

A093515


Numbers k such that either k or k1 is a prime.


14



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
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OFFSET

1,1


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 rightshift 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 k1 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 k1 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 k1 is in S} = S U (1 + S). (End)


LINKS

M. F. Hasler, Table of n, a(n) for n = 1..19998 (using prime(1..10^4)).
Ferenc Adorjan, Binary mapping of monotonic sequences  the Aronson and the CA functions
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Rule110 Elementary Cellular Automaton


FORMULA

{a(n)} = A000040 U (A000040 + 1), where A000040 are the primes.  M. F. Hasler, Jan 07 2019


MATHEMATICA

Select[Range[2, 150], !(!PrimeQ[#  1] && !PrimeQ[#]) &] (* Vincenzo Librandi, Jan 08 2019 *)


PROG

(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[ki+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) A093515=seqruletrans(primes(10^4), 110) \\ with:
/* 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]<i && c++; r[1+t=t%4*2+(v[c]==i)] && listput(L, i)); Set(L)} \\ 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(n1), ); n) \\ M. F. Hasler, Mar 01 2008
(PARI) is(n)=isprime(n)isprime(n1) \\ M. F. Hasler, Jan 07 2019
(MAGMA) [n: n in [2..180]  not(not IsPrime(n) and not IsPrime(n1))]; // Vincenzo Librandi, Jan 08 2019


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.
Sequence in context: A023778 A173016 A053577 * A249724 A084369 A167211
Adjacent sequences: A093512 A093513 A093514 * A093516 A093517 A093518


KEYWORD

easy,nonn


AUTHOR

Ferenc Adorjan (fadorjan(AT)freemail.hu)


EXTENSIONS

Name changed by Antti Karttunen, Jan 07 2019


STATUS

approved



