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A350826
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Number of prime sextuplets with n-digit initial term (A022008).
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5
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1, 1, 0, 0, 3, 0, 13, 64, 235, 1296, 7013, 41782, 253420, 1607418, 10520883, 70785653, 488096844
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OFFSET
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1,5
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COMMENTS
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Prime sextuplets are of the form (p, p+4, p+6, p+10, p+12, p+16), where p is the initial member, listed in A022008.
For n = 1 and n = 2 (see Example), the last member of the sextuplet has one digit more than the initial member (so the count would be 0 for these two, if all terms of the sextuplet had to have the same length). As far as we know, for all n > 2, all members of the sextuplets have the same length. A sufficient condition for this is that A033874(n) > 16.
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LINKS
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FORMULA
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a(n) = # { p in A022008 | 10^(n-1) < p < 10^n }.
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EXAMPLE
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For n = 1, p = 7 is the only 1-digit prime to be the initial term of a prime sextuplet, (7, 11, 13, 17, 19, 23), hence a(1) = 1.
For n = 2, p = 97 is the only 2-digit prime to be the initial term of a prime sextuplet, (97, 101, 103, 107, 109, 113), whence a(2) = 1.
For n = 3 and n = 4, there is no n-digit prime to be the initial term of a prime sextuplet, so a(n) = 0.
For n = 5, {16057, 19417, 43777} are the only 5-digit primes which are initial members of a prime sextuplet, therefore a(5) = 3.
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PROG
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(PARI) apply( {A350826(n, L=10^n)=n=L\10; for(c=0, oo, L<(n=next_A022008(n)) && return(c))}, [1..8])
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CROSSREFS
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Cf. A022008 (initial members of prime sextuplets), A033874 (10^n - precprime(10^n)).
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KEYWORD
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nonn,base,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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