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A093515
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Numbers k such that either k or k-1 is a prime.
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14
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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
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1,1
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COMMENTS
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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.
The "Rule110" transform as used here involves a right-shift of the sequence before applying the transform as described on the MathWorld page.
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)
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LINKS
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FORMULA
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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
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MATHEMATICA
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Select[Range[2, 150], !(!PrimeQ[# - 1] && !PrimeQ[#]) &] (* Vincenzo Librandi, Jan 08 2019 *)
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PROG
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(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]<i && c++; r[1+t=t%4*2+(v[c]==i)] && listput(L, i)); Set(L)}
(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
(Magma) [n: n in [2..180] | not(not IsPrime(n) and not IsPrime(n-1))]; // Vincenzo Librandi, Jan 08 2019
(Python)
from sympy import isprime
def ok(n): return isprime(n) or isprime(n-1)
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CROSSREFS
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Cf. A005381 (complement, apart from 1 which is in neither sequence), A323162.
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KEYWORD
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easy,nonn
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AUTHOR
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Ferenc Adorjan (fadorjan(AT)freemail.hu)
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EXTENSIONS
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STATUS
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approved
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