry on oeis.org

13, 37, 59, 91, 113, 137, 159, 191, 213, 237, 259, 291, 313, 337, 359, 391, 413, 437, 459, 491, 513, 537, 559, 591, 613, 637, 659, 691, 713, 737, 759, 791, 813, 837, 859, 891, 913, 937, 959, 991, 1013, 1037, 1059, 1091, 1113, 1137, 1159, 1191, 1213, 1237

Offset: 1

J. H. Conway

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

Cf. A045800-A045808, A045572, A045797, A045798.

import Data.List (findIndices)
a045809 n = a045809_list !! (n-1)
a045809_list = findIndices (`elem` [13,37,59,91]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

CoefficientList[Series[(13 + 24*x + 22*x^2 + 32*x^3 + 9*x^4)/(1 - x - x^4 + x^5), {x, 0, 80}], x] (* Wesley Ivan Hurt, Jan 23 2017 *)
LinearRecurrence[{1,0,0,1,-1},{13,37,59,91,113},50] (* Harvey P. Dale, Feb 03 2024 *)

G.f.: x*(13+24*x+22*x^2+32*x^3+9*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

a(n) = (50*n+4*i^(n*(n-1))+3*(-1)^n-25)/2, where i=sqrt(-1). - Bruno Berselli, Feb 22 2012

More terms from Erich Friedman.

A065502 Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Len Smiley, Nov 25 2001

Complement of A045572. - Reinhard Zumkeller, Nov 15 2009
Numbers that cannot be prefixed by a single digit to form a prime in decimal representation: A124665 is a subsequence. - Reinhard Zumkeller, Oct 22 2011
Up to 198, this is almost identical to "a(n) = n such that 3^n-1 is not squarefree", with the only exceptions being 39 and 117, which are not in this sequence. Why is that? - Felix Fröhlich, Oct 19 2014
The asymptotic density of this sequence is 3/5. - Amiram Eldar, Mar 09 2021

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A000035, A001622, A045572, A051628, A079998, A124665, A047229 (numbers divisible by 2 or 3).

a065502 n = a065502_list !! (n-1)
a065502_list = filter ((> 1) . (gcd 10)) [1..]
-- Reinhard Zumkeller, Oct 22 2011

A065502 := proc(n)
     option remember;
     if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if (a mod 2) =0 or (a mod 5) =0 then
                return a;
            end if;
        end do:
    end if;
end proc; # R. J. Mathar, Jul 20 2012

Select[Range[114], Mod[#, 2] == 0 || Mod[#, 5] == 0 &] (* T. D. Noe, Jul 13 2012 *)
Select[ Range@ 114, MemberQ[{0, 2, 4, 5, 6, 8}, Mod[#, 10]] &] (* Robert G. Wilson v, May 22 2014 *)

isok(m) = ! ((m%2) && (m%5)); \\ Michel Marcus, Mar 09 2021

A000035(a(n))*(1-A079998(a(n)))=0. - Reinhard Zumkeller, Nov 15 2009
G.f.: x*(2*x^4+x^2+2) / ((x-1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jul 18 2013
a(n) = 10*floor(n/6)+s(n mod 6)-floor(((n-1)mod 6)/5), where s(n) = n+1+floor((n+1)/3). - Gary Detlefs, Oct 05 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/5 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 28 2021

Offset changed from 0 to 1 by Harry J. Smith, Oct 20 2009

cp's OEIS Frontend

A047201 Numbers not divisible by 5.

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A002329 Periods of reciprocals of integers prime to 10.

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A017509 a(n) = 11*n + 10.

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A045798 Oddish numbers (prime to 10 and 10's digit is odd).

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A045797 Evenish numbers (prime to 10 and 10's digit is even).

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A017401 a(n) = 11n + 1.

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