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.

A186083 Values taken by A161903, sorted and duplicates removed.

Original entry on oeis.org

0, 3, 6, 7, 12, 13, 14, 15, 24, 25, 26, 27, 28, 30, 31, 48, 49, 50, 51, 52, 54, 55, 56, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 102, 103, 104, 107, 108, 109, 110, 111, 112, 115, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 192, 193, 194, 195, 196, 198, 199
Offset: 1

Views

Author

Ben Branman, Feb 12 2011

Keywords

Comments

The sequence contains 2k if and only if it contains k.
If the binary expansion of n has k bits, then A161903(n) will have k+1 bits. Thus, to determine if a number with m bits belongs to the sequence, it is sufficient to check A161903(i) up to i=2^(m-1)-1.

Links

Crossrefs

Cf. A161903.

Programs

  • Mathematica
    f[n_] := FromDigits[Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1], 2]; Union[Table[f[n], {n, 0, 2047}]]

A093515 Numbers k such that either k or k-1 is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 29, 30, 31, 32, 37, 38, 41, 42, 43, 44, 47, 48, 53, 54, 59, 60, 61, 62, 67, 68, 71, 72, 73, 74, 79, 80, 83, 84, 89, 90, 97, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 114, 127, 128, 131, 132, 137, 138, 139
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

Original name: Transform of the prime sequence by the Rule 110 cellular automaton.
As described in A051006, a monotonic sequence can be mapped into a fractional real. Then the binary digits of that real can be treated (transformed) by an elementary cellular automaton. Taking the resulting sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.
From M. F. Hasler, Mar 01 2008: (Start)
The "Rule110" transform as used here involves a right-shift of the sequence before applying the transform as described on the MathWorld page.
Robert G. Wilson v observed that this sequence contains exactly the indices for which A121561 equals 1. (End)
From M. F. Hasler, Jan 07 2019: (Start)
The correspondence of monotonic sequences with fractional reals mentioned in the first comment is not really relevant here: RuleX most naturally maps directly one characteristic sequence to another and thus one set (or increasing sequence) to another one. Interpreting the characteristic sequences as binary digits of a fractional real then yields a map from [0,1] into [0,1] rather as a consequence.
Antti Karttunen observed that this seems to be the complement of A005381 (k and k-1 are composite). This is indeed the case: The characteristic sequence of primes has no three subsequent 1's. In all other cases of the 8 possible inputs for Rule110, the output is 0 if and only if the cell itself and its neighbor to the right are zero, which here means "k and k+1 are composite", and with the right shift mentioned above, the complement of A005381, i.e., numbers such that k or k-1 is prime (or: primes U primes + 1). We have actually proved the more general
Theorem: Rule110 transforms any set S having no three consecutive integers into the set S' = {k | k or k-1 is in S} = S U (1 + S). (End)

Links

Crossrefs

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093514, A093516, A093517, A161903.
Cf. A005381 (complement, apart from 1 which is in neither sequence), A323162.
Cf. A121561.

Programs

  • Magma
    [n: n in [2..180] | not(not IsPrime(n) and not IsPrime(n-1))]; // Vincenzo Librandi, Jan 08 2019
  • Mathematica
    Select[Range[2, 150], !(!PrimeQ[# - 1] && !PrimeQ[#]) &] (* Vincenzo Librandi, Jan 08 2019 *)
  • PARI
    {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
j=0;l=matsize(v)[2];k=v[l];po=1;
for(i=1,k+2,j*=2;po=isin(i,v,l,po);j=(j+max(0,sign(po)))% 8;if(cav[j+1],r=concat(r,i)));
return(r) /* See the function "isin" at A092875 */}
  • PARI
    /* transform a sequence v by the rule r - Note: v could be replaced by a function, e.g. v[c] => prime(c) here */
seqruletrans(v,r)={my(c=1,L=List(),t=0); r=Vecrev(binary(r),8); for(i=1,v[#v], v[c]A093515=seqruletrans(primes(10^4),110) \\ M. F. Hasler, Mar 01 2008, updated Jan 07 2019
  • PARI
    A121561_is_1(N,n=0)=vector(N,i, while(!isprime(n+=1)&&!isprime(n-1),);n) \\ M. F. Hasler, Mar 01 2008
  • PARI
    is(n)=isprime(n)||isprime(n-1) \\ M. F. Hasler, Jan 07 2019
  • Python
    from sympy import isprime
def ok(n): return isprime(n) or isprime(n-1)
print(list(filter(ok, range(140)))) # Michael S. Branicky, Aug 10 2021

Formula

{a(n)} = A000040 U (A000040 + 1), where A000040 are the primes. - M. F. Hasler, Jan 07 2019
a(1) = 2, a(n) = a(n-1) + 1 if a(n-1) is prime, a(n) is the next prime after a(n-1) otherwise. - Luca Armstrong, Aug 10 2021

Extensions

Name changed by Antti Karttunen, Jan 07 2019

A269174 Formula for Wolfram's Rule 124 cellular automaton: a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 11, 24, 27, 30, 31, 28, 31, 22, 19, 48, 51, 54, 55, 60, 63, 62, 59, 56, 59, 62, 63, 44, 47, 38, 35, 96, 99, 102, 103, 108, 111, 110, 107, 120, 123, 126, 127, 124, 127, 118, 115, 112, 115, 118, 119, 124, 127, 126, 123, 88, 91, 94, 95, 76, 79, 70, 67, 192, 195, 198, 199, 204, 207, 206, 203, 216
Offset: 0

Views

Author

Antti Karttunen, Feb 22 2016

Keywords

Links

Crossrefs

Cf. A003986, A003987, A004198, A048724, A048725, A057889, A269173, A161903.
Cf. A269175.
Cf. A269176 (numbers not present in this sequence).
Cf. A269177 (same sequence sorted into ascending order, duplicates removed).
Cf. A269178 (numbers that occur only once).
Cf. A267357 (iterates from 1 onward).

Programs

Formula

a(n) = A163617(n) AND A269173(n).
a(n) = A163617(n) AND (A048724(n) OR A048725(n)).
a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A161903(A057889(n))). [Rule 124 is the mirror image of rule 110.]
G.f.: (-3*x^3 - 2*x^2 - 3*x)/(x^4 - 1) + Sum_{k>=1}((2^(k + 1)*x^(2^k) - 2^(k + 1)*x^(14*2^(k - 2)))/((x^(2^(k + 2)) - 1)*(x - 1))). - Miles Wilson, Jan 25 2025

A204371 Maximum period of cellular automaton rule 110 in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 14, 16, 7, 25, 110, 18, 351, 91, 295, 32, 578, 81, 285, 240, 630, 462, 1058, 552, 300, 351, 567, 2156, 1044, 1770, 2759, 2368, 1100, 969, 3920, 1584
Offset: 1

Views

Author

Ben Branman, Jan 14 2012

Keywords

Comments

a(n) >= A180001(n), and this sequence agrees with A180001 up to n=11.

Examples

			The 12 cell pattern
000100110111
001101111101
011111000111
110001001101
010011011111
110111110001
011100010011
110100110111
011101111100
110111000100
111101001101
000111011111
001101110001
011111010011
110001110111
010011011100
110111110100
111100011101
000100110111
Has period 18, which is the maximum possible, so a(12)=18

Links

Crossrefs

Cf. A180001, A161903, A006978, A332717, A332718.

Programs

  • Mathematica
    f[list_] := -Subtract @@ Flatten[Map[Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[110], list, Unequal, All], {0}]]; ma[n_] := Max[Table[f[IntegerDigits[i, 2, n]], {i, 0, 2^n - 1}]]; Table[ma[n], {n, 1, 10}]

Extensions

a(19)-a(36) from Lars Blomberg, Dec 24 2015
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