cp's OEIS Frontend

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 1-4 of 4 results.

A119445 Leading diagonal of triangle A119444.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 13, 27, 63, 109, 207, 331, 553, 931, 1531, 2527, 4093, 6673, 10831, 17563, 28561, 46227, 74883, 121219, 196239, 317607, 514047, 831823, 1346041, 2178079, 3524323, 5702619, 9227161, 14930019, 24157471, 39087823, 63245551
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Links

Crossrefs

Cf. A119444 for triangle corresponding to this sequence.
Cf. A100461 for powers of 2, A119446 for primes.

Programs

  • Magma
    function t(n,k) // t = A119444
  if k eq 1 then return Fibonacci(n+1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n): n in [1..60]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[1, n_]:= Fibonacci[n+1];  (* t = A119444 *)
t[m_, n_]/; 1, ]= 0;
A119445[n_]:= A119445[n]= t[n,n];
Table[A119445[n], {n,60}] (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k): # t = A119444
    if (k==1): return fibonacci(n+1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
def A119445(n): return t(n,n)
[A119445(n) for n in range(1,61)] # G. C. Greubel, Apr 07 2023

Formula

a(n) = A119444(n, n).

A100461 Triangle read by rows, based on array described below.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 3, 4, 6, 8, 7, 8, 9, 12, 16, 25, 26, 27, 28, 30, 32, 49, 50, 51, 52, 55, 60, 64, 109, 110, 111, 112, 115, 120, 126, 128, 229, 230, 231, 232, 235, 240, 245, 248, 256, 481, 482, 483, 484, 485, 486, 490, 496, 504, 512, 1003, 1004, 1005, 1008, 1010, 1014, 1015, 1016, 1017, 1020, 1024
Offset: 1

Views

Author

N. J. A. Sloane, Nov 22 2004

Keywords

Examples

			Array begins:
  1  2  4  8  16  32 ...
  *  1  2  6  12  30 ...
  *  *  1  4   9  28 ...
  *  *  *  3   8  27 ...
  *  *  *  *   7  26 ...
  *  *  *  *   *  25 ...
and triangle begins:
    1;
    1,   2;
    1,   2,   4;
    3,   4,   6,   8;
    7,   8,   9,  12,  16;
   25,  26,  27,  28,  30,  32;
   49,  50,  51,  52,  55,  60,  64;
  109, 110, 111, 112, 115, 120, 126, 128;

Links

Crossrefs

Cf. A100452, A100462, A119444.

Programs

  • Magma
    function t(n,k) // t = A100461
  if k eq 1 then return 2^(n-1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, 2^(n-1), (n-k+1)*Floor[(t[n, k-1] -1)/(n-k+1)]];
Table[t[n, n-k+1], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n,k): # t = A100461
    if (k==1): return 2^(n-1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n,n-k+1) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array t(m,n) (n >= 1, 1 <= m <= n) by: t(1,n) = 2^(n-1) for all n; t(m+1,n) = (n-m)*floor( (t(m,n) - 1)/(n-m) ) for 1 <= m <= n-1.

A119446 Triangle as described in A100461, except with t(1,n) = prime(n).

Original entry on oeis.org

2, 2, 3, 3, 4, 5, 3, 4, 6, 7, 3, 4, 6, 8, 11, 3, 4, 6, 8, 10, 13, 3, 4, 6, 8, 10, 12, 17, 3, 4, 6, 8, 10, 12, 14, 19, 3, 4, 6, 8, 10, 12, 14, 16, 23, 7, 8, 9, 12, 15, 18, 21, 24, 27, 29, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 31, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 37
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Examples

			Triangle begins as:
  2;
  2, 3;
  3, 4, 5;
  3, 4, 6,  7;
  3, 4, 6,  8, 11;
  3, 4, 6,  8, 10, 13;
  3, 4, 6,  8, 10, 12, 17;
  3, 4, 6,  8, 10, 12, 14, 19;
  3, 4, 6,  8, 10, 12, 14, 16, 23;
  7, 8, 9, 12, 15, 18, 21, 24, 27, 29;

Links

Crossrefs

Cf. A100461 for powers of 2, A119444 for Fibonacci and A119447 for leading diag. of this triangle.

Programs

  • Magma
    function t(n,k) // t = A119444
  if k eq 1 then return NthPrime(n);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-n+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, Prime[n], (n-k+1)*Floor[(t[n,k-1] -1)/(n -k+1)]];
Table[t[n, n-k+1], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k):
    if (k==1): return nth_prime(n)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array t(m,n) (n >= 1, 1 <= m <= n) by: t(1,n) = prime(n) for all n; t(m+1,n) = (n-m)*floor( (t(m,n) - 1)/(n-m) ) for 1 <= m <= n-1.

A119447 Leading diagonal of triangle A119446.

Original entry on oeis.org

2, 2, 3, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Comments

a(181) = 27 is the first term greater than 19. This is because prime(181)/181 > 6 for the first time. In general this sequence is determined by prime(n)/n: the pattern for each row of the triangle is that it ends with prime(n), preceded by multiples of k = prime(n)/n down to k^2, then the largest multiple of k-1 less than k^2 and the largest multiple of k-2 less than that and so on. This sequence gives the multiple of 1. See A000960 for the sequence that gives the ending value for each starting k.

Links

Crossrefs

Cf. A000960, A119444, A119446.

Programs

  • Magma
    function t(n,k) // t = A119446
  if k eq 1 then return NthPrime(n);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n): n in [1..100]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[1, n_]:= Prime[n];
t[m_, n_]/; 1, ]=0;
A119447[n_]:= A119447[n]= t[n,n];
Table[A119447[n], {n,100}] (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k): # t = A119446
    if (k==1): return nth_prime(n)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
def A119447(n): return t(n,n)
[A119447(n) for n in range(1,101)] # G. C. Greubel, Apr 07 2023

Formula

a(n) = A000960( prime(n)/n ).
a(n) = A119446(n, n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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