A284172 -id:A284172 - cp's OEIS frontend

A284188 a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i primes.

2, 3, 5, 8, 5, 9, 5, 10, 5, 11, 18, 11, 19, 28, 19, 29, 40, 29, 41, 54, 41, 55, 41, 56, 41, 57, 41, 58, 41, 59, 78, 59, 79, 100, 79, 101, 124, 101, 125, 101, 126, 101, 127, 154, 127, 155, 127, 156, 127, 157, 188, 157, 189, 157, 190, 157, 191, 226, 191, 227, 264, 227, 265

Offset: 1

Bob Selcoe, Mar 21 2017

nonn

Without repeated terms, the primes appear in order as A070865.
Variant of A284172; the difference is that in A284172, a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i composites (rather than i primes).
For n >= 3: When a(n) = prime p it is followed by an even number j at a(n+1); p repeats k-j times (where k is the smallest prime > j), appearing at a(n+2m) {m=1..k-j}. a(n+2m+1) = p+m until p+m = k (immediately following the final p); k now becomes "new p" immediately followed by a "new j", and the process repeats.

			a(10) = 11; there are 7 primes in the sequence up to and including a(10) so a(11) = 11+7 = 18. 18 is composite so a(12) = 18-7 = 11.  Now there are 8 primes in the sequence; and since 11 is prime, a(13) = 11+8 = 19 (the 9th prime in the sequence), so a(14) = 28.

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^16.
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^12, showing a(n) in red and A284172(n) in blue, accentuating the first 12 terms of A284172(n) by enlargement.

Cf. A070865, A284172.

c:= proc(n) option remember; `if`(n<1, 0,
      `if`(isprime(a(n)), 1, 0)+c(n-1))
    end:
a:= proc(n) option remember; `if`(n=1, 2, (m->
      `if`(isprime(m), 1, -1)*c(n-1)+m)(a(n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 15 2017

Block[{c = 1, m = 2, n}, {2}~Join~Reap[Do[If[PrimeQ[m], Set[n, m + c]; c++, Set[n, m - c + 1]]; Sow[n]; m = n, 63]][[-1, -1]]] (* Michael De Vlieger, Oct 20 2021 *)

lista(nn) = {print1(a=2, ", "); nbp = 1; for (n=2, nn, if (isprime(a), a += nbp, a -= nbp); print1(a, ", "); if (isprime(a), nbp++););} \\ Michel Marcus, Mar 24 2017

A284278 a(1)=2; for n >= 1, if n+2 is prime then a(2n+1) = 3n + 2 and a(2*n) = n + 2, otherwise all terms are 2.

Original entry on oeis.org

2, 3, 5, 4, 2, 5, 9, 6, 2, 7, 13, 8, 2, 9, 2, 10, 2, 11, 21, 12, 2, 13, 25, 14, 2, 15, 2, 16, 2, 17, 33, 18, 2, 19, 37, 20, 2, 21, 2, 22, 2, 23, 45, 24, 2, 25, 2, 26, 2, 27, 2, 28, 2, 29, 57, 30, 2, 31, 61, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 73, 38, 2, 39

Offset: 1

Vladimir Shevelev, Mar 24 2017

nonn

The sequence is motivated by A284172, by the message from B. Jubin dated Mar 23 2017 and by the classic open problem of showing that there are infinitely many primes p for which 2*p-1 is also prime. If there were only finitely many such primes, then there would be a place where this sequence is generated by the same rule as A284172.
The sequence of the first differences begins 1, 2, -1, -2, 3, 4, -3, -4, 5, 6, -5, -6, 7, -7, 8, -8, 9, 10, -9, -10, 11, 12, -11, -12, 13, -13, 14, -14, 15, 16, -15, -16, 17, 18, ...
From the definition it easily follows that, for a positive x, the sequence contains roughly equal numbers of prime and composite terms <= x.
A conditional property: if there is a maximal prime P such that 2*P-1 is also prime, then for n > P, every pair (a(2*n), a(2*n+1)) contains one prime and one composite. Indeed, if n+2 is prime, then a(2*n) = n + 2 is prime, while a(2*n+1) = 2*n + 3 = 2*(n+2) - 1 is composite; if n+2 is composite, then a(2*n) = n + 2 is composite, while a(2*n+1) = 2 is prime. - Vladimir Shevelev, Mar 26 2017

			For n=19, a(38) = a(2*19) = 19+2 = 21, a(39) = a(2*19+1) = 2, the latter since 19+2 is not prime;
for n=21, a(42) = a(2*21) = 21+2 = 23, a(43) = a(2*21+1) = 2*21+3 = 45 since 21+2 is prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^12.

Cf. A284172.

a[1]:=2;
a[n_?EvenQ]:=n/2+2;
a[n_?OddQ]:=If[PrimeQ[(n+1)/2+1], n+2, 2];
Map[a, Range[150]] (* Peter J. C. Moses, Mar 24 2017 *)

a(n) = if(n<2, 2, if(n%2, if(isprime((n + 1)/2 + 1), n + 2, 2), (n/2 + 2))); \\ Indranil Ghosh, Mar 25 2017

More terms from Peter J. C. Moses, Mar 24 2017

cp's OEIS Frontend

A284188 a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i primes.

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A284278 a(1)=2; for n >= 1, if n+2 is prime then a(2n+1) = 3n + 2 and a(2*n) = n + 2, otherwise all terms are 2.

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cp's OEIS Frontend

A284188 a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A284278 a(1)=2; for n >= 1, if n+2 is prime then a(2*n+1) = 3*n + 2 and a(2*n) = n + 2, otherwise all terms are 2.

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Examples

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Mathematica

PARI

Extensions

A284278 a(1)=2; for n >= 1, if n+2 is prime then a(2n+1) = 3n + 2 and a(2*n) = n + 2, otherwise all terms are 2.