A075323 -id:A075323 - cp's OEIS frontend

A247233 Smallest m such that A075323(m) = n-th odd prime, or zero, if no such m exists.

1, 2, 3, 4, 5, 11, 6, 7, 12, 8, 9, 29, 15, 10, 13, 16, 17, 14, 30, 23, 18, 19, 509, 24, 25, 20, 55, 21, 37, 26, 22, 35, 27, 31, 38, 33, 56, 28, 36, 43, 32, 34, 39, 41, 51, 45, 44, 53, 47, 40, 42, 65, 52, 46, 49, 67, 161, 48, 54, 63, 59, 66, 61, 50, 79, 57

Offset: 1

Reinhard Zumkeller, Nov 29 2014

nonn

Conjecture: a(388) = 0, i.e., A065091(388) = 2683 doesn't occur in A075323;

for n with a(n) > 0: A075323(a(n)) = A065091(n) = A000040(n+1).

			Also a(389) = 0 (presumably), whereas subsequent terms (n > 389) are > 0:
393,443,421,350,397,455,368,433,387,352,356,382,384,366,372,392,374, ...
with corresponding odd primes:
2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..387

Cf. A075323, A065091, A000040.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a247233 = (+ 1) . fromJust . (`elemIndex` a075323_list) . a065091

maxm = 3000;
A075321p[n_] := A075321p[n] = Module[{prevlist, i, p, q}, If[n == 1, Return[{3, 5}], prevlist = Array[A075321p, n-1] // Flatten]; For[i = 2, True, i++, p = Prime[i]; If[FreeQ[prevlist, p], q = p + 2*n; If[ PrimeQ[q] && FreeQ[prevlist, q], Return[{p, q}]]]]];
A075323[n_] := If[OddQ[n], A075321p[(n + 1)/2][[1]], A075321p[n/2][[2]]];
a[n_] := For[m = 1, m <= maxm, m++, If[A075323[m] == Prime[n + 1], Return[m]]] /. Null -> 0;
Array[a, 387] (* Jean-François Alcover, Feb 12 2018, after R. J. Mathar's program for A075321p *)

A075321 Pair the odd primes so that the n-th pair is (p, p+2n) where p is the smallest prime not included earlier such that p and p+2n are primes and p+2n also does not occur earlier: (3, 5), (7, 11), (13, 19), (23, 31), (37, 47), (17, 29), (53, 67) ... This is the sequence of the first member of every pair.

Original entry on oeis.org

3, 7, 13, 23, 37, 17, 53, 43, 61, 83, 109, 73, 101, 139, 41, 149, 157, 137, 113, 193, 197, 179, 211, 229, 263, 199, 227, 107, 331, 293, 311, 283, 241, 269, 349, 359, 383, 367, 401, 317, 379, 439, 491, 421, 409, 449, 463, 467

Offset: 1

Amarnath Murthy, Sep 14 2002

nonn

Question: Is every prime a member of some pair?
If the distance between the prime pairs is not required to be 2n, we get A031215. - R. J. Mathar, Nov 26 2014
a(n) = A075323(2*n-1).

			a(4)=23: For the 4th pair though 17 is the smallest prime not occurring earlier, 17+8 = 25 is not a prime and 23 + 8 = 31 is a prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A075322, A075323.

a075321 = a075323 . subtract 1 . (* 2)
-- Reinhard Zumkeller, Nov 29 2014

A075321p := proc(n)
    option remember;
    local prevlist,i,p,q ;
    if n = 1 then
        return [3,5];
    else
        prevlist := [seq(op(procname(i)),i=1..n-1)] ;
        for i from 2 do
            p := ithprime(i) ;
            if not p in prevlist then
                q := p+2*n ;
                if isprime(q) and not q in prevlist then
                    return [p,q] ;
                end if;
            end if;
        end do:
    end if;
end proc:
A075321 := proc(n)
    op(1,A075321p(n)) ;
end proc:
seq(A075321(n),n=1..60) ; # R. J. Mathar, Nov 26 2014

A075321p[n_] := A075321p[n] = Module[{prevlist, i, p, q }, If[n == 1, Return[{3, 5}], prevlist = Array[A075321p, n-1] // Flatten]; For[i = 2, True, i++, p = Prime[i]; If[FreeQ[prevlist, p], q = p + 2*n ; If[ PrimeQ[q] && FreeQ[ prevlist, q], Return[{p, q}]]]]];
A075321 [n_] := A075321p[n][[1]];
Array[A075321, 50] (* Jean-François Alcover, Feb 12 2018, translated from R. J. Mathar's program *)

Corrected by R. J. Mathar, Nov 26 2014

A075322 Pair the odd primes so that the k-th pair is (p, p+2n) where p is the smallest prime not included earlier such that p and p+2n are primes and p+2n also does not occur earlier: (3, 5), (7, 11), (13, 19), (23, 31), (37, 47), (17, 29), ... This is the sequence of the second member of every pair.

Original entry on oeis.org

5, 11, 19, 31, 47, 29, 67, 59, 79, 103, 131, 97, 127, 167, 71, 181, 191, 173, 151, 233, 239, 223, 257, 277, 313, 251, 281, 163, 389, 353, 373, 347, 307, 337, 419, 431, 457, 443, 479, 397, 461, 523, 577, 509, 499, 541, 557, 563

Offset: 1

Amarnath Murthy, Sep 14 2002

nonn

Question: Is every prime p a member of some pair?

a(n) = A075323(2*n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A075321, A075323.

a075322 = a075323 . (* 2)  -- Reinhard Zumkeller, Nov 29 2014

# A075321p() implemented in A075321.
A075322 := proc(n)
    op(2,A075321p(n)) ;
end proc:
seq(A075322(n),n=1..60) ; # R. J. Mathar, Nov 26 2014

A075321p[n_] := A075321p[n] = Module[{prevlist, i, p, q }, If[n == 1, Return[{3, 5}], prevlist = Array[A075321p, n - 1] // Flatten]; For[i = 2, True, i++, p = Prime[i]; If[FreeQ[prevlist, p], q = p + 2*n; If[PrimeQ[q] && FreeQ[ prevlist, q], Return[{p, q}]]]]];
a[n_] := A075321p[n][[2]];
Array[a, 50] (* Jean-François Alcover, Feb 12 2018, after R. J. Mathar *)

a(n) = A075321(n)+2*n.

a(n) = A075323(2*n).

Corrected by R. J. Mathar, Nov 26 2014

A147513 Numbers such that the n-th and (n+1)st terms are the successors of prime numbers and primes themselves and n+1 > n.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 31, 37, 47

Offset: 1

Jonathan Baca (jbakid(AT)gmail.com), Nov 05 2008

Is this the same as prime(k) such that prime(k+1)*prime(k+2) > 1 + prime(k)*prime(k+3)? - Arkadiusz Wesolowski, May 12 2018

a(n) = A075323(n) for 1 <= n <= 10. - Georg Fischer, Nov 02 2018

Cf. A075323.

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A247233 Smallest m such that A075323(m) = n-th odd prime, or zero, if no such m exists.

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A147513 Numbers such that the n-th and (n+1)st terms are the successors of prime numbers and primes themselves and n+1 > n.

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