A045797 -id:A045797 - cp's OEIS frontend

A045572 Numbers that are odd but not divisible by 5.

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153

Offset: 1

Jeff Burch, Dec 11 1999

Contains the repunits R_n, (A000042 or A002275): For any m in the sequence (divisible by neither 2 nor 5), Euler's theorem (i.e., m | 10^m - 1 = 9*R_n) guarantees that R_n is always some multiple of m (see A099679) and thus forms a subsequence. - Lekraj Beedassy, Oct 26 2004
Inverse formula: n = 4*floor(a(n)/10) + floor((a(n) mod 10)/3) + 1. - Carl R. White, Feb 06 2008
Numbers ending with 1, 3, 7 or 9. - Lekraj Beedassy, Apr 04 2009
Complement of A065502. - Reinhard Zumkeller, Nov 15 2009
Union of evenish and oddish numbers, cf. A045797, A045798. - Reinhard Zumkeller, Dec 10 2011
Numbers k such that k^(4*j) mod 10 = 1, for any j. - Gary Detlefs, Jan 03 2012
Numbers coprime to 10. - Charles R Greathouse IV, Sep 05 2013
This is also the sequence of numbers such that all their divisors are the sum of the proper divisors of some number (see A001065 (sum of proper divisors) and A078923 (possible values of sigma(n)-n)). This is due to the fact that in the set of untouchable numbers (A005114) there are only 2 prime numbers (2 and 5) and all other terms are even composite. - Michel Marcus, Jun 14 2014
Numbers n for which A001589(n) is divisible by 5. - Bruno Berselli, Jun 18 2014
For a(n) > 1, positive integers x such that the decimal representation of 1/x is purely periodic after the decimal point (1/x is a repeating decimal with no non-repeating portion). - Doug Bell, Aug 05 2015
The asymptotic density of this sequence is 2/5. - Amiram Eldar, Oct 18 2020

			a(18) = 10*floor(17/4) + 2*floor( (4*(17 mod 4) + 1)/3 ) + 1
      = 10*4 + 2*floor( (4*(1)+1)/3 ) + 1
      = 40 + 2*floor(5/3) + 1
      = 40 + 2*1 + 1
      = 43.
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + ...

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

Relative complement of A017329 in A005408.

Cf. A000035, A000042, A001065, A001589, A002275, A005114, A045797, A045798, A065502, A078923, A079998, A082768 (numbers that begin with 1, 3, 7 or 9), A085820, A099679.

a045572 n = a045572_list !! (n-1)
a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list
-- Reinhard Zumkeller, Dec 10 2011

[ 2*n + 2*Floor((n-3)/4) + 1: n in [1..70] ]; // Vincenzo Librandi, Aug 01 2011

A045572:=n->2*n + 2*floor((n-3)/4) + 1; seq(A045572(n), n=1..50); # Wesley Ivan Hurt, Jun 14 2014

Flatten[Table[10n + {1, 3, 7, 9}, {n, 0, 19}]] (* Alonso del Arte, Jan 13 2012 *)
Select[2Range@78 -1, Mod[#, 5] > 0 &] (* Robert G. Wilson v, Apr 02 2017 *)
Map[(1/2*(5*# + Mod[3*# + 2, 4] - 4))&, Range[10^3]] (* Mikk Heidemaa, Nov 23 2017 *)

a(n)=10*(n>>2)+[-1,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jul 31 2011

is(n)=gcd(n,10)==1 \\ Charles R Greathouse IV, Sep 05 2013

{a(n) = 2*n - 1 + (n+1) \ 4 * 2}; /* Michael Somos, Jun 15 2014 */

def A045572(n): return 2*(n+(n+1)//4) - 1 # Chai Wah Wu, Jan 08 2022

a(n) = 10*floor((n-1)/4) + 2*floor( (4*((n-1) mod 4) + 1)/3 ) + 1; a(n) = a(n-1) + 2 + 2*floor(((x+6) mod 10)/9). - Carl R. White, Feb 06 2008
a(n) = 2*n + 2*floor((n-3)/4) + 1. - Kenneth Hammond (weregoose(AT)gmail.com), Mar 07 2008
a(n) = -1 + 2*n + 2*floor((n+1)/4). - Kenneth Hammond (weregoose(AT)gmail.com), Mar 25 2008
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1+x) * (x^2+1) * (x-1)^2). (End)
A000035(a(n))*(1 - A079998(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2009
a(n) = (10*n + 2*(-1)^(n*(n+1)/2) - (-1)^n - 5)/4. - Bruno Berselli, Nov 06 2011
G.f.: x * (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x) * (1 - x^4)). - Michael Somos, Jun 15 2014
a(1 - n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2014
0 = (a(n) - 2*a(n+1) + a(n+2)) * (a(n) - 4*a(n+2) + 3*a(n+3)) for all n in Z. - Michael Somos, Jun 15 2014
From Mikk Heidemaa, Nov 22 2017: (Start)
a(n) = (1/2)*(5*n + ((3*n + 2) mod 4) - 4);
a(n) = (1/4)*((-1)^(n + 1) + 10*n + 2*cos((n*Pi)/2) - 2*sin((n*Pi)/2) - 5);
a(n) = (1/4)*((-1)^(1 + n) + (1 - i)*exp(-(1/2)*i*n*Pi) + (1 + i)*exp(i*n*Pi/2) + 10*n - 5) (for n > 0), where i is the imaginary unit. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 12 2021
E.g.f.: (2 + cos(x) + (5*x - 3)*cosh(x) - sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Dec 07 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.

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.

A289741 a(n) = Kronecker symbol (-20/n).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0

Offset: 0

Jianing Song, Dec 27 2018

Period 20: repeat [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1].
This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and {((-100)/n)} (A185276).
Note that (Sum_{i=0..19} i*a(i))/(-20) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-5)). The fact Q(sqrt(-5)) has class number 2 implies that Q(sqrt(-5)) is not a unique factorization domain.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Cf. A035170 (inverse Moebius transform).
Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), this sequence (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).
Cf. A045797, A045798, A185276.

Array[KroneckerSymbol[-20, #]&, 100, 0] (* Amiram Eldar, Jan 10 2019 *)

PARI
```
a(n) = kronecker(-20, n)
```

a(n) = 1 for n in A045797; -1 for n in A045798; 0 for n that are not coprime with 20.
Completely multiplicative with a(p) = a(p mod 20) for primes p.
a(n) = A080891(n)*A101455(n).
a(n) = -a(n+10) = -a(-n) for all n in Z.
Multiplicative with a(2) = a(5) = 0, a(p) = (-1)^floor(p/10) otherwise; equivalently: a(n) = (-1)^floor(n/10) if n is coprime to 2*5, 0 otherwise. - M. F. Hasler, Feb 28 2022

A045801 1-ish numbers (end in 11, 39, 73, 77).

Original entry on oeis.org

11, 39, 73, 77, 111, 139, 173, 177, 211, 239, 273, 277, 311, 339, 373, 377, 411, 439, 473, 477, 511, 539, 573, 577, 611, 639, 673, 677, 711, 739, 773, 777, 811, 839, 873, 877, 911, 939, 973, 977, 1011, 1039, 1073, 1077, 1111, 1139, 1173, 1177, 1211, 1239

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-A045809, A045572, A045797, A045798.

import Data.List (findIndices)
a045801 n = a045801_list !! (n-1)
a045801_list = findIndices (`elem` [11,39,73,77]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Table[100n+k,{n,0,20},{k,{11,39,73,77}}]//Flatten (* or *) LinearRecurrence[ {1,0,0,1,-1},{11,39,73,77,111},80] (* Harvey P. Dale, Jul 05 2019 *)

G.f.: x*(11+28*x+34*x^2+4*x^3+23*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

a(n) = (50*n-12*i^(n*(n-1))-9*(-1)^n-25)/2, where i=sqrt(-1). - Bruno Berselli, Feb 23 2012

More terms from Erich Friedman.

A045808 8-ish numbers (end in 67, 69, 81, 83).

Original entry on oeis.org

67, 69, 81, 83, 167, 169, 181, 183, 267, 269, 281, 283, 367, 369, 381, 383, 467, 469, 481, 483, 567, 569, 581, 583, 667, 669, 681, 683, 767, 769, 781, 783, 867, 869, 881, 883, 967, 969, 981, 983, 1067, 1069, 1081, 1083, 1167, 1169, 1181, 1183, 1267, 1269

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045808 n = a045808_list !! (n-1)
a045808_list = findIndices (`elem` [67,69,81,83]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1300],MemberQ[{67,69,81,83},Mod[#,100]]&] (* Harvey P. Dale, Jul 03 2013 *)

Empirical g.f.: x*(67+2*x+12*x^2+2*x^3+17*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman

A045802 2-ish numbers (end in 03, 21, 29, 47).

Original entry on oeis.org

3, 21, 29, 47, 103, 121, 129, 147, 203, 221, 229, 247, 303, 321, 329, 347, 403, 421, 429, 447, 503, 521, 529, 547, 603, 621, 629, 647, 703, 721, 729, 747, 803, 821, 829, 847, 903, 921, 929, 947, 1003, 1021, 1029, 1047, 1103, 1121, 1129, 1147, 1203, 1221

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-A045809, A045572, A045797, A045798.

import Data.List (findIndices)
a045802 n = a045802_list !! (n-1)
a045802_list = findIndices (`elem` [3,21,29,47]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Join[{3},Select[Range[10,1300],MemberQ[{{0,3},{2,1},{2,9},{4,7}}, Take[ IntegerDigits[#],-2]]&]] (* or *) LinearRecurrence[{1,0,0,1,-1},{3,21,29,47,103},50] (* Harvey P. Dale, May 05 2012 *)

G.f.: x*(3+18*x+8*x^2+18*x^3+53*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012
a(n) = (50*n-24*i^(n*(n+1))-7*(-1)^n-75)/2, where i=sqrt(-1). - Bruno Berselli, Feb 23 2012
a(1)=3, a(2)=21, a(3)=29, a(4)=47, a(5)=103, a(n)=a(n-1)+a(n-4)-a(n-5) .[Harvey P. Dale, May 05 2012]

More terms from Erich Friedman.

A045803 3-ish numbers (end in 17, 19, 31, 33).

Original entry on oeis.org

17, 19, 31, 33, 117, 119, 131, 133, 217, 219, 231, 233, 317, 319, 331, 333, 417, 419, 431, 433, 517, 519, 531, 533, 617, 619, 631, 633, 717, 719, 731, 733, 817, 819, 831, 833, 917, 919, 931, 933, 1017, 1019, 1031, 1033, 1117, 1119, 1131, 1133, 1217, 1219

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-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045803 n = a045803_list !! (n-1)
a045803_list = findIndices (`elem` [17,19,31,33]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1300],MemberQ[{17,19,31,33},Mod[#,100]]&] (* or *) LinearRecurrence[{1,0,0,1,-1},{17,19,31,33,117},50] (* Harvey P. Dale, Dec 17 2014 *)

a(n) = -75/2 - (23*(-1)^n)/2 - (9-9*I)*(-I)^n - (9+9*I)*I^n + 25*n \\ Colin Barker, Oct 16 2015

Vec(x*(17+2*x+12*x^2+2*x^3+67*x^4)/(1-x-x^4+x^5) + O(x^100)) \\ Colin Barker, Oct 16 2015

G.f.: x*(17+2*x+12*x^2+2*x^3+67*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

a(n) = -75/2 - (23*(-1)^n)/2 - (9-9*i)*(-i)^n - (9+9*i)*i^n + 25*n where i=sqrt(-1). - Colin Barker, Oct 16 2015

More terms from Erich Friedman

A045804 4-ish numbers (end in 09, 41, 63, 87).

Original entry on oeis.org

9, 41, 63, 87, 109, 141, 163, 187, 209, 241, 263, 287, 309, 341, 363, 387, 409, 441, 463, 487, 509, 541, 563, 587, 609, 641, 663, 687, 709, 741, 763, 787, 809, 841, 863, 887, 909, 941, 963, 987, 1009, 1041, 1063, 1087, 1109, 1141, 1163, 1187, 1209, 1241

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-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045804 n = a045804_list !! (n-1)
a045804_list = findIndices (`elem` [9,41,63,87]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Select[Range[1250],MemberQ[{9,41,63,87},Mod[#,100]]&] (* Harvey P. Dale, Aug 01 2020 *)

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

More terms from Erich Friedman.

cp's OEIS Frontend

A045572 Numbers that are odd but not divisible by 5.

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

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

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

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A289741 a(n) = Kronecker symbol (-20/n).

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A045801 1-ish numbers (end in 11, 39, 73, 77).

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A045808 8-ish numbers (end in 67, 69, 81, 83).

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