A058868 -id:A058868 - cp's OEIS frontend

A054342 First occurrence of distances of equidistant lonely primes. Each equidistant prime is at the same distance (or has the same gap) from the preceding prime and the next prime.

5, 53, 211, 20201, 16787, 69623, 255803, 247141, 3565979, 6314447, 4911311, 12012743, 23346809, 43607429, 34346287, 36598607, 51042053, 460475569, 652576429, 742585297, 530324449, 807620777, 2988119339, 12447231899, 383204683, 4470608101, 5007182863, 36589015601

Offset: 1

Harvey P. Dale, May 06 2000

nonn

Or, least balanced primes: starting with 2nd term, 53, the smallest prime such that the distances to the next smallest and next largest primes are both equal to 6n.
The distances corresponding to the above terms are 2, 6, 12, 18, 24, ..., 192, 198, 204, 210, 218, 224.
a(1) is the smallest prime p such that {p-2, p, p+2} are three consecutive primes. For n>1, a(n) is the smallest prime p such that {p-6*(n-1), p, p+6*(n-1)} are three consecutive primes. - Jeppe Stig Nielsen, Apr 16 2022

			211 is an equidistant lonely prime with distance 12. This is the first occurrence of the distance 12, thus 211 is in the sequence.
20201 is a least balanced prime because it is the third term in the sequence and is separated from both the next lower and next higher primes by 3 * 6 = 18.
Here is the beginning of the table of equidistant lonely primes.
Equivalent to 3 consecutive primes in arithmetic progression.
* indicates a maximal gap. This table gives rise to A058867, A058868 and the present sequence.
  Gap  First occurrence
  ---  ----------------
    2*         5
    6*        53
   12*       211
   18      20201
   24*     16787
   30*     69623
   36     255803
   42*    247141
   48*   3565979
   54    6314447
   60*   4911311
   66*  12012743
   72*  23346809
   78   43607429
   84*  34346287
   90*  36598607
   96*  51042053
  102  460475569
  108  652576429

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..53 (based on A052187 b-file)

Cf. A006562, A052187, A058867, A058868, A103709.

a(1) = A052187(1) + 2. For n>1, a(n) = A052187(n) + 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

More terms from Jud McCranie, Jun 13 2000
Further terms from Harvey Dubner (harvey(AT)dubner.com), Sep 11 2004
Entry revised by N. J. A. Sloane, Jul 23 2006
4 further terms from Walter Neumann (neumann(AT)math.columbia.edu), Aug 14 2006
a(28) corrected, and terms after a(28) moved from Data section to b-file by Jeppe Stig Nielsen, Apr 16 2022

A058867 Equidistant lonely primes. Each prime is the same distance (gap) from the preceding prime and the next prime. These distances are maximal: each distance is larger than all such previous distances.

Original entry on oeis.org

5, 53, 211, 16787, 69623, 247141, 3565979, 4911311, 12012743, 23346809, 34346287, 36598607, 51042053, 383204683, 4470608101, 5007182863, 5558570491, 48287689717, 50284155289, 178796541817, 264860525507, 374787490919, 1521870804107, 2093308790851, 4228611064537, 6537587646671, 17432065861517, 22546768250359, 26923643849953, 187891466722913

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 07 2000; extended Sep 11 2004

nonn

			47, 53 and 59 are primes. There are no other primes between 47 and 59 and 59-53=53-47=6. There are no other such primes with a smaller distance so 53 is included in the sequence.

The distances are in A058868. First occurrences of distances are in A054342.

Primes:= select(isprime,[2,seq(2*i+1,i=1..10^7)]):
g:= 0: count:= 0:
for i from 2 to nops(Primes)-1 do
  if Primes[i+1]+Primes[i-1] = 2*Primes[i] and Primes[i+1]-Primes[i] > g then
     count:= count+1;
     a[count]:= Primes[i];
     g:= Primes[i+1]-Primes[i];
  fi
od:
seq(a[i],i=1..count); # Robert Israel, Sep 20 2015

a(21)-a(30) from Dmitry Petukhov, Sep 22 2015

cp's OEIS Frontend

A054342 First occurrence of distances of equidistant lonely primes. Each equidistant prime is at the same distance (or has the same gap) from the preceding prime and the next prime.

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