A185359 -id:A185359 - 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

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 *)

A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Peter Luschny, Jan 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011

A380088 The largest unitary divisor of n that is a term in A207481.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 12 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A377518, A380085, A380089, A380090.

f[p_, e_] := If[e <= p, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] <= f[i, 1], f[i, 1]^f[i, 2], 1));}

Multiplicative with a(p^e) = p^e if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < n if and only if n is in A185359.
a(n) = n if and only if n is in A207481.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(2*(p+1)) + p^(2*p+1) - p^(p+1) - p^p + 1)/(p^(2*p+1) * (p+1)) = 0.87453068804586281444... .

A380089 The number of unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 12 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A377519, A380086, A380088, A380090.

f[p_, e_] := If[e <= p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= f[i,1], 2, 1));}

a(n) = A034444(A380088(n)).
Multiplicative with a(p^e) = 2 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034444(n) if and only if n is in A185359.
a(n) = A034444(n) if and only if n is in A207481.
a(n) = A377519(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((p+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(p+1))) = 0.93168306734008028353...

A380090 The sum of the unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Jan 12 2025

First differs from A371242 at n = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

f[p_, e_] := If[e <= p, p^e, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], f[i,1]^f[i,2], 0) + 1);}

a(n) = A034448(A380088(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034448(n) if and only if n is in A185359.
a(n) = A034448(n) if and only if n is in A207481.
a(n) = A377520(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .

A253782 Numbers n for which A075167(n) <> A252464(n).

Original entry on oeis.org

16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 376, 378, 384, 392, 400, 405, 408, 416, 424, 432, 440, 448, 456, 459

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Complement: A253781.
Subsequence of A013929 (numbers that are not squarefree).
Cf. A075167, A252464.
Differs from A185359(n+1) for the first time at n=17, where a(17) = 135, while A185359(18) = 136.

A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416

Offset: 1

Amiram Eldar, Sep 22 2023

First differs from A185359 at n = 22.
Numbers k such that A020639(k) < A051904(k).
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/prime(n)^(prime(n)+1)) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/8, 1/162, 1/46875, 4/86472015 and 8/109844993185235.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13119421909731920416... .

			8 = 2^3 is a term since its least prime factor, 2, is smaller than its exponent, 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020639, A051904.
Subsequences: A008590 \ {0}, A365887, A365888.
Subsequence of A185359.

q[n_] := Less @@ FactorInteger[n][[1]]; Select[Range[2, 420], q]

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}

A185358 The period of the sequence i^i (mod n) starts from i=a(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1

Offset: 1

José María Grau Ribas, Jan 21 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000
R. Hampel, The length of the shortest period of rests of numbers n^n, Ann. Polon. Math. 1 (1955), 360-366.

Cf. A185359.

a[p_,e_]:=1- p*(1+Floor[-e/p]);a[n_]:=Max@Module[{fa=FactorInteger[n]},Table[a[fa[[i,1]],fa[[i,2]]],{i,1,Length[fa]}]];Table[a[n],{n,1,84}]

from sympy import factorint, floor
def a(n):
    f=factorint(n)
    return 1 if n==1 else max(1 - i*(1 + (-f[i])//i) for i in f)
print([a(n) for n in range(1, 201)]) # Indranil Ghosh, Jun 29 2017

If n = Product_{pi^ei} then a(n) = Max_{1- pi*(1+floor[-ei/pi])}.

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A008590 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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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

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A380088 The largest unitary divisor of n that is a term in A207481.

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A380089 The number of unitary divisors of n that are terms in A207481.

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A380090 The sum of the unitary divisors of n that are terms in A207481.

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A253782 Numbers n for which A075167(n) <> A252464(n).

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A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

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