A081502 -id:A081502 - cp's OEIS frontend

A081503 Number of steps to reach a single digit when map in A081502 is iterated.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3

Offset: 0

N. J. A. Sloane, Apr 22 2003

			19 -> 3+9 = 12 -> 3+2 = 5, taking 2 steps, so a(19)=2.

A081503 := proc(n)
    local nitr,a ;
    nitr := n ;
    a := 0 ;
    while nitr > 9 do
        nitr := A081502(nitr) ;
        a := a+1 ;
    end do;
    a ;
end proc: # R. J. Mathar, Oct 03 2014

a(n) = O(log n).

More terms from Matthew Conroy, Jan 16 2006

Formula from Charles R Greathouse IV, Aug 02 2010

A081594 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 2x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 20

Offset: 0

N. J. A. Sloane, Apr 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Differs from A028897, A156230 and A244158 for the first time at n=100, which here is a(100) = 20.

[(n+4*y)/5 where y is n mod 10: n in [0..100]]; // Bruno Berselli, Jun 24 2014

A081594:=n->n-8*floor(n/10); seq(A081594(n), n=0..100); # Wesley Ivan Hurt, Jun 25 2014

CoefficientList[Series[-x (7 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x+1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,2},110] (* or *) Table[Range[n,n+9],{n,0,26,2}]//Flatten (* Harvey P. Dale, Jul 22 2021 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 2*x+y) \\ Colin Barker, Jun 24 2014

[n-8*floor(n/10) for n in (0..100)] # Bruno Berselli, Jun 24 2014

a(n) = (2 * floor(n/10)) + (n modulo 10). - Antti Karttunen, Jun 22 2014
G.f.: -x*(7*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 23 2014
a(n) = n - 8*floor(n/10). [Bruno Berselli, Jun 24 2014]

Terms up to n=100 added by Antti Karttunen, Jun 22 2014

G.f. revised by Vincenzo Librandi, Jun 25 2014

A028898 Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27

Offset: 0

N. J. A. Sloane

If n is a multiple of 7, then a(n) is also a multiple of 7. See the Bhattacharyya link. - Michel Marcus, May 11 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Malay Bhattacharyya, Sanghamitra Bandyopadhyay, Ujjwal Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis, No 19/2009.

Cf. A008589 (multiples of 7).

Different from A081502 for n>=100.

a:= proc(n) option remember;   n mod 10 + 3*procname(floor(n/10))
end proc:
a(0):= 0:
seq(a(i),i=0..100); # Robert Israel, May 11 2016

a = {1}; Do[AppendTo[a, If[Mod[n, 10] == 0, 3 a[[n/10]], a[[n - 1]] + 1]], {n, 2, 76}]; {0}~Join~a (* Michael De Vlieger, May 10 2016 *)

a(n)=if(n<1,0,if(n%10,a(n-1)+1,3*a(n/10)))

a(n) = subst(Pol(digits(n)), x, 3); \\ Michel Marcus, May 10 2016

a(0)=0, a(n)=3*a(n/10) if n==0 (mod 10), a(n)=a(n-1)+1 otherwise. - Benoit Cloitre, Dec 21 2002

G.f.: G(x) = (1-x)^(-1) * Sum_{i>=0} 3^i*p(x^(10^i)) where p(t) = (t+2*t^2+3*t^3+4*t^4+5*t^5+6*t^6+7*t^7+8*t^8+9*t^9)/(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9) satisfies (1-x)*G(x) = p(x) + 3*(1-x^10)*G(x^10). - Robert Israel, May 11 2016

More terms from Erich Friedman

Moved Wesley Ivan Hurt's formula to A081502 where it applies. - Kevin Ryde, Dec 03 2019

A081600 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 9x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 108, 109, 110

Offset: 0

N. J. A. Sloane, Apr 22 2003

More than the usual number of terms are displayed in order to distinguish this from some closely related sequences. - N. J. A. Sloane, Mar 22 2014

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Different from A028904.

k:=9; [n-(10-k)*Floor(n/10): n in [0..150]]; // Bruno Berselli, Jun 24 2014

f1:=proc(n) local x,y;
y:= (n mod 10);
x:=(n-y)/10;
9*x+y;
end;
[seq(f1(n),n=0..200)];

my(n, x, y); vector(500, n, y=(n-1)%10; x=(n-1-y)\10; 9*x+y) \\ Colin Barker, Jun 23 2014

a(n)=n - n\10 \\ Charles R Greathouse IV, Sep 01 2015

G.f.: x*(x^2 +x +1)*(x^6 +x^3 +1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - floor(n/10). - Bruno Berselli, Jun 24 2014

A081595 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 4x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 28, 29, 30, 31, 32, 33, 34, 35

Offset: 0

N. J. A. Sloane, Apr 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Starts to differ from A028899 at a(100).

k:=4; [n-(10-k)*Floor(n/10): n in [0..100]]; // Bruno Berselli, Jun 24 2014

CoefficientList[Series[-x (5 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,4},80] (* Harvey P. Dale, Sep 17 2023 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 4*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(5*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - 6*floor(n/10). [Bruno Berselli, Jun 24 2014]

A081596 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 5x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 35, 36, 37, 38, 39, 40, 41

Offset: 0

N. J. A. Sloane, Apr 22 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Different from A028900.

k:=5; [n-(10-k)*Floor(n/10): n in [0..100]]; // Bruno Berselli, Jun 24 2014

CoefficientList[Series[-x (4 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 5*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(4*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - 5*floor(n/10). [Bruno Berselli, Jun 24 2014]

A081597 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 6x + y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 42, 43, 44, 45, 46, 47

Offset: 0

N. J. A. Sloane, Apr 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Different from A028901.

k:=6; [n-(10-k)*Floor(n/10): n in [0..10]]; // Bruno Berselli, Jun 24 2014

A081597:=n->n-4*floor(n/10): seq(A081597(n), n=0..150); # Wesley Ivan Hurt, Apr 25 2017

CoefficientList[Series[-x (3 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 6*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(3*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014
a(n) = n - 4*floor(n/10). [Bruno Berselli, Jun 24 2014]
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10. - Chai Wah Wu, Apr 25 2017

A081598 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 7x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 49, 50, 51, 52, 53, 54

Offset: 0

N. J. A. Sloane, Apr 22 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A081502. Different from A028902.

k:=7; [n-(10-k)*Floor(n/10): n in [0..100]]; // Bruno Berselli, Jun 24 2014

CoefficientList[Series[-x (2 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)
Table[n-3*Floor[n/10],{n,0,80}] (* Harvey P. Dale, Apr 22 2019 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 7*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(2*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - 3*floor(n/10). [Bruno Berselli, Jun 24 2014]

A081599 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 8x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 56, 57, 58, 59, 60, 61

Offset: 0

N. J. A. Sloane, Apr 22 2003

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Different from A028898.

k:=8; [n-(10-k)*Floor(n/10): n in [0..100]]; // Bruno Berselli, Jun 24 2014

A081599 := proc(n)
    local x,y ;
    x := floor(n/10) ;
    y := modp(n,10) ;
    8*x+y ;
end proc:
seq(A081599(n),n=0..100) ; # R. J. Mathar, May 25 2023

Table[n-2*Floor[n/10],{n,0,80}] (* Harvey P. Dale, Nov 07 2017 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 8*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - 2*floor(n/10). - Bruno Berselli, Jun 24 2014

cp's OEIS Frontend

A081503 Number of steps to reach a single digit when map in A081502 is iterated.

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A081594 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 2x+y.

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A028898 Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.

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A081600 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 9x+y.

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A081595 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 4x+y.

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A081596 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 5x+y.

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A081597 Let n = 10*x + y where 0 <= y <= 9, x >= 0. Then a(n) = 6*x + y.

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A081597 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 6x + y.