A076312 -id:A076312 - cp's OEIS frontend

A008601 Multiples of 19.

0, 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 380, 399, 418, 437, 456, 475, 494, 513, 532, 551, 570, 589, 608, 627, 646, 665, 684, 703, 722, 741, 760, 779, 798, 817, 836, 855, 874, 893, 912, 931, 950, 969, 988

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 331.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008599, A008600, A008602.

Range[0, 1500, 19] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=19*n; \\ Michel Marcus, Dec 27 2014

(floor(a(n)/10) + 2*(a(n) mod 10)) == 0 modulo 19, see A076312. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 19*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 19*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 19*x*exp(x).
a(n) = (A008600(n) + A008602(n))/2. (End)

A076314 a(n) = floor(n/10) + (n mod 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Reinhard Zumkeller, Oct 06 2002

For n<100 this is equal to the digital sum of n (see A007953). - Hieronymus Fischer, Jun 17 2007

			a(15) = floor(15 / 10) + (15 mod 10) = 1 + 5 = 6. - _Indranil Ghosh_, Feb 13 2017

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

Cf. A076313, A010879, A076309, A076310, A076311, A076312, A055017, A007953, A003132.

a076314 = uncurry (+) . flip divMod 10 -- Reinhard Zumkeller, Jun 01 2013

[n-9*Floor(n/10):n in [0..100]]; // Vincenzo Librandi, Dec 10 2016

Table[n - 9 Floor[n / 10], {n, 0, 100}] (* Vincenzo Librandi Dec 10 2016 *)

a(n)=n\10+n%10 \\ Charles R Greathouse IV, Sep 24 2015

def A076314(n): return (n/10)+(n%10) # Indranil Ghosh, Feb 13 2017

From Hieronymus Fischer, Jun 17 2007: (Start)
a(n) = n - 9*floor(n/10).
a(n) = (n + 9*(n mod 10))/10.
a(n) = n - 9*A002266(A004526(n)) = n - 9*A004526(A002266(n)).
a(n) = (n + 9*A010879(n))/10.
a(n) = (n + 9*A000035(n) + 18*A010874(A004526(n)))/10.
a(n) = (n + 9*A010874(n) + 45*A000035(A002266(n)))/10.
G.f.: x*(8*x^10 - 9*x^9 + 1)/((1 - x^10)*(1 - x)^2). (End)
a(n) = A033930(n) for 1 <= n < 100. - R. J. Mathar, Sep 21 2008
a(n) = +a(n-1) + a(n-10) - a(n-11). - R. J. Mathar, Feb 20 2011

A076313 a(n) = floor(n/10) - (n mod 10).

Original entry on oeis.org

0, -1, -2, -3, -4, -5, -6, -7, -8, -9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, 8, 7, 6, 5, 4

Offset: 0

Reinhard Zumkeller, Oct 06 2002

For n<100 equal to the negated alternating digital sum of n (see A055017). - Hieronymus Fischer, Jun 17 2007

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

Cf. A076314, A010879, A076309, A076310, A076311, A076312, A055017, A076314, A007953, A003132.

a076313 = uncurry (-) . flip divMod 10 -- Reinhard Zumkeller, Jun 01 2013

Table[Floor[n/10]-Mod[n,10],{n,0,100}] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,-1,-2,-3,-4,-5,-6,-7,-8,-9,1},100] (* Harvey P. Dale, Nov 02 2022 *)

a(n)=n\10-n%10 \\ Charles R Greathouse IV, Jan 30 2012

From Hieronymus Fischer, Jun 17 2007: (Start)
a(n) = 11*floor(n/10)-n.
a(n) = (n-11*(n mod 10))/10.
a(n) = 11*A002266(A004526(n))-n=11*A004526(A002266(n))-n.
a(n) = (n-11*A010879(n))/10.
a(n) = (n-11*A000035(n)-22*A010874(A004526(n)))/10.
a(n) = (n-11*A010874(n)-55*A000035(A002266(n)))/10.
G.f.: x*(-8*x^10+11*x^9-1)/((1-x^10)*(1-x)^2). (End)

A076309 a(n) = floor(n/10) - 2*(n mod 10).

Original entry on oeis.org

0, -2, -4, -6, -8, -10, -12, -14, -16, -18, 1, -1, -3, -5, -7, -9, -11, -13, -15, -17, 2, 0, -2, -4, -6, -8, -10, -12, -14, -16, 3, 1, -1, -3, -5, -7, -9, -11, -13, -15, 4, 2, 0, -2, -4, -6, -8, -10, -12, -14, 5, 3, 1, -1, -3, -5, -7, -9, -11, -13, 6, 4, 2, 0, -2, -4, -6, -8, -10

Offset: 0

Reinhard Zumkeller, Oct 06 2002

Delete the last digit from n and subtract twice this digit from the shortened number. - N. J. A. Sloane, May 25 2019

(n==0 modulo 7) iff (a(n)==0 modulo 7); applied recursively, this property provides a useful test for divisibility by 7.

			695591 is not a multiple of 7, as 695591 -> 69559-2*1=69557 -> 6955-2*7=6941 -> 694-2*1=692 -> 69-2*2=65=7*9+2, therefore the answer is NO.
Is 3206 divisible by 7? 3206 -> 320-2*6=308 -> 30-2*8=14=7*2, therefore the answer is YES, indeed 3206=2*7*229.

Paul Erdős and János Surányi. Topics in the Theory of Numbers. New York: Springer, 2003. Problem 6, page 3.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.2 Fundamental Operations, p. 121.
Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Divisibility Tests.
Wikipedia, Divisibility rule.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008589, A076310, A076311, A076312.

a076309 n =  n' - 2 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

Table[Floor[n/10] - 2*Mod[n, 10], {n, 0, 100}] (* G. C. Greubel, Apr 07 2016 *)

a(n) = n\10 - 2*(n % 10); \\ Michel Marcus, Apr 07 2016

From R. J. Mathar, Nov 23 2010: (Start)
a(n) = a(n-1) + a(n-10) - a(n-11).
G.f.: x*(-2 -2*x -2*x^2 -2*x^3 -2*x^4 -2*x^5 -2*x^6 -2*x^7 -2*x^8 +19*x^9)/((1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^2). (End)

A076311 a(n) = floor(n/10) - 5*(n mod 10).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Oct 06 2002

(n==0 modulo 17) iff (a(n)==0 modulo 17); applied recursively, this property provides a divisibility test for numbers given in base 10 notation.

			12808 is not a multiple of 17, as 12808 -> 1280-5*8=1240 -> 124-5*0=124 -> 12-5*4=-8=17*(-1)+9, therefore the answer is NO.
Is 9248 divisible by 17? 9248 -> 924-5*8=884 -> 88-5*4=68=17*4, therefore the answer is YES.

Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Divisibility Tests.
Wikipedia, Divisibility rule
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008599, A076309, A076310, A076312.

a076311 n =  n' - 5 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

[Floor(n/10)-5*(n mod 10): n in [0..50]]; // Vincenzo Librandi, Jun 23 2015

Table[Floor[n/10]-5Mod[n,10],{n,0,60}] (* or *) LinearRecurrence[ {1,0,0,0,0,0,0,0,0,1,-1},{0,-5,-10,-15,-20,-25,-30,-35,-40,-45,1},60] (* Harvey P. Dale, Dec 21 2014 *)

a(n)=n\10 - n%10*5 \\ Charles R Greathouse IV, Oct 07 2015

a(n)= +a(n-1) +a(n-10) -a(n-11). G.f. x *(-5-5*x-5*x^2-5*x^3-5*x^4-5*x^5-5*x^6-5*x^7-5*x^8+46*x^9) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x-1)^2 ). - R. J. Mathar, Feb 20 2011

A076310 a(n) = floor(n/10) + 4*(n mod 10).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Oct 06 2002

nonn

(n==0 modulo 13) iff (a(n)==0 modulo 13); applied recursively, this property provides a divisibility test for numbers given in base 10 notation.

			435598 is not a multiple of 13, as 435598 -> 43559+4*8=43591 -> 4359+4*1=4363 -> 436+4*3=448 -> 44+4*8=76 -> 7+4*6=29=13*2+3, therefore the answer is NO.
Is 8424 divisible by 13? 8424 -> 842+4*4=858 -> 85+4*8=117 -> 11+4*7=39=13*3, therefore the answer is YES.

Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Divisibility Tests.
Wikipedia, Divisibility rule
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008595, A076309, A076311, A076312.

a076310 n =  n' + 4 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

[Floor(n/10)+4*(n mod 10): n in [0..75]]; // Vincenzo Librandi, Feb 27 2016

A076310:=n->floor(n/10) + 4*(n mod 10); seq(A076310(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Table[Floor[n/10] + 4*Mod[n, 10], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,4,8,12,16,20,24,28,32,36,1},80] (* Harvey P. Dale, Sep 30 2015 *)

a(n) = n\10 + 4*(n % 10); \\ Michel Marcus, Jan 31 2014

a(n) = +a(n-1) +a(n-10) -a(n-11). G.f.: -x*(-4-4*x-4*x^2-4*x^3-4*x^4-4*x^5-4*x^6-4*x^7-4*x^8+35*x^9) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x-1)^2 ). - R. J. Mathar, Feb 20 2011

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A008601 Multiples of 19.

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A076314 a(n) = floor(n/10) + (n mod 10).

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A076313 a(n) = floor(n/10) - (n mod 10).

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A076309 a(n) = floor(n/10) - 2*(n mod 10).

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A076311 a(n) = floor(n/10) - 5*(n mod 10).

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A076310 a(n) = floor(n/10) + 4*(n mod 10).

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