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A045572 Numbers that are odd but not divisible by 5. 56
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

A000035(a(n))*(1 - A079998(a(n))) = 1; 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. - 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

LINKS

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).

FORMULA

a(n) = -1 + Sum_{k=0..n}((1/12)*(5*(k mod 4) + 11*((k+1) mod 4) - ((k+2) mod 4) + 5*((k+3) mod 4))). - Paolo P. Lava, Dec 20 2007

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)

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

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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 *)

PROG

(GNU bc) scale=0; for(n=1; n<=100; n++) 10*((n-1)/4)+2*((4*((n-1)%4)+1)/3)+1 /* Carl R. White, Feb 06 2008 */

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

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

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

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

(Haskell)

a045572 n = a045572_list !! (n-1)

a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list

-- Reinhard Zumkeller, Dec 10 2011

CROSSREFS

Complement of A017329.

Cf. A001589, A005408, A082768 (numbers that begin with 1, 3, 7 or 9), A085820.

Sequence in context: A005818 A288444 A085820 * A069254 A105585 A080903

Adjacent sequences:  A045569 A045570 A045571 * A045573 A045574 A045575

KEYWORD

nonn,easy

AUTHOR

Jeff Burch, Dec 11 1999

STATUS

approved

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Last modified September 24 04:14 EDT 2017. Contains 292402 sequences.