A076309 -id:A076309 - cp's OEIS frontend

A008589 Multiples of 7.

0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378

Offset: 0

N. J. A. Sloane, Mar 15 1996

Also the Engel expansion of exp(1/7); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
Complement of A047304; A082784(a(n))=1; A109720(a(n))=0. - Reinhard Zumkeller, Nov 30 2009
The most likely sum of digits to occur when randomly tossing n pairs of (fair) six-sided dice. - Dennis P. Walsh, Jan 26 2012

			For n=2, a(2)=14 because 14 is the most likely sum (of the possible sums 4, 5, ..., 24) to occur when tossing 2 pairs of six-sided dice. - _Dennis P. Walsh_, Jan 26 2012

Ivan Panchenko, Table of n, a(n) for n = 0..200
Edward Brooks, Divisibility by Seven, The Analyst, Vol. 2, No. 5 (Sep., 1875), pp. 129-131.
Tanya Khovanova, Recursive Sequences
Michael Penn, The Luckiest Divisibility Test, YouTube video, 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 319
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008587, A008588, A016993.

a008589 = (* 7)
a008589_list = [0, 7 ..]  -- Reinhard Zumkeller, Jan 25 2013

[7*n : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

A008589:=n->7*n; seq(A008589(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014

Range[0, 1000, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
7*Range[0,60] (* Harvey P. Dale, Feb 28 2023 *)

makelist(7*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=7*n \\ Charles R Greathouse IV, Jul 10 2016

(floor(a(n)/10) - 2*(a(n) mod 10)) == 0 modulo 7, see A076309. - Reinhard Zumkeller, Oct 06 2002
a(n) = 7*n = 2*a(n-1)-a(n-2); G.f.: 7*x/(x-1)^2. - Vincenzo Librandi, Dec 24 2010
E.g.f.: 7*x*exp(x). - Ilya Gutkovskiy, May 11 2016

A082784 Characteristic function of multiples of 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Reinhard Zumkeller, May 22 2003

This sequence is the Euler transformation of A185017. - Jason Kimberley, Oct 14 2011

			a(14) = a(2*7) = 1; a(41) = a(5*7+6) = 0.

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

Cf. A008589, A076309.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), this sequence (g=7). - Jason Kimberley, Oct 14 2011
Cf. A000007, A047304, A109720, A185017.

a082784 = a000007 . (`mod` 7)
a082784_list = cycle [1,0,0,0,0,0,0]
-- Reinhard Zumkeller, Oct 27 2012

[Binomial(n-1,6) mod 7 : n in [0..100]]; // Wesley Ivan Hurt, Oct 07 2014

A082784:=n->0^(n mod 7): seq(A082784(n), n=0..100); # Wesley Ivan Hurt, Oct 07 2014

Table[Mod[Binomial[n - 1, 6], 7], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 07 2014 *)
Table[Boole[Divisible[n, 7]], {n, 0, 100}] (* Amiram Eldar, Oct 31 2023 *)

a(n)=!(n%7) \\ Charles R Greathouse IV, Dec 03 2012

a(n) = 0^(n mod 7).
a(0)=1, a(n)=0 for 1<=n<7, a(n+7)=a(n).
a(n) = 1 - (n^6 mod 7). - Paolo P. Lava, Oct 02 2006
a(n) = 1 - A109720(n); a(A008589(n)) = 1; a(A047304(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(n/7)-floor((n-1)/7). - Tani Akinari, Oct 26 2012
a(n) = C(n-1,6) mod 7. - Wesley Ivan Hurt, Oct 07 2014
From Wesley Ivan Hurt, Jul 11 2016: (Start)
G.f.: 1/(1-x^7).
a(n) = a(n-7) for n>6.
a(n) = (gcd(n,7) - 1)/6. (End)

Wrong formula and keyword mult removed by Amiram Eldar, Oct 31 2023

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)

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

A076312 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, 3, 5, 7, 9, 11, 13, 15, 17, 19, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 7, 9, 11, 13, 15, 17, 19, 21

Offset: 0

Reinhard Zumkeller, Oct 06 2002

nonn

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

(n==0 modulo 19) iff (a(n)==0 modulo 19); applied recursively, this property provides a useful test for divisibility by 19.

			26468 is not a multiple of 19, as 26468 -> 2646+2*8=2662 -> 266+2*2=270 -> 27+2*0=27=19*1+8, therefore the answer is NO.
Is 12882 divisible by 19? 12882 -> 1288+2*2=1292 -> 129+2*2=133 -> 13+2*3=19, therefore the answer is YES.

Erdős, Paul, and János Surányi. Topics in the Theory of Numbers. New York: Springer, 2003. Problem 6, page 3.
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. A008601, A076309, A076310, A076311.

a076312 n =  n' + 2 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

[Floor(n/10) + 2*(n mod 10): n in [0..100]]; // Vincenzo Librandi, Mar 05 2020

f[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Most[idn]]+2idn[[-1]]]; Array[ f,80,0] (* Harvey P. Dale, Mar 01 2020 *)

G.f.: -x(17x^9-2-2x-2x^2-2x^3-2x^4-2x^5-2x^6-2x^7-2x^8)/((1-x)^2(1+x)(x^4+x^3+x^2+x+1)(x^4-x^3+x^2-x+1)). a(n)=A059995(n)+2*A010879(n). [R. J. Mathar, Jan 24 2009]

A336483 a(n) = floor(n/10) + (5 times last digit of n).

Original entry on oeis.org

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

Offset: 0

Michel Marcus, Aug 11 2020

If the resulting number is divisible by 7, then n is divisible by 7; (re)discovered by 12-year-old Nigerian Chika Ofili.

L. E. Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. Chelsea Publishing Co., New York 1966.

D. B. Eperson, Puzzles, Pastimes, Problems, Mathematics in School Vol. 16, No. 5 (Nov., 1987), pp. 18-19, 34-35.
OB360 Media, 12-year-old Nigerian Chika Ofili wins special award for discovering a new Mathematics formula, November 2019.
Skeptics Stack Exchange, Is Chika Ofili's method for checking divisibility for 7 a “new discovery” in math?.
A. Zbikowski, Note sur la divisibilité des nombres, Bull. Acad. Sci. St. Petersbourg 3 (1861) pp. 151-153.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008589, A076309.

Table[Floor[n/10]+5Mod[n,10],{n,0,80}] (* or  *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,5,10,15,20,25,30,35,40,45,1},80] (* Harvey P. Dale, Nov 01 2023 *)

PARI
```
a(n) = 5*(n % 10) + (n\10);
```

From Stefano Spezia, Aug 11 2020: (Start)
O.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 - 44*x^9)/(1 - x - x^10 + x^11).
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10. (End)

cp's OEIS Frontend

A008589 Multiples of 7.

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A082784 Characteristic function of multiples of 7.

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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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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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