A164917 -id:A164917 - cp's OEIS frontend

A164920 Primes which are obtained at least by two ways using the iterations of the Bertrand operator (see A164917) beginning with primes of A164368.

113, 139, 199, 211, 223, 277, 293, 397, 421, 443

Offset: 1

Vladimir Shevelev, Aug 31 2009

The sequence is connected with our sieve selecting the primes of A164368 from all primes.

			113 is in the sequence since it is obtained either by iterations of Bertrand operator beginning from 17 (17=>31=>61=>113) or by such iterations beginning with 59 (59=>113), and both of primes 17 and 59 are in A164368.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A164917, A164918, A104272, A164368, A164288.

A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

Original entry on oeis.org

2, 2, 2, 2, 11, 2, 17, 11, 2, 29, 17, 11, 41, 2, 47, 29, 59, 17, 67, 71, 11, 41, 2, 47, 97, 101, 29, 107, 109, 17, 127, 67, 137, 11, 149, 151, 41, 2, 167, 47, 179, 181, 191, 97, 197, 29, 107, 17, 227, 229, 233, 239, 241, 127, 67, 263, 269, 137, 11, 281, 283, 149, 307, 311, 41

Offset: 1

Vladimir Shevelev, Aug 31 2009

nonn

a(n) is the starting value of the prime chain described in A164917 which contains (touches) prime(n).

By construction, each member of this sequence here is one of the values of A164368, the head elements of all chains of this map.

			The first four values are 2 because prime(1)=2, prime(2)=3, prime(3)=5 and prime(4)=7 are all in the prime chain starting at 2.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A164917, A104272, A164368, A164288.

A060308 := proc(n) prevprime(2*n+1) ; end:
isA164368 := proc(p) local q ; q := nextprime(floor(p/2)) ; return (numtheory[pi](2*q) -numtheory[pi](p) >= 1); end proc:
A164368 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1)) ; while not isA164368(a) do a := nextprime(a) ; end do : RETURN(a) ; end if; end proc:
A164918 := proc(n) local p, a, j, q, itr ; p := ithprime(n) ; a := 1000000000000000 ; for j from 1 do q := A164368(j) ; if q > p then break; end if; itr := 0 ; while q < p do q := A060308(q) ; itr := itr+1 ; end do; if q = p then return A164368(j) ; end if; end do: end proc:
seq(A164918(n), n=1..120) ; # R. J. Mathar, Mar 12 2010

lp[n_] := NextPrime[2n, -1];
a[n_] := For[pn = Prime[n]; p = 2, p <= pn, p = NextPrime[p], nwl = NestWhileList[lp, p, # <= Prime[n]&]; If[MemberQ[nwl, pn], Return[p]]];
Array[a, 120] (* Jean-François Alcover, Dec 01 2017 *)

Edited and extended by R. J. Mathar, Mar 12 2010

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Original entry on oeis.org

2, 3, 2, 3, 2, 13, 3, 19, 2, 13, 31, 3, 19, 43, 2, 53, 13, 61, 31, 71, 73, 3, 19, 43, 2, 101, 103, 53, 109, 113, 13, 131, 31

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union {2,3} and A164333 from all primes. Note that a(n)=n iff p_n is in the considered union, which corresponds to 0's iterations of g.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A164333, A055496, A164960, A104272, A080359, A164918, A006992, A164917, A164368, A164288.

A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

Original entry on oeis.org

127, 149, 211, 223, 257, 307, 431, 449

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union of {2,3} and A164333 from all primes.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A055496, A164333 A164960, A164962, A164920, A006992, A164917, A164918, A104272, A080359, A164368, A164288.

cp's OEIS Frontend

A164920 Primes which are obtained at least by two ways using the iterations of the Bertrand operator (see A164917) beginning with primes of A164368.

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A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

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A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

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A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

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