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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

Original entry on oeis.org

1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010
Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011
2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
a(n) is not divisible by 3, 5, 7, or 11. - Vladimir Shevelev, Feb 03 2014
With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015
Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015
Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016
a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017
a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018
From Klaus Purath, Dec 07 2020: (Start)
a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199.
Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.
If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)
Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021
The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the external perimeter and the perimeter of inscribed squares having the cell (1,1) as a unique common vertex. See Spezia link. - Stefano Spezia, May 28 2025

Examples

			From _Bruno Berselli_, Mar 09 2015: (Start)
By the definition (first formula):
----------------------------------------------------------------------
  1       4         8           13            19              26
----------------------------------------------------------------------
                                                              X
                                              X              X X
                                X            X X            X X X
                    X          X X          X X X          X X X X
          X        X X        X X X        X X X X        X X X X X
  X      X X      X X X      X X X X      X X X X X      X X X X X X
          X        X X        X X X        X X X X        X X X X X
----------------------------------------------------------------------
(End)
From _Klaus Purath_, Dec 07 2020: (Start)
Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.
(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.
(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.
(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)
From _Omar E. Pol_, Aug 08 2021: (Start)
Illustration of initial terms:                             _ _
.                                           _ _           |_|_|_
.                              _ _         |_|_|_         |_|_|_|_
.                   _ _       |_|_|_       |_|_|_|_       |_|_|_|_|_
.          _ _     |_|_|_     |_|_|_|_     |_|_|_|_|_     |_|_|_|_|_|_
.   _     |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.  |_|    |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.
.   1       4         8          13            19              26
------------------------------------------------------------------------ (End)

References

  • A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
  • G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
  • Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.

Links

Crossrefs

Subsequence of A165157.
Triangular numbers (A000217) minus two.
Third diagonal of triangle in A059317.
Cf. A000096, A000124, A000217, A000290, A005408, A005843, A008277, A024206, A027379.
Cf. A028347, A038889, A038890, A049600, A091823, A101881, A113452, A113453, A113454.
Cf. A113455, A124199, A130296, A131818, A134225, A161680, A168244, A253909, A267633.

Programs

  • Haskell
    a034856 = subtract 1 . a000096 -- Reinhard Zumkeller, Feb 20 2015
  • Magma
    [Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011
  • Maple
    a := n -> hypergeom([-2, n-1], [1], -1);
seq(simplify(a(n)), n=1..53); # Peter Luschny, Aug 02 2014
  • Mathematica
    f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *)
Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)
  • Maxima
    A034856(n) := block(
        n-1+(n+1)*n/2
)$ /* R. J. Mathar, Mar 19 2012 */
  • PARI
    A034856(n)=(n+3)*n\2-1 \\ M. F. Hasler, Jan 21 2015
  • Python
    def A034856(n): return n*(n+3)//2 -1 # G. C. Greubel, Jun 15 2025

Formula

G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
a(n) = A049600(3, n-2).
a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003
With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003
Row sums of triangle A131818. - Gary W. Adamson, Jul 27 2007
Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
Row sums of triangle A134225. - Gary W. Adamson, Oct 14 2007
a(n) = A000217(n+1) - 2. - Omar E. Pol, Apr 23 2008
From Jaroslav Krizek, Sep 05 2009: (Start)
a(n) = a(n-1) + n + 1 for n >= 1.
a(n) = n*(n-1)/2 + 2*n - 1.
a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)
a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014
a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - Richard R. Forberg, Jan 17 2015
a(n) = A101881(2*(n-1)). - Reinhard Zumkeller, Feb 20 2015
a(n) = A253909(n+3) - A000217(n+3). - David Neil McGrath, May 23 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - David Neil McGrath, May 23 2015
For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016
From Klaus Purath, Dec 07 2020: (Start)
a(n) = A024206(n) + A024206(n+1).
a(2*n-1) = -A168244(n+1).
a(2*n) = A091823(n). (End)
Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021
a(n) + a(n+1) = A028347(n+2). - R. J. Mathar, Mar 13 2021
a(n) = A000290(n) - A161680(n-1). - Omar E. Pol, Mar 26 2021
E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 05 2021
a(n) = A024916(n) - A244049(n). - Omar E. Pol, Aug 01 2021
a(n) = A000290(n) - A000217(n-2). - Omar E. Pol, Aug 05 2021

Extensions

More terms from Zerinvary Lajos, May 12 2006

A275660 Numbers n such that sigma(n) = Sum_{j=1..k} d(n^j) for some k, where sigma(n) is the sum of the divisors of n and d(n) is the number of divisors of n.

Original entry on oeis.org

1, 13, 19, 34, 43, 53, 58, 89, 103, 151, 229, 251, 254, 329, 341, 349, 404, 433, 463, 593, 674, 701, 739, 859, 1033, 1223, 1429, 1483, 1506, 1670, 1709, 1826, 1846, 1886, 1889, 1948, 1951, 2067, 2091, 2143, 2255, 2308, 2431, 2699, 3001, 3079, 3319, 3739, 4003, 4093
Offset: 1

Views

Author

Paolo P. Lava, Aug 04 2016

Keywords

Comments

The primes in this sequence are A124199. - Robert Israel, Feb 20 2024

Examples

			d(53^1) + d(53^2) + d(53^3) + d(53^4) + d(53^5) + d(53^6) + d(53^7) + d(53^8) + d(53^9) = 54 = sigma(53).

Links

Crossrefs

Cf. A000005, A000203, A124199, A270389, A275661.

Programs

  • Maple
    P:= proc(q) local a,k,n;
for n from 1 to q do a:=sigma(n); k:=0;
while a>0 do k:=k+1; a:=a-tau(n^k); od;
if a=0 then print(n); fi; od; end: P(10^9);

A357219 Primes of the form T(p) - 2 where T(p) is the triangular number (A000217) with prime index p in A357218.

Original entry on oeis.org

13, 89, 151, 433, 701, 859, 1429, 1889, 2699, 4003, 4751, 11173, 12401, 18719, 19501, 27259, 33151, 36313, 38501, 39619, 49139, 56951, 75853, 80599, 83843, 104651, 129793, 135979, 146609, 188189, 205759, 226799, 246049, 318001, 367651, 385001, 388519, 431983, 454579
Offset: 1

Views

Author

Bernard Schott, Sep 18 2022

Keywords

References

  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 496, page 142.

Links

Crossrefs

Cf. A000217, A357218.
Subsequence of A124199.

Programs

  • Maple
    f:= n -> n*(n+1)/2-2:
select(isprime, map(f, [seq(ithprime(i),i=1..200)])); # Robert Israel, Sep 20 2022
  • Mathematica
    Select[(#*(# + 1)/2 - 2) & /@ Prime[Range[165]], PrimeQ] (* Amiram Eldar, Sep 18 2022 *)
  • PARI
    isok(p) = my(q); isprime(p) && ispolygonal(p+2, 3, &q) && isprime(q); \\ Michel Marcus, Sep 19 2022

Formula

a(n) = A000217(A357218(n))-2.

Extensions

More terms from David A. Corneth, Sep 18 2022

A175452 a(n) = smallest prime such that a(n)+2 is multiple of 2n+1.

Original entry on oeis.org

7, 3, 5, 7, 31, 11, 13, 83, 17, 19, 67, 23, 79, 317, 29, 31, 103, 109, 37, 367, 41, 43, 139, 47, 151, 157, 53, 283, 293, 59, 61, 193, 199, 67, 211, 71, 73, 229, 709, 79, 911, 83, 433, 443, 89, 277, 283, 677, 97, 503, 101, 103, 2459, 107, 109, 337, 113, 349, 593, 1087
Offset: 1

Views

Author

Zak Seidov, May 16 2010

Keywords

Comments

Terms appearing twice: a(1)=a(4)=7, a(5)=a(16)=31,...,
terms appearing thrice: a(28)=a(47)=a(142)=283, a(20)=a(61)=a(184)=367, etc.

Examples

			n=1: 7+2 is multiple of 3, n=2: 3+2 is multiple of 5, n=5: 31+2 is multiple of 11, n=8: 83+2 is multiple of 17.

Links

Crossrefs

Cf. A124199.

Programs

  • Mathematica
    s={};Do[k=2;While[Mod[2+(p=Prime[k]),n]>0,k++ ];AppendTo[s,p],{n,3,2001,2}];s
  • PARI
    a(n) = my(p=2); while ((p+2) % (2*n+1), p = nextprime(p+1)); p; \\ Michel Marcus, Jul 03 2021
  • Python
    from sympy import nextprime
def a(n):
    p, m = 2, 2*n+1
    while (p+2)%m: p = nextprime(p)
    return p
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jul 03 2021

A295265 Numbers m such that sum of its i first divisors equals the sum of its j first non-divisors for some i, j.

Original entry on oeis.org

4, 8, 10, 13, 14, 16, 19, 20, 21, 22, 26, 28, 30, 32, 34, 38, 39, 40, 43, 44, 46, 50, 52, 53, 56, 58, 60, 62, 63, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 89, 90, 92, 94, 98, 99, 100, 103, 104, 106, 110, 111, 112, 116, 117, 118, 122, 124, 128, 130, 132, 134, 135
Offset: 1

Views

Author

Michel Lagneau, Feb 22 2018

Keywords

Comments

Or numbers m such that Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) for some i, j where d(k) are the i first divisors and nd(k) the j non-divisors of m.
The corresponding sums are 3, 3, 3, 14, 3, 3, 20, 3, 11, 3, 3, (3 or 14), 11, 3, 3, 3, 17, 3, 44, 3, 3, 3, 3, 54, 3, 3, 15, 3, 11, 3, 3, 3, 33, 3, 3, 3, ... containing the set of primes {3, 11, 17, 23, 29, 37, 41, 43, 53, 59, 61, 71, 79, ...}.
The equality Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) is not always unique, for instance for a(12) = 28, we find 1 + 2 = 3 and 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14.
The primes of the sequence are 13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, ... (primes of the form k(k+1)/2 - 2; see A124199).
+-----+-----+-----+------+-----------------------------------------+
| n | i | j | a(n) | Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) |
+-----+-----+-----+------+-----------------------------------------+
| 1 | 2 | 1 | 4 | 1 + 2 = 3 |
| 2 | 2 | 1 | 8 | 1 + 2 = 3 |
| 3 | 2 | 1 | 10 | 1 + 2 = 3 |
| 4 | 2 | 4 | 13 | 1 + 13 = 2 + 3 + 4 + 5 = 14 |
| 5 | 2 | 1 | 14 | 1 + 2 = 3 |
| 6 | 2 | 1 | 16 | 1 + 2 = 3 |
| 7 | 2 | 5 | 19 | 1 + 19 = 2 + 3 + 4 + 5 + 6 = 20 |
| 8 | 2 | 1 | 20 | 1 + 2 = 3 |
| 9 | 3 | 3 | 21 | 1 + 3 + 7 = 2 + 4 + 5 = 11 |
| 10 | 2 | 1 | 22 | 1 + 2 = 3 |
| 11 | 2 | 1 | 26 | 1 + 2 = 3 |
| 12 | 2 | 1 | 28 | 1 + 2 = 3 |
| | 4 | 3 | 28 | 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14 |
| 13 | 4 | 2 | 30 | 1 + 2 + 3 + 5 = 4 + 7 = 11 |
| 14 | 2 | 1 | 32 | 1 + 2 = 3 |

Examples

			30 is in the sequence because d(1) + d(2) + d(3) + d(4) = 1 + 2 + 3 + 5 = 11 and nd(1) + nd(2) = 4 + 7 = 11.

Links

Crossrefs

Cf. A064510, A124199, A185729, A240698.

Programs

  • Maple
    with(numtheory):nn:=300:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):lst:={}:ii:=0:
  for i from 1 to n do:
   lst:=lst union {i}:
  od:
    lst:=lst minus d:n1:=nops(lst):
     for m from 1 to n0 while(ii=0) do:
      s1:=sum(‘d[i]’, ‘i’=1..m):
       for j from 1 to n1 while(ii=0) do:
        s2:=sum(‘lst[i]’, ‘i’=1..j):
         if s1=s2
          then
          ii:=1:printf(`%d, `,n):
         else
         fi:
        od:
     od:
  od:
  • Mathematica
    fQ[n_] := Block[{d = Divisors@ n}, nd = nd = Complement[Range@ n, d]; Intersection[Accumulate@ d, Accumulate@ nd] != {}]; Select[ Range@135, fQ] (* Robert G. Wilson v, Mar 06 2018 *)
  • PARI
    isok(n) = {d = divisors(n); psd = vector(#d, k, sum(j=1, k, d[j])); nd = setminus([1..n], d); psnd = vector(#nd, k, sum(j=1, k, nd[j])); #setintersect(psd, psnd) != 0;} \\ Michel Marcus, May 05 2018
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