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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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.

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A072867 A034606 A233839 * A303447 A107617 A234980

Adjacent sequences:  A059041 A059042 A059043 * A059045 A059046 A059047

KEYWORD

nonn

AUTHOR

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

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

STATUS

approved

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Last modified March 7 16:49 EST 2021. Contains 341897 sequences. (Running on oeis4.)