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 1-6 of 6 results.

A239580 Numbers k such that A227364(k) = 1 + 2*3 + 4*5*6 + 7*8*9*10 + ... + ...*k is a prime.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 13, 14, 15, 18, 30, 32, 54, 58, 59, 81, 85, 128, 140, 203, 204, 206, 209, 223, 286, 305, 343, 350, 367, 397, 399, 451, 453, 506, 534, 656, 676, 698, 730, 756, 845, 849, 878, 944, 1020, 1040, 1091, 1248, 1256, 1300, 1310, 1326, 1364, 1406, 1535
Offset: 1

Views

Author

Alex Ratushnyak, Mar 21 2014

Keywords

Crossrefs

Cf. A227364.

Programs

  • Python
    from sympy import isprime
for n in range(10000):
  sum_ = 0
  i = k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= i
      i += 1
      if i>n: break
    sum_ += product
    k += 1
  if isprime(sum_):  print(n, end=', ')

A227363 a(n) = n + (n-1)*(n-2) + (n-3)*(n-4)*(n-5) + (n-6)*(n-7)*(n-8)*(n-9) + ... + ...*(n-n).

Original entry on oeis.org

0, 1, 2, 5, 10, 17, 32, 61, 110, 185, 316, 557, 986, 1705, 2840, 4661, 7702, 12881, 21620, 35965, 58706, 94217, 150016, 239045, 382670, 614401, 984332, 1564301, 2458810, 3826745, 5918936, 9136597, 14115686, 21842225, 33803620, 52181021, 80128082, 122221801, 185211440
Offset: 0

Views

Author

Alex Ratushnyak, Jul 07 2013

Keywords

Comments

From a question by Jonathan Vos Post dated Jul 09 2013, the indices of a(n) which are prime begin: 2, 3, 5, 7, 11, 41, 111, 205, 211, 215, 341, 345, 395, 581, 585, 1221, ..., . - Robert G. Wilson v, Jul 10 2013

Examples

			a(2) = 2 + 1*0 = 2.
a(3) = 3 + 2*1 = 5.
a(9) = 9 + 8*7 + 6*5*4 + 3*2*1*0 = 9 + 56 + 120 = 185.
a(11) = 11 + 10*9 + 8*7*6 + 5*4*3*2 = 557.
a(18) = 18 + 17*16 + 15*14*13 + 12*11*10*9 + 8*7*6*5*4 = 21620.

Links

Crossrefs

Cf. A227364, A227365, A227366, A227367.

Programs

  • Mathematica
    f[n_] := Sum[ Product[ n - k (k - 1)/2 - i + 1, {i, k}], {k, Sqrt[ 2n]}]; Array[f, 39, 0] (* Robert G. Wilson v, Jul 10 2013 *)
  • PARI
    a(n)=sum(k=1,sqrtint(2*n)+1,prod(i=1,k,max(n-k*(k-1)/2-i+1,0))) \\ Charles R Greathouse IV, Jul 09 2013
  • Python
    for n in range(55):
  sum = i = 0
  k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= n-i
      i += 1
      if i>n: break
    sum += product
    k += 1
  print(str(sum), end=',')

A227365 a(n) = 0 + 1*2 + 3*4*5 + 6*7*8*9 + ... + ...*n.

Original entry on oeis.org

0, 1, 2, 5, 14, 62, 68, 104, 398, 3086, 3096, 3196, 4406, 20246, 243326, 243341, 243566, 247406, 316766, 1638686, 28150526, 28150547, 28150988, 28161152, 28405550, 34526126, 193916126, 4503821726, 4503821754, 4503822538, 4503846086, 4504576886, 4527986846, 5301270686
Offset: 0

Views

Author

Alex Ratushnyak, Jul 07 2013

Keywords

Examples

			a(4) = 0 + 1*2 + 3*4 = 14.
a(5) = 0 + 1*2 + 3*4*5 = 62.
a(6) = 0 + 1*2 + 3*4*5 + 6 = 68.

Links

Crossrefs

Cf. A227363, A227364, A227366, A227367.

Programs

  • Python
    for n in range(55):
  sum = i = 0
  k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= i
      i += 1
      if i>n: break
    sum += product
    k += 1
  print(str(sum), end=',')

A227367 a(0)=1, a(n+1) = a(0) + a(1)*a(2) + a(3)*a(4)*a(5) + a(6)*a(7)*a(8)*a(9) + ... + ...*a(n).

Original entry on oeis.org

1, 1, 2, 3, 6, 21, 381, 762, 290703, 84397476747, 7122934049104967061783, 14245868098209934123566, 101472378935797762635619628499635817245339961, 10296643686890133479148472187437767614663729545766251948487237380959682684821520304732841
Offset: 0

Views

Author

Alex Ratushnyak, Jul 08 2013

Keywords

Examples

			a(1) = a(0) = 1
a(2) = a(0) + a(1) = 2
a(3) = a(0) + a(1)*a(2) = 3
a(4) = a(0) + a(1)*a(2) + a(3) = 1 + 2 + 3 = 6
a(5) = a(0) + a(1)*a(2) + a(3)*a(4) = 1 + 2 + 18 = 21
a(6) = a(0) + a(1)*a(2) + a(3)*a(4)*a(5) = 1 + 2 + 18*21 = 381

Crossrefs

Cf. A227363, A227364, A227365, A227366.

Programs

  • Python
    a = [1]*99
for n in range(20):
  sum = i = 0
  k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= a[i]
      i += 1
      if i>n: break
    sum += product
    k += 1
  a[n+1] = sum
  print(str(a[n]),end=',')

A227366 a(0)=1, a(n+1) = a(n) + a(n-1)*a(n-2) + a(n-3)*a(n-4)*a(n-5) + a(n-6)*a(n-7)*a(n-8)*a(n-9) + ... + ...*a(0).

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 33, 118, 584, 4714, 76206, 2879841, 364389490, 220150411628, 1049813737275512, 80222580570107370160, 231117086585854944888597249, 84218767584329653007205530276477742, 18540809099930664963747242025045529905738135516
Offset: 0

Views

Author

Alex Ratushnyak, Jul 08 2013

Keywords

Examples

			a(1) = a(0) = 1
a(2) = a(1) + a(0) = 2
a(3) = a(2) + a(1)*a(0) = 3
a(4) = a(3) + a(2)*a(1) + a(0) = 3 + 2 + 1 = 6
a(5) = a(4) + a(3)*a(2) + a(1)*a(0) = 6 + 3*2 + 1 = 13

Crossrefs

Cf. A227363, A227364, A227365, A227367.

Programs

  • Python
    a = [1]*99
for n in range(20):
  sum = i = 0
  k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= a[n-i]
      i += 1
      if i>n: break
    sum += product
    k += 1
  a[n+1] = sum
  print(str(a[n]),end=',')

A336502 Partial sums of A057003.

Original entry on oeis.org

1, 7, 127, 5167, 365527, 39435607, 6006997207, 1226103906007, 322796982334807, 106460296033918807, 42980408446129381207, 20846482682939051365207, 11959807608801430284133207, 8010447502346968140207973207, 6193994326661240674349352805207, 5476021766725276671842502543205207
Offset: 1

Views

Author

Seiichi Manyama, Jul 23 2020

Keywords

Comments

Inspired by doubly triangular numbers (A002817).

Examples

			a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3 + 4*5*6 = 127.
a(4) = 1 + 2*3 + 4*5*6 + 7*8*9*10 = 5167.

Links

Crossrefs

Cf. A000217, A002817, A007489, A057003, A227364, A336513.

Programs

  • Mathematica
    Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* Amiram Eldar, Jul 23 2020 *)
  • PARI
    {a(n) = sum(i=1, n, prod(j=(i-1)*i/2+1, i*(i+1)/2, j))}

Formula

a(n) = Sum_{i=1..n} Product_{j=T(i-1)+1..T(i)} j where T(n) is n-th triangular number.
a(n) = A227364(T(n)) where T(n) is n-th triangular number.
a(n) ~ n^(2*n) / 2^n. - Vaclav Kotesovec, Nov 20 2021
Showing 1-6 of 6 results.

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

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