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A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements. 9
30, 60, 90, 120, 180, 240, 270, 300, 330, 360, 390, 450, 480, 510, 540, 570, 600, 660, 690, 720, 750, 780, 810, 870, 900, 930, 960, 990, 1020, 1080, 1110, 1140, 1170, 1200, 1230, 1290, 1320, 1350, 1380, 1410, 1440, 1500, 1530, 1560, 1590, 1620, 1650, 1710, 1740 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The 6 elements are not necessarily consecutive primes.
A342309(d) gives the first element of the smallest AP with 6 elements whose common difference is a(n) = d.
All the terms are positive multiples of 30 (A249674) but are not multiples of 7 and also must not belong to A206041; indeed, terms d' in A206041 correspond to the longest possible APs of primes that have exactly 7 elements with this common difference d'.
LINKS
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
FORMULA
m is a term iff A123556(m) = 6.
EXAMPLE
d = 30 is a term because the longest possible APs of primes with common difference d = 30 all have 6 elements; the first such APs start with 7, 107, 359, .... The smallest one is (7, 37, 67, 97, 127, 157); then 187 = 11*17.
d = 60 is another term because the longest possible APs of primes with common difference d = 60 all have 6 elements; the first such APs start with 11, 53, 641, .... The smallest one is (11, 71, 131, 191, 251, 311); then 371 = 7*53.
d = 150 is not a term because the longest possible AP of primes with common difference d = 150 is (7, 157, 307, 457, 607, 757, 907) which has 7 elements; this last one is unique.
MAPLE
filter := d -> (irem(d, 30) = 0) and (irem(d, 7) <> 0) and not (isprime(7+d) and isprime(7+2*d) and isprime(7+3*d) and isprime(7+4*d) and isprime(7+5*d) and isprime(7+6*d)): select(filter, [$(1 .. 1740)]);
CROSSREFS
Subsequence of A249674.
Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), this sequence (k=6), A206041 (k=7), no sequence for (k=8) and (k=9), A360146 (k=10), A206045 (k=11).
Sequence in context: A358756 A069819 A143207 * A108454 A235483 A064783
KEYWORD
nonn
AUTHOR
Bernard Schott, Jan 29 2023
STATUS
approved

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Last modified May 18 15:10 EDT 2024. Contains 372653 sequences. (Running on oeis4.)