A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787

Offset: 1

Author

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

Keywords

• nonn

Comments

Each set has a constant difference of 30, for all of the terms listed so far.

It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.

The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018

About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.

Links

• Zak Seidov, Table of n, a(n) for n = 1..241 (all terms up to 3*10^9)
• Jens K. Andersen, The Largest Known CPAP's, updated Sept. 2018
• OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).

Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.

Cf. A052239: start of first CPAP-4 with common difference 6n.

Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Cf. A006560: first prime to start a CPAP-n.

Programs

Mathematica

Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)

PARI

A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018

Formula

Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.

A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - M. F. Hasler, Oct 27 2018, edited Jan 02 2020.

Extensions

a(16)-a(22) from Donovan Johnson, Sep 05 2008

Reference added by Harvey P. Dale, Jun 22 2013

Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018