A103517 -id:A103517 - cp's OEIS frontend

A299174 The positive even integers.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144

Offset: 1

Joss Langford, Feb 04 2018

Possible periods of Post's {00, 1101} tag system. - Charles R Greathouse IV, Dec 13 2021

Numbers m such that 2^m - m is divisible by 2. - Bernard Schott, Dec 15 2021

Joss Langford, Simple Integer Sequences
James B. Shearer, Periods of strings, American Scientist Vol. 84, No. 3 (May-June 1996), p. 207. (See final page of pdf, and see also pp. 3-4 for an introduction.)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Equals A005843 without the leading zero.
Bisection of A000027. Complement of A004273. - Omar E. Pol, Feb 25 2018
First row of A083140.
Cf. A005408.
Cf. A000325, A047257.
Essentially the same as A163300, A103517, A051755, A005843 and A004277.

Maple
```
A299174 := n->2*n;
```
Mathematica
```
Range[2, 180, 2]
```

a(n)=2*n \\ Charles R Greathouse IV, Dec 13 2021

R
```
seq(2, 180, 2)
```

a(n) = 2*n, n >= 1.
G.f.: 2*x/(1 - x)^2; corrected by Ilya Gutkovskiy, Mar 29 2018
a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Jul 17 2025

A272651 The no-3-in-line problem: maximal number of points from an n X n square grid so that no three lie on a line.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92

Offset: 1

N. J. A. Sloane, May 10 2016

a(47) is the first open case.
It is conjectured that a(n) < 2n for all sufficiently large n.
A000769 has an extensive list of references and links.
Comment from Warren D. Smith, May 10 2015: 2n is a trivial upper bound, because if you pick 2n+1 points, then some three must lie on a horizontal line.
Apart from the offset this may be the same as A103517. - R. J. Mathar, May 13 2016

A. Flammenkamp, Progress in the no-three-in-line problem
A. Flammenkamp, Solutions of the no-three-in-line problem

Cf. A000769.

A326478 a(n) = ndenominator(nBernoulli(n-1))/denominator(Bernoulli(n-1)).

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 3, 10, 1, 12, 1, 14, 5, 16, 1, 18, 1, 20, 7, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 32, 11, 34, 35, 36, 1, 38, 13, 40, 1, 42, 1, 44, 3, 46, 1, 48, 7, 50, 17, 52, 1, 54, 55, 56, 19, 58, 1, 60, 1, 62, 21, 64, 13, 66, 1, 68, 23, 70, 1, 72, 1

Offset: 1

Peter Luschny, Jul 16 2019

nonn

Empirical: a(2*n) = [x^n] x*(2/(x - 1)^2 - 1) for n >= 1, implying the conjecture that a(2*n) = A103517(n+1) and/or A272651(n).

Conjectural, the odd fixed points > 1 of this sequence are A121707; in other words, for n > 1, denominator(n*Bernoulli(n-1)) = denominator(Bernoulli(n-1)) <=> n | Sum_{k=1..n-1} k^(n-1). (See the conjectures of Thomas Ordowski in A121707.)

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A326577, A326578, A103517, A272651, A027641/A027642 (Bernoulli), A121707.

A326478 := n -> n*denom(n*bernoulli(n-1))/denom(bernoulli(n-1)):
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> n/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..73);

a[n_] := Module[{b =  BernoulliB[n - 1]}, n * Denominator[n * b] / Denominator[b]]; Array[a, 100] (* Amiram Eldar, Apr 26 2024 *)

a(n) = n*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

a(prime(n)) = 1.

a(n) = n/gcd(n*N(n-1), D(n-1)), with N(k)/D(k) = B(k) the k-th Bernoulli number.

A103516 Triangle read by rows: count in a vee.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 0, 0, 4, 5, 0, 0, 0, 5, 6, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 0, 0, 0, 0, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 0, 9, 10, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Feb 09 2005

Row sums are A103517, antidiagonal sums are A187012.

			Triangle begins
1,
2, 2,
3, 0, 3,
4, 0, 0, 4,
5, 0, 0, 0, 5,
6, 0, 0, 0, 0, 6,
7, 0, 0, 0, 0, 0, 7,
8, 0, 0, 0, 0, 0, 0, 8,
...

Join[{1},Flatten[Table[Flatten[Join[{n,PadRight[{},n-2,0],n}]],{n,2,15}]]] (* Harvey P. Dale, Mar 23 2013 *)

tabl(nn) = {for (i=1, nn, for (j=1, i, if (j == 1, t = i, if (j==i, t = i, t = 0)); print1(t, ", ");); print(););} \\ Michel Marcus, Aug 30 2013

Number triangle T(n, k)=if(k<=n, 0^(k(n-k))*(n+1), 0)

A135853 A103516 * A007318 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 4, 2, 6, 6, 3, 8, 12, 12, 4, 10, 20, 30, 20, 5, 12, 30, 60, 60, 30, 6, 14, 42, 105, 140, 105, 42, 7, 16, 56, 168, 280, 280, 168, 56, 8, 18, 72, 252, 504, 630, 504, 252, 72, 9, 20, 90, 360, 840, 1260, 1260, 840, 360, 90, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle are:
   1;
   4,   2;
   6,   6,  3;
   8,  12,  12,   4;
  10,  20,  30,  20,   5;
  12,  30,  60,  60,  30,   6;
  14,  42, 105, 140, 105,  42,   7;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A007318, A103516.
Cf. A103517 (1st column), A135854 (row sums).
Cf. A135852 (= A007318 * A103516).

T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//flatten (* G. C. Greubel, Dec 07 2016 *)

def A135853(n,k):
    if (n==0): return 1
    elif (k==0): return 2*(n+1)
    else: return (k+1)*binomial(n+1, k+1)
flatten([[A135853(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022

T(n, k) = (A103516 * A007318)(n, k).
Sum_{k=0..n} T(n, k) = A135854(n).
T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - G. C. Greubel, Dec 07 2016
T(n, 0) = A103517(n). - G. C. Greubel, Feb 06 2022

A284359 Double triangle (2n+2 terms by row). Every row is 2n + 1 followed by 2n + 1 times 2n + 2.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Mar 25 2017

In essence the same as A167991. - R. J. Mathar, Mar 27 2017

			1,  2,
3,  4,  4,  4,
5,  6,  6,  6,  6,  6,
7,  8,  8,  8,  8,  8,  8,  8,
9, 10, 10, 10, 10, 10, 10, 10, 10, 10,
... .
The row sum is A000466(n+1).

Cf. A000466, A005408, A103517 (main diagonal), A167381.

Table[2 n + 2 - Boole[k == 1], {n, 0, 8}, {k, 2 n + 2}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

for(n=0, 10, for(k=1, 2*n + 2, print1(2*n + 2 - (k==1), ", ");); print();) \\ Indranil Ghosh, Mar 26 2017, translated from Mathematica code

for n in range(0, 11):
    print([2*n + 2 -(k==1) for k in range(1, 2*n + 3)])
# Indranil Ghosh, Mar 26 2017

a(n) = A167381(n+1) - A167381(n).

A237285 Lexicographically earliest sequence of primes such that a(n)*n / sum(i=1..n, a(n) ) is strictly increasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 29, 37, 47, 59, 71, 89, 107, 127, 149, 173, 199, 227, 257, 293, 331, 373, 419, 467, 521, 577, 641, 709, 787, 859, 937, 1019, 1103, 1193, 1289, 1399, 1511, 1621, 1741, 1867, 1997, 2129, 2267, 2411, 2579, 2741, 2909, 3079, 3257, 3449

Offset: 1

Jaroslav Krizek, Feb 24 2014

nonn

If we replace in name of sequence:
primes -> nonprime numbers, then a(n) = A103517(n-1),
primes -> composite numbers, then a(n) = A103517(n),
primes -> noncomposite numbers, then a(n) = A237288(n),
primes -> natural numbers, then a(n) = A000027(n).

			For n=6: prime a(6) = 17 > a(5) = 11 is the smallest prime such that (6*17 / 45) > (5*11 / 28); a(6) is not 13 because (6*13 / (45-4)) < (5*11 / 28).

Cf. A000040 (primes).

A237288 Lexicographically earliest sequence of noncomposite numbers such that a(n)*n / sum(i=1..n, a(n) ) is strictly increasing.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 17, 23, 31, 41, 53, 67, 83, 101, 127, 151, 179, 211, 251, 293, 337, 389, 443, 503, 569, 641, 719, 809, 907, 1009, 1117, 1229, 1361, 1493, 1637, 1787, 1949, 2129, 2309, 2503, 2707, 2917, 3137, 3371, 3613, 3877, 4153, 4441, 4751, 5059, 5381

Offset: 1

Jaroslav Krizek, Feb 28 2014

nonn

If we replace in name of sequence:
noncomposite numbers -> nonprime numbers, then a(n) = A103517(n-1),
noncomposite numbers -> composite numbers, then a(n) = A103517(n),
noncomposite numbers -> primes, then a(n) = A237285(n),
noncomposite numbers -> natural numbers, then a(n) = A000027(n).

			For n=8: noncomposite number a(8) = 23 > a(7) = 17 is the smallest noncomposite number such that (8*23 / 69) > (7*17 / 46), a(8) is not 19 because (8*19 / (69-4)) < (7*17 / 46).

Cf. A008578 (noncomposite numbers).

A361871 The smallest order of a non-abelian group with an element of order n.

Original entry on oeis.org

6, 6, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 1

Yue Yu, Apr 01 2023

Wikipedia, Lagrange's theorem (group theory)

Essentially the same as A163300, A103517, A051755.

a(n) = 2*n for n >= 3.

Proof: By Lagrange's theorem in group theory we have that n divides a(n) for all n. A group of order n and with an element of order n is the cyclic group of order n, hence being abelian. On the other hand, the dihedral group D_{2n} is non-abelian for n >= 3 and contains an element of order n. - Jianing Song, Aug 11 2023

cp's OEIS Frontend

A299174 The positive even integers.

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A272651 The no-3-in-line problem: maximal number of points from an n X n square grid so that no three lie on a line.

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A326478 a(n) = n*denominator(n*Bernoulli(n-1))/denominator(Bernoulli(n-1)).

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Mathematica

PARI

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A103516 Triangle read by rows: count in a vee.

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PARI

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A135853 A103516 * A007318 as an infinite lower triangular matrix.

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Mathematica

Sage

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A284359 Double triangle (2*n+2 terms by row). Every row is 2*n + 1 followed by 2*n + 1 times 2*n + 2.

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Mathematica

PARI

Python

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A237285 Lexicographically earliest sequence of primes such that a(n)*n / sum(i=1..n, a(n) ) is strictly increasing.

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A237288 Lexicographically earliest sequence of noncomposite numbers such that a(n)*n / sum(i=1..n, a(n) ) is strictly increasing.

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A326478 a(n) = ndenominator(nBernoulli(n-1))/denominator(Bernoulli(n-1)).

A284359 Double triangle (2n+2 terms by row). Every row is 2n + 1 followed by 2n + 1 times 2n + 2.