A047304 -id:A047304 - 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

A020658 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 99

Offset: 1

David W. Wilson

nonn

This is different from A047304: note the gap between 41 and 50. - M. F. Hasler, Oct 07 2014

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A047304.
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

Noap:= proc(N,m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A,forbid,n,c,ds,j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
od:
convert(A,list)
end proc:
Noap(100, 7); # Robert Israel, Jan 04 2016

Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] + 1 (* Jean-François Alcover, Aug 18 2023, after M. F. Hasler *)

a(n) = A020657(n)+1. - M. F. Hasler, Oct 07 2014

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

A109720 Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.

Original entry on oeis.org

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

Offset: 0

Bruce Corrigan (scentman(AT)myfamily.com), Aug 09 2005

This sequence also represents n^12 mod 7; n^18 mod 7; (exponents are = 0 mod 6).
Characteristic sequence for numbers n>=1 to be relatively prime to 7. - Wolfdieter Lang, Oct 29 2008
a(n+4), n>=0, (periodic 1,1,1,0,1,1,1) is also the characteristic sequence for mod m reduced positive odd numbers (i.e., gcd(2*n+1,m)=1, n>=0) for each modulus m from 7*A003591 = [7,14,28,49,56,98,112,196,...]. [Wolfdieter Lang, Feb 04 2012]

Index entries for characteristic functions - _Reinhard Zumkeller_, Nov 30 2009
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1)

Cf. A010876 = n mod 7; A053879 = n^2 mod 7; A070472 = m^3 mod 7; A070512 = n^4 mod 7; A070593 = n^5 mod 7.

Cf. A168185, A145568, A168184, A168182, A168181, A097325, A011558, A166486, A011655, A000035.

PadRight[{},120,{0,1,1,1,1,1,1}] (* Harvey P. Dale, Jul 09 2018 *)

a(n)=n^6%7 \\ Charles R Greathouse IV, Sep 24 2015

[power_mod(n,6,7)for n in range(0, 105)] # Zerinvary Lajos, Nov 06 2009

a(n) = 0 if n=0 mod 7; a(n)= 1 else.
G.f. = (x+x^2+x^3+x^4+x^5+x^6)/(1-x^7)= -x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( (x-1)*(1+x+x^2+x^3+x^4+x^5+x^6) ).
a(n)=1-A082784(n); a(A047304(n))=1; a(A008589(n))=0; A033439(n) = SUM(a(k)*(n-k): 0<=k<=n). - Reinhard Zumkeller, Nov 30 2009
Multiplicative with a(p) = (if p=7 then 0 else 1), p prime. - Reinhard Zumkeller, Nov 30 2009
Dirichlet g.f. (1-7^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

A319528 a(n) = 8 * sigma(n).

Original entry on oeis.org

8, 24, 32, 56, 48, 96, 64, 120, 104, 144, 96, 224, 112, 192, 192, 248, 144, 312, 160, 336, 256, 288, 192, 480, 248, 336, 320, 448, 240, 576, 256, 504, 384, 432, 384, 728, 304, 480, 448, 720, 336, 768, 352, 672, 624, 576, 384, 992, 456, 744, 576, 784, 432, 960, 576, 960, 640, 720, 480, 1344, 496, 768, 832

Offset: 1

Omar E. Pol, Sep 22 2018

8 times the sum of the divisors of n.

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) in which the structure of every 45-degree three-dimensional sector arises after the 45-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is an eight-pointed star formed by eight rhombuses (see Links section).

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle A237593 before the 45-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

k times sigma(n), k=1..7: A000203, A074400, A272027, A239050, A274535, A274536, A319527.

Cf. A008589, A047304, A237593, A283078.

List([1..70],n->8*Sigma(n)); # Muniru A Asiru, Sep 28 2018

with(numtheory): seq(8*sigma(n), n=1..64);

8*DivisorSigma[1,Range[70]] (* Harvey P. Dale, Dec 24 2018 *)

PARI
```
a(n) = 8 * sigma(n);
```

a(n) = 8*A000203(n) = 4*A074400(n) = 2*A239050(n).
a(n) = A000203(n) + A319527(n).
Dirichlet g.f.: 8*zeta(s-1)*zeta(s). (After Ilya Gutkovskiy)
Conjecture: a(n) = sigma(7*n) = A283078(n) iff n is not a multiple of 7.
Conjecture is true, since sigma is multiplicative, so if (7,n) = 1 then sigma(7*n) = sigma(7)*sigma(n) = 8*sigma(n). - Charlie Neder, Oct 02 2018

A123866 a(n) = n^6 - 1.

Original entry on oeis.org

0, 63, 728, 4095, 15624, 46655, 117648, 262143, 531440, 999999, 1771560, 2985983, 4826808, 7529535, 11390624, 16777215, 24137568, 34012223, 47045880, 63999999, 85766120, 113379903, 148035888, 191102975, 244140624, 308915775, 387420488

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 7 = 0 iff n mod 7 > 0: a(A008589(n))=6; a(A047304(n)) = 0; a(n) mod 7 = 6*(1-A082784(n)).

a(n) = A005563(n-1)*A059826(n) = A068601(n)*A001093(n). - Reinhard Zumkeller, Feb 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A024004, A001014, A005563, A123865, A123867, A123868.

List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019

a123866 = (subtract 1) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014

[n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014

Table[n^6-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,63,728,4095, 15624,46655, 117648}, 30] (* Harvey P. Dale, Nov 18 2012 *)

A123866(n):=n^6 -1 $ makelist(A123866(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015

[n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(63 + 287*x + 322*x^2 + 42*x^3 + 7*x^4 - x^5)/(1-x)^7. - Colin Barker, May 08 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=0, a(2)=63, a(3)=728, a(4)=4095, a(5)=15624, a(6)=46655, a(7)=117648. - Harvey P. Dale, Nov 18 2012
Sum_{n>=2} 1/a(n) = 11/12 - Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/6. - Vaclav Kotesovec, Feb 14 2015
E.g.f.: 1 + (-1 + x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 6*Pi^2*sech(sqrt(3)*Pi/2)^2. - Amiram Eldar, Jan 20 2021

A125704 Table read by antidiagonals: row n contains the positive integers (in order) which are coprime to the n-th prime.

Original entry on oeis.org

1, 1, 3, 1, 2, 5, 1, 2, 4, 7, 1, 2, 3, 5, 9, 1, 2, 3, 4, 7, 11, 1, 2, 3, 4, 6, 8, 13, 1, 2, 3, 4, 5, 7, 10, 15, 1, 2, 3, 4, 5, 6, 8, 11, 17, 1, 2, 3, 4, 5, 6, 8, 9, 13, 19, 1, 2, 3, 4, 5, 6, 7, 9, 11, 14, 21, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 16, 23, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 25, 1, 2, 3, 4, 5

Offset: 1

Leroy Quet, Jan 31 2007

			Beginning of table:
  1,  3,  5,  7,  9, 11, 13, ...
  1,  2,  4,  5,  7,  8, 10, 11, ...
  1,  2,  3,  4,  6,  7,  8,  9, 11, ...
  1,  2,  3,  4,  5,  6,  8,  9, 10, ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).

A125704 := proc(n,m) local p,i,a ; p := ithprime(n) ; a := 1 ; for i from 2 to m do a := a+1 ; while gcd(a,p) <> 1 do a := a+1 ; od ; od ; RETURN(a) ; end : maxdiag := 15 ; for d from 1 to maxdiag do for n from d to 1 by -1 do printf("%d,",A125704(n,d-n+1)) ; od ; od; # R. J. Mathar, Feb 02 2007

Table[Function[n, k - 1 + Floor[(k + Prime[n] - 2)/(Prime[n] - 1)]][m - k + 1], {m, 14}, {k, m}] // Flatten (* Michael De Vlieger, Oct 10 2017, after PARI by Benoit Cloitre *)

T(n,m)=m-1+floor((m+prime(n)-2)/(prime(n)-1)) \\ Benoit Cloitre, Jul 11 2009

T(1,m) = A005408(m). T(2,m) = A001651(m). T(3,m) = A047201(m). T(4,m) = A047304(m). - R. J. Mathar, Feb 02 2007

T(n,m) = m - 1 + floor((m+prime(n)-2)/(prime(n)-1)) where prime(n) = n-th prime. - Benoit Cloitre, Jul 11 2009

More terms from R. J. Mathar, Feb 02 2007

A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, -2, 3, 4, 5, -6, 10, -11, 13, -15, 15, 18, -22, 24, 25, 29, -31, 33, -37, -45, -55, 55, 59, -67, -72, 74, 80, -81, 85, -86, 88, -90, -95, 99, -101, -102, 108, -116, 118, -122, 129, -130, 143, 148, -151, -155, -157, 158, 159, -162, 164, 165

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 7 = (6*k+1)^2 + 6, which is six more than a square.

			a(2)=-2 since 36*(-2)^2 + 12*(-2) + 7 = 127, which is prime (as well as being six more than a square).

This sequence and formula generate all primes of the form k^2+6, i.e., A056909. Except for the first term, none of the a(n) are a multiple of 7 and so the rest of this sequence is a subsequence of A047304. Cf. A056900, A056902, A056904, A056906, A056907, A056908.

a(n) = (-1 +- sqrt(A056909(n) - 6))/6, choosing +- to give an integer result for each n.

A067761 Positive integers divisible by 5 but not by 7.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 40, 45, 50, 55, 60, 65, 75, 80, 85, 90, 95, 100, 110, 115, 120, 125, 130, 135, 145, 150, 155, 160, 165, 170, 180, 185, 190, 195, 200, 205, 215, 220, 225, 230, 235, 240, 250, 255, 260, 265, 270, 275, 285, 290, 295, 300, 305, 310, 320

Offset: 1

Jonathan A. Heese (macgyver86(AT)aol.com), Feb 06 2002

Kenneth H. Rosen, Discrete Mathematics and Its Applications, 4th Ed., p. 79, 1.7.32.b.

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

Cf. A047304.

Select[Range[65]*5, Mod[5#, 7] != 0 &] (* corrected by Georg Fischer, Jun 17 2020 *)

a(n) = 5*A047304(n). - Zak Seidov, Mar 19 2014
a(n) = a(n - 6) + 35. - David A. Corneth, Jun 17 2020
G.f.: 5*x*(x^6+x^5+x^4+x^3+x^2+x+1)/(x^7-x^6-x+1). - Alois P. Heinz, Jun 17 2020

Edited by Robert G. Wilson v, Feb 07 2002

a(36)=205 corrected by Georg Fischer, Jun 17 2020

cp's OEIS Frontend

A008589 Multiples of 7.

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A020658 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 7.

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A082784 Characteristic function of multiples of 7.

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A109720 Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.

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A319528 a(n) = 8 * sigma(n).

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A123866 a(n) = n^6 - 1.

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A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).