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

A271979 G_8(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 211, 3325, 555551, 77777775, 20000000211, 30000003325, 50000555551, 70077777775, 100000000211, 100000003325, 100000555551, 100077777775
Offset: 3

Views

Author

Natan Arie Consigli, Apr 30 2016

Keywords

Comments

At least half of the digits of every term (except a(14)) are the same.
Let n > 0:
a(4n) mod 100 = 211;
a(4n+1) mod 1000 = 3325;
a(4n+2) mod 1000000 = 555551;
a(4n+3) mod 100000000 = 77777775;
Proof for a(4n):
If x is divisible by 4 its hereditary representation in base 2 has all summands divisible by 4 and it cannot have the summands 1 and 2.
If we calculate G_1(x) we would end with:
G_1(x) = B_2(x)-1.
Clearly, B_2(x) = 3^a + 3^b + ... is divisible by 3^3 = 27 and that would mean that the representation of B_2(x)-1 would be B_2(x)-1 = X_3 + 2*3^2+2*3+2.
From now on, let X_n be a sum of powers of n (greater than the right term).
We finish proving the statement by calculating G_8(x):
G_2(x) = B_3(X_3 +2*3^2+2*3+2)-1 = X_4 + 2*4^2+2*4+2-1;
G_3(x) = B_4(X_4 +2*4^2+2*4-1)-1 = X_5 + 2*5^2+2*5+1-1;
G_4(x) = B_5(X_5 +2*5^2+2*5)-1 = X_6 + 2*6^2+2*6-1;
G_5(x) = B_6(X_6 +2*6^2+6+5)-1 = X_7 + 2*7^2+7+5-1;
G_6(x) = B_7(X_7 +2*7^2+7+4)-1 = X_8 + 2*8^2+8+4-1;
G_7(x) = B_8(X_8 +2*8^2+8+3)-1 = X_9 + 2*9^2+9+3-1;
G_8(x) = B_9(X_9 +2*9^2+9+2)-1 = X_10 + 2*10^2+10+2-1 = X_10 + 211;
So finally G_8(x) mod 100 = 211.
The other cases can be proved using the same reasoning.
a(17) = 3.3330...*10^3333, a(18) = 5.555550...*10^555555. - Pontus von Brömssen, Sep 25 2020

Examples

			Calculate G_8(5):
G_1(5) = B_2(5)-1 = B_2(2^2+1)-1 = 27;
G_2(5) = B_3(3^3)-1 = 4^4-1 = 255;
G_3(5) = B_4(3*4^3 + 3*4^2 + 3*4 + 3)-1 = 3*5^3 + 3*5^2 + 3*5 + 3-1 = 467;
G_4(5) = B_5(3*5^3 + 3*5^2 + 3*5 + 2)-1 = 3*6^3 + 3*6^2 + 3*6 + 2-1 = 775;
G_5(5) = B_6(3*6^3 + 3*6^2 + 3*6 + 1)-1 = 3*7^3 + 3*7^2 + 3*7 + 1-1 = 1197;
G_6(5) = B_7(3*7^3 + 3*7^2 + 3*7)-1 = 3*8^3 + 3*8^2 + 3*8-1 = 1751;
G_7(5) = B_8(3*8^3 + 3*8^2 + 2*8 + 7)-1 = 3*9^3 + 3*9^2 + 2*9 + 7-1 = 2454;
G_8(5) = B_9(3*9^3 + 3*9^2 + 2*9 + 6)-1 = 3*10^3 + 3*10^2 + 2*10 + 6-1 = 3325.

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); this sequence: 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 A271979(n):
  if n==3: return 0
  for i in range(2,10):
    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

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.

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

Content is available under The OEIS End-User License Agreement.