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.

A093094 "Products into digits": start with a(1)=2, a(2)=2; adjoin digits of product of a(k) and a(k+1) for k from 1 to infinity.

Original entry on oeis.org

2, 2, 4, 8, 3, 2, 2, 4, 6, 4, 8, 2, 4, 2, 4, 3, 2, 1, 6, 8, 8, 8, 1, 2, 6, 2, 6, 4, 8, 6, 4, 6, 4, 8, 2, 1, 2, 1, 2, 1, 2, 2, 4, 3, 2, 4, 8, 2, 4, 2, 4, 2, 4, 3, 2, 1, 6, 2, 2, 2, 2, 2, 2, 4, 8, 1, 2, 6, 8, 3, 2, 1, 6, 8, 8, 8, 8, 8, 1, 2, 6, 2, 6, 1, 2, 4, 4, 4, 4, 4, 8, 3, 2, 8, 2, 1, 2, 4, 8, 2, 4
Offset: 1

Views

Author

Bodo Zinser, Mar 20 2004

Keywords

Comments

Only the digits 1,2,3,4,6,8 occur, infinitely often. The sequence is not periodic. Around a(800) there are many 8's.
From Giovanni Resta, Mar 16 2006: (Start)
Proof that sequence is not periodic:
Let us assume that somewhere in the sequence there is a subsequence of 3 adjacent 8': ...,8,8,8,....(which is true).
Then we know that in the following there will be the subsequence ...,6,4,6,4.. (i.e. 8x8, 8x8) again, there will be somewhere ...,2,4,2,4,2,4,... (i.e. 6x4, 4x6, 6x4) and finally ...,8,8,8,8,8,...
Analogously, starting from 8,8,8,8 we obtain 6,4,6,4,6,4 then 2,4,2,4,2,4,2,4,2,4 and finally 8,8,8,8,8,8,8,8,8.
Generalizing, if somewhere appears a run of k>2 8's, then in some future position will appear a run of at least 4*k-7 8's (where since k>2, 4*k-7>k).
So the sequence will contain arbitrary long runs of 8's, without being constantly equal to 8, thus it cannot be periodic. (End)
Essentially the same as A045777. [R. J. Mathar, Sep 08 2008]

Examples

			a(3)=a(1)*a(2), a(4)=a(2)*a(3), a(5)=first digit of (a(3)*a(4)), a(6)=2nd digit of (a(3)*a(4)), a(9)=a(6)*a(7)

Links

Crossrefs

Cf. A045777, A093086, A093087, A093088, A093089, A093090, A093091, A093095, A093096, A093097, A096381.

Programs

  • Haskell
    a093094 n = a093094_list !! (n-1)
a093094_list = f [2,2] where
   f (u : vs@(v : _)) = u : f (vs ++
     if w < 10 then [w] else uncurry ((. return) . (:)) $ divMod w 10)
        where w = u * v
-- Reinhard Zumkeller, Aug 08 2013
  • Mathematica
    Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {2, 2}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)
  • Python
    from itertools import islice
from collections import deque
def agen(): # generator of terms
    a = deque([2, 2])
    while True:
        a.extend(list(map(int, str(a[0]*a[1]))))
        yield a.popleft()
print(list(islice(agen(), 101))) # Michael S. Branicky, Feb 15 2024

Extensions

Definition revised by Franklin T. Adams-Watters, Mar 16 2006

A093097 Start with a(1) = 3, a(2) = 7; then apply the rule of A093094.

Original entry on oeis.org

3, 7, 2, 1, 1, 4, 2, 1, 4, 8, 2, 4, 3, 2, 1, 6, 8, 1, 2, 6, 2, 6, 4, 8, 8, 2, 1, 2, 1, 2, 1, 2, 2, 4, 3, 2, 6, 4, 1, 6, 2, 2, 2, 2, 2, 2, 4, 8, 1, 2, 6, 1, 2, 2, 4, 4, 6, 1, 2, 4, 4, 4, 4, 4, 8, 3, 2, 8, 2, 1, 2, 6, 2, 4, 8, 1, 6, 2, 4, 6, 2, 8, 1, 6, 1, 6, 1, 6, 1, 6, 3, 2, 2, 4, 6, 1, 6, 1, 6, 2, 2, 1, 2, 1, 2
Offset: 1

Views

Author

Bodo Zinser, Mar 20 2004

Keywords

Examples

			a(3)=first digit of (a(1)*a(2)), a(4)=2nd digit of (a(1)*a(2)), a(5)=first digit of (a(2)*a(3)), a(6)=2nd digit of (a(2)*a(3))

Links

Crossrefs

Cf. A045777, A060310, A093094, A093095, A093096, A096381.

Programs

  • Mathematica
    Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {3, 7}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)
  • Python
    from itertools import islice
from collections import deque
def agen(): # generator of terms
    a = deque([3, 7])
    while True:
        a.extend(list(map(int, str(a[0]*a[1]))))
        yield a.popleft()
print(list(islice(agen(), 105))) # Michael S. Branicky, Aug 18 2025

A093095 Start with a(1) = 2, a(2) = 3, then apply the rule of A093094.

Original entry on oeis.org

2, 3, 6, 1, 8, 6, 8, 4, 8, 4, 8, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 4, 8, 2, 4, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 2, 4, 3, 2, 1, 6, 8, 1, 2, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 2, 8, 1, 2, 6, 2, 6, 4, 8, 8, 2, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
Offset: 1

Views

Author

Bodo Zinser, Mar 20 2004

Keywords

Examples

			a(3)=a(1)*a(2), a(4)=first digit of (a(2)*a(3)), a(5)=2nd digit of (a(2)*a(3)), a(6)=a(3)*a(4)

References

  • Hugo Steinhaus, Studentenfutter, Urania, Leipzig 1991, #1.

Links

Crossrefs

Cf. A045777, A060310, A093094, A093096, A093097, A096381.

Programs

  • Haskell
    a093095 n = a093095_list !! (n-1)
a093095_list = f [2,3] where
   f (u : vs@(v : _)) = u : f (vs ++
     if w < 10 then [w] else uncurry ((. return) . (:)) $ divMod w 10)
        where w = u * v
-- Reinhard Zumkeller, Aug 08 2013
  • Mathematica
    Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {2, 3}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)
  • Python
    from itertools import islice
from collections import deque
def agen(): # generator of terms
    a = deque([2, 3])
    while True:
        a.extend(list(map(int, str(a[0]*a[1]))))
        yield a.popleft()
print(list(islice(agen(), 105))) # Michael S. Branicky, Aug 18 2025

A093096 Start with a(1) = a(2) = 3; then apply the rule of A093094.

Original entry on oeis.org

3, 3, 9, 2, 7, 1, 8, 1, 4, 7, 8, 8, 4, 2, 8, 5, 6, 6, 4, 3, 2, 8, 1, 6, 4, 0, 3, 0, 3, 6, 2, 4, 1, 2, 6, 1, 6, 8, 6, 2, 4, 0, 0, 0, 0, 1, 8, 1, 2, 8, 4, 2, 1, 2, 6, 6, 4, 8, 4, 8, 1, 2, 8, 0, 0, 0, 0, 0, 8, 8, 2, 1, 6, 3, 2, 8, 2, 2, 1, 2, 3, 6, 2, 4, 3, 2, 3, 2, 3, 2, 8, 2, 1, 6, 0, 0, 0, 0, 0, 0, 6, 4, 1, 6, 2
Offset: 1

Views

Author

Bodo Zinser, Mar 20 2004

Keywords

Comments

Essentially the same as A060310. [From R. J. Mathar, Sep 08 2008]

Examples

			a(3)=a(1)*a(2), a(4)=first digit of (a(2)*a(3)), a(5)=2nd digit of (a(2)*a(3)), a(6)=first digit of (a(3)*a(4)), a(7)=2nd digit of (a(3)*a(4))

References

  • Hugo Steinhaus, Studentenfutter, Urania, Leipzig 1991, #1.

Links

Crossrefs

Cf. A045777, A060310, A093094, A093095, A093097, A096381.

Programs

  • Haskell
    a093096 n = a093096_list !! (n-1)
a093096_list = f [3,3] where
   f (u : vs@(v : _)) = u : f (vs ++
     if w < 10 then [w] else uncurry ((. return) . (:)) $ divMod w 10)
        where w = u * v
-- Reinhard Zumkeller, Aug 13 2013
  • Mathematica
    Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {3, 3}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)
  • Python
    from itertools import islice
from collections import deque
def agen(): # generator of terms
    a = deque([3, 3])
    while True:
        a.extend(list(map(int, str(a[0]*a[1]))))
        yield a.popleft()
print(list(islice(agen(), 105))) # Michael S. Branicky, Aug 18 2025

A060310 a(1)=1, a(2)=3; then append digits of a(n-1)*a(n).

Original entry on oeis.org

1, 3, 3, 9, 2, 7, 1, 8, 1, 4, 7, 8, 8, 4, 2, 8, 5, 6, 6, 4, 3, 2, 8, 1, 6, 4, 0, 3, 0, 3, 6, 2, 4, 1, 2, 6, 1, 6, 8, 6, 2, 4, 0, 0, 0, 0, 1, 8, 1, 2, 8, 4, 2, 1, 2, 6, 6, 4, 8, 4, 8, 1, 2, 8, 0, 0, 0, 0, 0, 8, 8, 2, 1, 6, 3, 2, 8, 2, 2, 1, 2, 3, 6, 2, 4, 3, 2, 3, 2, 3, 2, 8, 2, 1, 6, 0, 0, 0, 0, 0, 0, 6, 4, 1, 6
Offset: 1

Views

Author

Jason Earls, Mar 27 2001

Keywords

Examples

			1*3=3, 3*3=9, 3*9=27, 9*2=18, ...

Links

Crossrefs

Cf. A045777, A093094, A093095, A093096, A093097, A096381.

Programs

  • Mathematica
    Fold[Join[#, IntegerDigits[Times @@ #[[#2;; #2+1]]]] &, {1, 3}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)
  • Python
    from itertools import islice
from collections import deque
def agen(): # generator of terms
    a = deque([1, 3])
    while True:
        a.extend(list(map(int, str(a[0]*a[1]))))
        yield a.popleft()
print(list(islice(agen(), 105))) # Michael S. Branicky, Aug 18 2025
Showing 1-5 of 5 results.

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