A350827
Number of prime septuplets (i.e.: 7-tuples) with initial member (A022009 or A022010) between 10^(n-1) and 10^n.
Original entry on oeis.org
0, 1, 0, 1, 1, 4, 5, 21, 70, 370, 1862, 9634
Offset: 1
a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime to start a prime septuplet.
a(2) = a(4) = a(5) = 1 because 11 = A022009(1), 5639 = A022010(1) and 88799 = A022010(2) are the only prime with 2, 4 resp. 5 digits to start a prime septuplet.
Then there are a(6) = 4 six-digit primes, 165701, 284729, 626609 and 855719, which start a prime septuplet.
Cf.
A022009,
A022010: initial members p of prime septuplets (p, p+2, p+6, ...) resp. (p, p+2, p+8, ...).
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apply( {A350827(n,v=vector(6),c)=forprime(p=10^(n-1),10^n, v[n=1+n%#v]+20==(v[n]=p) && c++);c}, [1..8]) \\ becomes slow for n > 8. - M. F. Hasler, Apr 12 2022
A350828
Number of prime octuplets with initial member (A065706) between 10^(n-1) and 10^n.
Original entry on oeis.org
0, 2, 0, 1, 1, 3, 3, 9, 28, 136, 541, 2936
Offset: 1
a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime initial member of a prime octuplet.
a(2) = 2 because 11 and 17 are the only 2-digit members of A065706, i.e., primes to start a prime octuplet.
a(4) = a(5) = 1 because 1277 (resp. 88793) is the only prime with 4 (resp. 5) digits to start a prime octuplet.
Then there are a(6) = 3 six-digit primes, 113147, 284723 and 855713, which start a prime octuplet.
Cf.
A065706 (initial members p of prime octuplets (p, ..., p+26)),
A022011,
A022012,
A022013 (idem, specifically for each of the three possible patterns).
A350829
Number of prime 9-tuples (or: nonuplets) with initial member (A257125) between 10^(n-1) and 10^n.
Original entry on oeis.org
1, 3, 0, 1, 1, 3, 0, 1, 8, 30, 88
Offset: 1
a(1) = 1 because 7 is the only single-digit prime to start a prime nonuplet, i.e., member of A257125. (All other members of this nonuplet have 2 digits.)
a(2) = 3 because 11, 13 and 17 are the three 2-digit primes to start a prime nonuplet.
a(3) = 0 because there is no 3-digit prime initial member of a prime nonuplet.
Cf.
A257125 (initial members p of prime nonuplets (p, ..., p+30)),
A022545 -
A022548 (idem, specifically for each of the four possible patterns).
Cf.
A350825 -
A350828: similar for quintuplets, sextuplets, septuplets and octuplets.
A350830
Number of prime 10-tuples (or decaplets) with initial member (A257127) between 10^(n-1) and 10^n.
Original entry on oeis.org
0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 4, 15, 81, 357, 1685, 8256
Offset: 1
a(2) = 1 because 11 is the only two-digit prime to start a prime decaplet, i.e., member of A257127.
a(n) = 0 for all other n < 10 because the next larger prime decaplet is made of 10-digit primes, A257127(2) = 9853497737 and successors.
a(10) = 1 because there is only one prime decaplet made of 10-digit primes.
a(11) = 4 because there are only four terms in A257127 (for indices n = 3..6) which have 11 digits.
Cf.
A257127 (initial members p of prime 9-tuples (p, ..., p+32)),
A027569,
A027570 (idem, specifically for each of the two possible patterns).
Cf.
A350825 -
A350829: similar for quintuples, sextuples, septuples, octuples and 9-tuples.
Showing 1-4 of 4 results.
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