A008589 -id:A008589 - cp's OEIS frontend

A048730 Differences between A008589 (multiples of 7) and A048727, a(n) = ((n*7)-Xmult(n,7)).

0, 0, 0, 12, 0, 8, 24, 28, 0, 0, 16, 28, 48, 56, 56, 60, 0, 0, 0, 12, 32, 40, 56, 60, 96, 96, 112, 124, 112, 120, 120, 124, 0, 0, 0, 12, 0, 8, 24, 28, 64, 64, 80, 92, 112, 120, 120, 124, 192, 192, 192, 204, 224, 232, 248

Offset: 0

Antti Karttunen, Apr 26 1999

For n = binary n[k],n[k-1],...,n[0], bits a(n) = binary b[k+1],b[k],...,b[0] are b[i] = 1 when n[i-1] + n[i-2] + n[i-3] >= 2, so the majority bit 0 or 1 among the 3 bits of n below position i (with 0 bits below the radix point of n as necessary). This is since 7*n = 4*n + 2*n + n is n[i-1] + n[i-2] + n[i-3] at position i-1, and 4*n XOR 2*n XOR n is the same but no carry, so b[i] is the carry only. - Kevin Ryde, Mar 26 2021

Paolo Xausa, Table of n, a(n) for n = 0..8191

Positions of zeros are given by A048715. Cf. A048733, A342697.

Diagonal 7 of A061858.

A048730[n_] := 7*n - BitXor[n, 2*n, 4*n];
Array[A048730,100, 0] (* Paolo Xausa, Aug 06 2025 *)

a(n)=7*n - bitxor(n, bitxor(2*n, 4*n)) \\ Charles R Greathouse IV, Oct 03 2016

def A048730(n): return 7*n-(n^n<<1^n<<2) # Chai Wah Wu, Jun 29 2022

A008591 Multiples of 9: a(n) = 9*n.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477

Offset: 0

N. J. A. Sloane

An Iraqi tablet dating from the Middle Babylonian period (1400-1100 BC) gives a(1)-a(20), a(30), a(40), and a(50). See CDLI link for images and more information. - Charles R Greathouse IV, Jan 21 2017
Apart from 0, numbers whose digital root is 9. - Halfdan Skjerning, Mar 15 2018
Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 9. - Stefano Spezia, Jul 08 2025

Ivan Panchenko, Table of n, a(n) for n = 0..200
CDLI, MKT 1, 039, HS 0217a
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 321
Tanya Khovanova, Recursive Sequences
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Triangular Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008588, A008589, A008590.
Cf. A168182, A168183.
Cf. A002283, A007953.
Cf. A060544.

Range[0, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

makelist(9*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=9*n \\ Charles R Greathouse IV, Jan 30 2012

Complement of A168183; A168182(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = A007953(A002283(n)). - Reinhard Zumkeller, Aug 06 2010
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 9*n = 2*a(n-1) - a(n-2).
G.f.: 9x/(x-1)^2. (End)
a(n) = A060544(n+1) - A060544(n). - Leo Tavares, Jul 17 2022
E.g.f.: 9*x*exp(x). - Stefano Spezia, Oct 08 2022

A121027 Multiples of 7 containing a 7 in their decimal representation.

Original entry on oeis.org

7, 70, 77, 147, 175, 217, 273, 287, 357, 371, 378, 427, 476, 497, 567, 574, 637, 672, 679, 700, 707, 714, 721, 728, 735, 742, 749, 756, 763, 770, 777, 784, 791, 798, 847, 875, 917, 973, 987, 1057, 1071, 1078, 1127, 1176, 1197, 1267, 1274, 1337, 1372, 1379

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Intersection of A008589 and A011537.

Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121026, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[7*Range[500], MemberQ[IntegerDigits[#], 7] &] (* Paolo Xausa, Feb 25 2024 *)

is(n)=n%7==0 && setsearch(Set(digits(n)), 7) \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 7n. - Charles R Greathouse IV, Feb 12 2017

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

A016993 a(n) = 7*n + 1.

Original entry on oeis.org

1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 218, 225, 232, 239, 246, 253, 260, 267, 274, 281, 288, 295, 302, 309, 316, 323, 330, 337, 344, 351, 358, 365, 372, 379

Offset: 0

N. J. A. Sloane

For n > 3, also the number of (not necessarily maximal) cliques in the n-web graph. - Eric W. Weisstein, Nov 29 2017

The number of notes in a musical scale of n octaves. - Geoffrey Trueman Falk, Feb 16 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093564 (column 1).

Cf. A131844, A008589.

a016993 = (+ 1) . (* 7)
a016993_list = [1, 8 ..]  -- Reinhard Zumkeller, Jan 25 2013

[7*n+1: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A016993:=n->7*n+1: seq(A016993(n), n=0..70); # Wesley Ivan Hurt, Nov 01 2014

7*Range[0, 55] + 1 (* Alonso del Arte, Oct 26 2014 *)
Table[7 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {8, 15}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(1 + 6 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=7*n+1 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 7*n + 1.
G.f.: (1+6*x)/(1-x)^2.
From Elmo R. Oliveira, Mar 07 2024: (Start)
a(n) = 2*a(n-1) - a(n-2).
E.g.f.: (1 + 7*x)*exp(x). (End)

A017089 a(n) = 8*n + 2.

Original entry on oeis.org

2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 226, 234, 242, 250, 258, 266, 274, 282, 290, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 378, 386, 394, 402, 410, 418, 426

Offset: 0

N. J. A. Sloane, Dec 11 1996

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 33 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 81 ).
First differences of A002939. - Aaron David Fairbanks, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002939, A008589, A008590, A016813, A016993, A017005, A017017, A017077.

a017089 = (+ 2) . (* 8)
a017089_list = [2, 10 ..]  -- Reinhard Zumkeller, Jun 07 2015

[8*n+2: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017089:=n->8*n+2; seq(A017089(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[2, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n) = 8*n+2; \\ Michel Marcus, Sep 17 2015

a(n) = 8*n+2; a(n) = 2*a(n-1)-a(n-2). - Vincenzo Librandi, May 28 2011
Sum_{n>=0} (-1)^n/a(n) = (Pi + 2*log(cot(Pi/8)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
From Elmo R. Oliveira, Mar 17 2024: (Start)
G.f.: 2*(1+3*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(1 + 4*x).
a(n) = 2*A016813(n) = A008590(n) + 2. (End)

A017029 a(n) = 7*n + 4.

Original entry on oeis.org

4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 221, 228, 235, 242, 249, 256, 263, 270, 277, 284, 291, 298, 305, 312, 319, 326, 333, 340, 347, 354, 361, 368, 375, 382

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005, A017017.

Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

[7*n + 4: n in [0..60]]; // Vincenzo Librandi, Jun 18 2011

Range[4,1000,7] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)
CoefficientList[Series[(3*x + 4)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 27 2013 *)
LinearRecurrence[{2,-1},{4,11},60] (* Harvey P. Dale, Mar 27 2025 *)

a(n)=7*n+4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (3*x + 4)/(1-x)^2. - Vincenzo Librandi, Jan 27 2013
From Elmo R. Oliveira, Apr 12 2025: (Start)
E.g.f.: exp(x)*(4 + 7*x).
a(n) = 2*a(n-1) - a(n-2). (End)

Extended by Ray Chandler, Jan 25 2005

A017017 a(n) = 7*n + 3.

Original entry on oeis.org

3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 325, 332, 339, 346, 353, 360, 367, 374, 381

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005.

[7*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017017:=n->7*n + 3; seq(A017017(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Range[3, 600, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n)=7*n+3 \\ Charles R Greathouse IV, Jul 02 2013

[7*n+3 for n in range(80)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (3 + 4*x)/(1 - x)^2.
E.g.f.: (3 + 7*x)*exp(x). (End)

A113801 Numbers that are congruent to {1, 13} mod 14.

Original entry on oeis.org

1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379

Offset: 1

Giovanni Teofilatto, Jan 22 2006

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010

More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

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

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

a113801 n = a113801_list !! (n-1)
a113801_list = 1 : 13 : map (+ 14) a113801_list
-- Reinhard Zumkeller, Jan 07 2012

LinearRecurrence[{1,1,-1},{1,13,15},60] (* or *) Select[Range[500], MemberQ[{1,13},Mod[#,14]]&] (* Harvey P. Dale, May 11 2011 *)

a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

a(n) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010
From Bruno Berselli, Oct 26 2010: (Start)
a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.
G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2)+14 for n>2.
a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)
a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 5*exp(-x))/2. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/14).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/14)*cosec(Pi/14). (End)

Corrected and extended by Giovanni Teofilatto, Nov 14 2008

Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

A195159 Multiples of 7 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 7, 3, 14, 5, 21, 7, 28, 9, 35, 11, 42, 13, 49, 15, 56, 17, 63, 19, 70, 21, 77, 23, 84, 25, 91, 27, 98, 29, 105, 31, 112, 33, 119, 35, 126, 37, 133, 39, 140, 41, 147, 43, 154, 45, 161, 47, 168, 49, 175, 51, 182, 53, 189, 55, 196, 57, 203, 59, 210, 61

Offset: 0

Omar E. Pol, Sep 10 2011

This is 7*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 11-gonal (or hendecagonal) numbers A195160.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 11-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008589 and A005408 interleaved.
Column k=7 of A195151.
Cf. Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, this sequence, A195161.
Cf. A056020, A195160.

&cat[[7*n, 2*n+1]: n in [0..40]]; // Vincenzo Librandi, Sep 27 2011

Table[If[EvenQ[n], 7(n/2), n], {n, 0, 61}] (* Alonso del Arte, Sep 14 2011 *)
With[{nn=40},Riffle[7*Range[0,nn],Range[1,2nn,2]]] (* Harvey P. Dale, Aug 01 2019 *)

a(n)=(5*(-1)^n+9)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 7n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
From Bruno Berselli, Sep 14 2011: (Start)
G.f.: x*(1+7*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = (5*(-1)^n+9)*n/4.
a(n) + a(n-1) = A056020(n). (End)
Multiplicative with a(2^e) = 7*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 5/2^s). - Amiram Eldar, Oct 25 2023

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

cp's OEIS Frontend

A048730 Differences between A008589 (multiples of 7) and A048727, a(n) = ((n*7)-Xmult(n,7)).

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A008591 Multiples of 9: a(n) = 9*n.

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Maxima

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A121027 Multiples of 7 containing a 7 in their decimal representation.

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A016993 a(n) = 7*n + 1.

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A017089 a(n) = 8*n + 2.

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A017029 a(n) = 7*n + 4.

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A017017 a(n) = 7*n + 3.

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