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 1-5 of 5 results.

A271977 G_6(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 139, 1751, 187243, 16777215, 33554571, 50333399, 84073323, 134217727, 134217867, 134219479, 134404971, 150994943
Offset: 3

Views

Author

Natan Arie Consigli, Apr 24 2016

Keywords

Comments

The next term (line break for better formatting) is a(16) = \
1619239197880733074062994004113160848331305687934176134326809 \
538279709713884753268291640071900343455846003089194770060104834018705547.
a(17) = 2.870...*10^1585, a(18) = 6.943...*10^169099. - Pontus von Brömssen, Sep 24 2020

Examples

			Find G_6(7):
G_1(7) = B_2(7)-1= B_2(2^2+2+1)-1 = 3^3+3+1-1 = 30;
G_2(7) = B_3(G_1(7))-1 = B_3(3^3+3)-1 =  4^4+4-1 = 259;
G_3(7) = B_4(G_2(7))-1 = 5^5+3-1 = 3127;
G_4(7) = B_5(G_3(7))-1 = 6^6+2-1 = 46657;
G_5(7) = B_6(G_4(7))-1 = 7^7+1-1 = 823543;
G_6(7) = B_7(G_5(7))-1 = 8^8-1 = 16777215.

Links

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); this sequence: G_6(n); A271978: G_7(n); A271979: G_8(n); A271985: G_9(n); A271986: G_10(n); A266201: G_n(n).

Programs

  • Python
    from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A271977(n):
  if n==3: return 0
  for i in range(2,8):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 24 2020

Extensions

a(10) corrected by Pontus von Brömssen, Sep 24 2020

A271978 G_7(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 173, 2454, 332147, 37665879, 774841151, 1162263921, 1937434592, 2749609302, 3486784574, 3486786855, 3487116548, 3524450280
Offset: 3

Views

Author

Natan Arie Consigli, Apr 30 2016

Keywords

Comments

a(16) is too big to include - see b-file. a(17) = 9.221...*10^2347, a(18) = 2.509...*10^316952. - Pontus von Brömssen, Sep 25 2020

Examples

			Find G_7(7):
G_1(7) = B_2(7)-1= B[2](2^2+2+1)-1 = 3^3+3+1-1 = 30;
G_2(7) = B_3(G_1(7))-1 = B[3](3^3+3)-1 =  4^4+4-1 = 259;
G_3(7) = B_4(G_2(7))-1 = 5^5+3-1 = 3127;
G_4(7) = B_5(G_3(7))-1 = 6^6+2-1 = 46657;
G_5(7) = B_6(G_4(7))-1 = 7^7+1-1 = 823543;
G_6(7) = B_7(G_5(7))-1 = 8^8-1 = 16777215;
G_7(7) = B_8(G_6(7))-1 = 7*9^7+7*9^6+7*9^5+7*9^4+7*9^3+7*9^2+7*9+7-1 = 37665879.

Links

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); A271977: G_6(n); this sequence: G_7(n); A271979: G_8(n); A271985: G_9(n); A271986: G_10(n); A266201: G_n(n).

Programs

  • Python
    from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A271978(n):
  if n==3: return 0
  for i in range(2,9):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Extensions

a(9) corrected by Pontus von Brömssen, Sep 25 2020

A271985 G_9(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 253, 4382, 885775, 150051213, 570623341475, 855935016215, 1426559238830, 1997331745490, 3138428376974, 3138428381103, 3138429262496, 3138578427934
Offset: 3

Views

Author

Natan Arie Consigli, Apr 30 2016

Keywords

Comments

a(17) = 2.066...*10^4574. - Pontus von Brömssen, Sep 25 2020

Examples

			Compute G_9(10):
G_1(10)= B_2(10)-1 = B_2(2^(2+1)+2)-1 = 3^(3+1)+3-1 = 83;
G_2(10) = B_3(3^(3+1)+2)-1 = 4^(4+1)+2-1 = 1025;
G_3(10) = B_4(4^(4+1)+1)-1 = 5^(5+1)+1-1 = 15625;
G_4(10) = B_5(5*5^(5+1))-1 = 6^(6+1)-1= 279935;
G_5(10) = B_6(5*6^6+5*6^5+5*6^4+5*6^3+5*6^2+5*6+5)-1 = 5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+5-1 = 4215754;
G_6(10) = B_7(5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+4)-1 = 5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+4-1 = 84073323;
G_7(10) = B_8(5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+3)-1 = 5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+3-1 = 1937434592;
G_8(10) = B_9(5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+2)-1 = 5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+2-1 = 50000555551;
G_9(10) = B_10(5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+1)-1 = 5*11^11+5*11^5+5*11^4+5*11^3+5*11^2+5*11+1-1 = 1426559238830.

Links

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); A271977: G_6(n); A271978: G_7(n); A271979: G_8(n); this sequence: G_9(n); A271986: G_10(n); A266201: G_n(n).

Programs

  • Python
    from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A271985(n):
  if n==3: return 0
  for i in range(2,11):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Extensions

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

A271986 G_10(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 299, 5643, 1357259, 273624711, 17832200896811, 26748301350411, 44580503598539, 62412976762503, 106993205379371, 106993205384715, 106993206736331, 106993479003783
Offset: 3

Views

Author

Natan Arie Consigli, May 01 2016

Keywords

Comments

a(17) = 1.926...*10^6103. - Pontus von Brömssen, Sep 25 2020

Links

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); A271977: G_6(n); A271978: G_7(n); A271979: G_8(n); A271985: G_9(n); this sequence: G_10(n); A266201: G_n(n).

Programs

  • Python
    from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A271986(n):
  if n==3: return 0
  for i in range(2,12):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Extensions

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

A296441 Array A(n, k) = G_k(n) where G_k(n) is the k-th term of the Goodstein sequence of n, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 0, 0, 1, 3, 4, 0, 0, 0, 3, 26, 5, 0, 0, 0, 2, 41, 27, 6, 0, 0, 0, 1, 60, 255, 29, 7, 0, 0, 0, 0, 83, 467, 257, 30, 8, 0, 0, 0, 0, 109, 775, 3125, 259, 80, 9, 0, 0, 0, 0, 139, 1197, 46655, 3127, 553, 81, 10, 0, 0, 0, 0, 173, 1751, 98039, 46657, 6310, 1023, 83, 11
Offset: 0

Views

Author

Iain Fox, Dec 12 2017

Keywords

Comments

G_0(n) = n. To get to the second term in the row, convert n to hereditary base 2 representation (see links), replace each 2 with a 3, and subtract 1. For the third term, convert the second term (G_1(n)) into hereditary base 3 notation, replace each 3 with a 4, and subtract one. This pattern continues until the sequence converges to 0, which, by Goodstein's Theorem, occurs for all n.

Examples

			| n\k |  0   1    2     3      4      5       6       7       8       9  ...
|-----|------------------------------------------------------------------------
|  0  |  0,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  1  |  1,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  2  |  2,  2,   1,    0,     0,     0,      0,      0,      0,      0, ...
|  3  |  3,  3,   3,    2,     1,     0,      0,      0,      0,      0, ...
|  4  |  4, 26,  41,   60,    83,   109,    139,    173,    211,    253, ...
|  5  |  5, 27, 255,  467,   775,  1197,   1751,   2454,   3325,   4382, ...
|  6  |  6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, ...
| ... |

Links

Crossrefs

n-th row: A000004 (n=0), A000007 (n=1), A215409 (n=3), A056193 (n=4), A266204 (n=5), A266205 (n=6), A271554 (n=7), A271555 (n=8), A271556 (n=9), A271557 (n=10), A271558 (n=11), A271559 (n=12), A271560 (n=13), A271561 (n=14), A222117 (n=15), A059933 (n=16), A271562 (n=17), A271975 (n=18) A211378 (n=19), A271976 (n=20).
k-th column: A001477 (k=0), A056004 (k=1), A057650 (k=2), A059934 (k=3), A059935 (k=4), A059936 (k=5), A271977 (k=6), A271978 (k=7), A271979 (k=8), A271985 (k=9), A271986 (k=10).
G_n(n) = A266201(n) (main diagonal of array).

Programs

  • PARI
    B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n
Showing 1-5 of 5 results.

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