A047592 -id:A047592 - cp's OEIS frontend

A008590 Multiples of 8.

0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432

Offset: 0

N. J. A. Sloane

For n > 3, the number of squares on the infinite 4-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
First differences of odd squares: a(n) = A016754(n) - A016754(n-1) for n > 0. - Reinhard Zumkeller, Nov 08 2009
Complement of A047592; A168181(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
For n >= 1, number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) = n. - Michel Marcus, Nov 28 2014
These terms are the area of square frames (using integer lengths), with specific instances where the area equals the sum of inner and outer perimeters (see example and formula below). The thickness of the frames are always 2, which is of further significance when considering that all regular polygons have an area that is equal to perimeter when apothem is 2. - Peter M. Chema, Apr 03 2016
From Lechoslaw Ratajczak, Sep 03 2017: (Start)
Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers.
a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even).
a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even).
a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even).
The conjecture was checked for 5*10^4 consecutive integers. (End)

			Beginning with n = 2, illustration of the terms as the area of square frames, where area equals the sum of inner and outer perimeters:
                                                                _ _ _ _ _ _ _ _
                                              _ _ _ _ _ _ _    |               |
                              _ _ _ _ _ _    |             |   |    _ _ _ _    |
                _ _ _ _ _    |           |   |    _ _ _    |   |   |       |   |
   _ _ _ _     |         |   |    _ _    |   |   |     |   |   |   |       |   |
  |       |    |    _    |   |   |   |   |   |   |     |   |   |   |       |   |
  |       |    |   |_|   |   |   |_ _|   |   |   |_ _ _|   |   |   |_ _ _ _|   |
  |       |    |         |   |           |   |             |   |               |
  |_ _ _ _|    |_ _ _ _ _|   |_ _ _ _ _ _|   |_ _ _ _ _ _ _|   |_ _ _ _ _ _ _ _|
  a(2) = 16      a(3) = 24     a(4) = 32        a(5) = 40          a(6) = 48
The inner square has side n-2 and outer square side n+2, pursuant to the above and related formula. Note that a(2) is simply the square 4*4, with the inner square having side 0; considering the inner square as a center point, this frame also has thickness of 2.
E.g., for a(4), the square frame is formed by a 6 X 6 outer square and a 2 X 2 inner square, with the area (6 X 6 minus 2 X 2) equal to the perimeter (4*6 + 4*2) at 32. - _Peter M. Chema_, Apr 03 2016

Ivan Panchenko, Table of n, a(n) for n = 0..200
Ch. Berdellé, Démonstration élémentaire d’un théorème énoncé par M. E. Catalan, Bulletin de la S. M. F., tome 17 (1889), p. 102. [Every positive multiple of 8 is the sum of 8 odd squares.]
E. Catalan, Extrait d’une lettre, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 320.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Leo Tavares, Illustration: Square Ray Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Cf. A010014.
Essentially the same as A022144.
Subsequence of A185359, apart initial 0.
Cf. A000567, A016754, A045944, A047592, A168181.

a008590 = (* 8)
a008590_list = [0,8..]  -- Reinhard Zumkeller, Apr 02 2012

Table[8*n,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *)

a(n) = 8*n; \\ Altug Alkan, Apr 08 2016

def A008590(n): return n<<3 # Chai Wah Wu, Mar 11 2025

a(n) = (2*n+1)^2 - (2*n-1)^2. - Xavier Acloque, Oct 22 2003
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 8*n = 2*a(n-1) - a(n-2).
G.f.: 8*x/(x-1)^2. (End)
a(n) = Sum_{k=1..4n} (i^k + 1)*(i^(4n-k) + 1), where i=sqrt(-1). - Bruno Berselli, Mar 19 2012
a(n) = (n+2)^2 - (n-2)^2 = 4*(n+2) + 4*(n-2), as exemplified below. - Peter M. Chema, Apr 03 2016
a(n) = A000567(n+1) - A045944(n-1). - Leo Tavares, Mar 25 2022
E.g.f.: 8*x*exp(x). - Stefano Spezia, Apr 03 2023

A168181 Characteristic function of numbers that are not multiples of 8.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Nov 30 2009

Multiplicative with a(p^e) = (if p=2 then A019590(e) else 1), p prime and e>0.

Period 8 Repeat: [0, 1, 1, 1, 1, 1, 1, 1]. - Wesley Ivan Hurt, Jun 21 2014

			G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168184, A168182, A109720, A097325, A011558, A166486, A011655, A000035, A010877, A244413, A253513.

[Sign(n mod 8) : n in [0..100]]; // Wesley Ivan Hurt, Jun 21 2014

with(numtheory); A168181:=n->signum(n mod 8); seq(A168181(n), n=0..100); # Wesley Ivan Hurt, Jun 21 2014

Table[Sign[Mod[n,8]], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 21 2014 *)

a(n)=n%8 > 0 \\ Felix Fröhlich, Aug 11 2014

def A168181(n): return int(bool(n&7)) # Chai Wah Wu, Jul 09 2022

a(n+8) = a(n);
a(n) = A000007(A010877(n));
a(A047592(n)) = 1; a(A008590(n)) = 0;
A033440(n) = Sum_{k=0..n} a(k)*(n-k).
Dirichlet g.f. (1-1/8^s)*zeta(s). - R. J. Mathar, Feb 19 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
a(n) = sign(n mod 8). - Wesley Ivan Hurt, Jun 21 2014
a(n) = sign( 1 - floor(cos(Pi*n/4)) ). - Wesley Ivan Hurt, Jun 21 2014
Euler transform of length 8 sequence [ 1, 0, 0, 0, 0, 0, -1, 1]. - Michael Somos, Jun 24 2014
Moebius transform is length 8 sequence [ 1, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Jun 24 2014
G.f.: x * (1 - x^7) / ((1 - x) * (1 - x^8)). - Michael Somos, Jun 24 2014
a(n) = 1-A253513(n). - Antti Karttunen, Oct 08 2017

A207481 Numbers such that e <= p for all p^e in their prime factorization, p prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Reinhard Zumkeller, Feb 18 2012

nonn

Proper subsequence of A047592, a(n) = A047592(n) for n <= 70.

The asymptotic density of this sequence is Product_{p prime} 1 - 1/p^(p+1) = 0.86414207232219778408... - Amiram Eldar, Nov 24 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A185359 (complement), A054743 (same construction, but with e > p)

a207481 n = a207481_list !! (n-1)
a207481_list = [x | x <- [1..], and $ zipWith (<=)
                    (map toInteger $ a124010_row x) (a027748_row x) ]

okQ[n_] := AllTrue[FactorInteger[n], #[[2]] <= #[[1]]&];
Select[Range[100], okQ] (* Jean-François Alcover, Jun 08 2016 *)

A255805 Numbers with no zeros in base-8 representation.

Original entry on oeis.org

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A047592, A207481.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Octal
Index entries for 2-automatic sequences.

Cf. A007094, A100970 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255808 (base 9), A052382 (base 10).

a255805 n = a255805_list !! (n-1)
a255805_list = iterate f 1 where
   f x = 1 + if r < 7 then x else 8 * f x'  where (x', r) = divMod x 8

Select[Range[100],DigitCount[#,8,0]==0&] (* Harvey P. Dale, Jun 08 2015 *)

isok(m) = vecmin(digits(m,8)) > 0; \\ Michel Marcus, Jan 23 2022

def ok(n): return '0' not in oct(n)[2:]
print([k for k in range(85) if ok(k)]) # Michael S. Branicky, Jan 23 2022

from sympy import integer_log
def A255805(n):
    m = integer_log(k:=6*n+1,7)[0]
    return sum(1+(k-7**m)//(6*7**j)%7<<3*j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Jan 30 2025

First differs from A047592, A187320, A207481 and A255805 at n = 48: A047592(48) = A187320(48) = A207481(48) = A255805(48) = 54 is not a term of this sequence.
Numbers k such that A020639(k) >= A051903(k).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = nextprime(k) = A007918(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.84999238500582943243..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p < i} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p < i} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

			6 = 2^1 * 3^1 is a term since 2 >= 1.
8 = 2^3 is not a term since 2 < 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Wikipedia, Rough number.

Cf. A020639, A051903, A007918.
Cf. A047592, A187320, A207481, A255805.
Subsequences: A380692, A380694, A380695.

q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Max[f[[;; , 2]]]]; Select[Range[100], q]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] >= vecmax(e));

A187320 a(n) = floor((Pi-2)*n); complement of A186544.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128, 130, 131, 132, 133, 134, 135, 136

Offset: 0

Clark Kimberling, Mar 08 2011

nonn

a(n) = A047592(n) for 1 <= n <= 112, but a(113) = 128 and A047592(113) = 129. - Georg Fischer, Oct 28 2018

Georg Fischer, Table of n, a(n) for n = 0..1000

Cf. A000796, A186544.

seq(floor((Pi-2)*n),n=0..120); # Muniru A Asiru, Oct 28 2018

Mathematica
```
Table[Floor[(Pi-2)n], {n,0,120}]
```

a(n) = (Pi-2)*n\1 \\ Charles R Greathouse IV, Jan 27 2015

a(n) = floor((Pi-2)*n).

A225055 Irregular triangle which lists the three positions of 2*n-1 in A060819 in row n.

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 5, 10, 20, 7, 14, 28, 9, 18, 36, 11, 22, 44, 13, 26, 52, 15, 30, 60, 17, 34, 68, 19, 38, 76, 21, 42, 84, 23, 46, 92, 25, 50, 100, 27, 54, 108, 29, 58, 116, 31, 62, 124, 33, 66, 132, 35, 70, 140, 37, 74, 148

Offset: 1

Paul Curtz, Apr 26 2013

There are no multiples of 8 in the triangle.
A047592 contains a sorted list of all elements of the triangle.
The triangle is a member of a family of triangles with parameter k that list the k positions of 2*n-1: 2*n-1 in A000027 (k=1), A043547 the k=2 positions in A026741, the triangle 1,2,4,8; 3,6,12,24;... with the k=4 positions in A106609, or the triangle 1,2,4,8,16; 3,6,12,24,48;... with the k=5 positions in A106617.

			1, 2, 4;  # 1 at A060819(1), A060819(2) and A060819(4)
3, 6, 12;  # 3 at A060819(3), A060819(6) and A060819(12)
5, 10, 20;
7, 14, 28;
9, 18, 36;
11, 22, 44;
13, 26, 52;
15, 30, 60;

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A147587. A042968.

numberOfTriplets = 19; A060819 = Table[n/GCD[n, 4], {n, 1, 8*numberOfTriplets}]; Table[Position[A060819, 2*n-1], {n, 1, numberOfTriplets}] // Flatten (* Jean-François Alcover, Apr 30 2013 *)

T(n,1) = 2*n-1. T(n,2) = 4*n-2. T(n,3) = 8*n-4.

A294116 Fibonacci sequence beginning 2, 21.

Original entry on oeis.org

2, 21, 23, 44, 67, 111, 178, 289, 467, 756, 1223, 1979, 3202, 5181, 8383, 13564, 21947, 35511, 57458, 92969, 150427, 243396, 393823, 637219, 1031042, 1668261, 2699303, 4367564, 7066867, 11434431, 18501298, 29935729, 48437027, 78372756, 126809783, 205182539, 331992322, 537174861

Offset: 0

Bruno Berselli, Oct 23 2017

Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045.
Subsequence of A047201, A047592, A113763.
Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2*A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32).

a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];

Mathematica
```
LinearRecurrence[{1, 1}, {2, 21}, 40]
```

Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017

a = BinaryRecurrenceSequence(1, 1, 2, 21)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017

G.f.: (2 + 19*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) =     Lucas(n) + g(0,20;n), see A022354;
a(n) = Fibonacci(n) + g(2,20;n), see A022372;
a(n) =  2*g(1,21;n) - g(0,21;n);
a(n) =     g(1,k;n) + g(1,21-k;n) for all k in Z.
a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
a(n) = 2*Fibonacci(n-1) + 21*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 21.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017

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A008590 Multiples of 8.

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A168181 Characteristic function of numbers that are not multiples of 8.

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A207481 Numbers such that e <= p for all p^e in their prime factorization, p prime.

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A255805 Numbers with no zeros in base-8 representation.

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A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

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A187320 a(n) = floor((Pi-2)*n); complement of A186544.

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A225055 Irregular triangle which lists the three positions of 2*n-1 in A060819 in row n.

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