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A211681
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Numbers such that all the substrings of length <= 2 are primes.
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55
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2, 3, 5, 7, 23, 37, 53, 73, 237, 373, 537, 737, 2373, 3737, 5373, 7373, 23737, 37373, 53737, 73737, 237373, 373737, 537373, 737373, 2373737, 3737373, 5373737, 7373737, 23737373, 37373737, 53737373, 73737373, 237373737, 373737373, 537373737, 737373737
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OFFSET
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1,1
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COMMENTS
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The terms are primes for n= 1, 2, 3, 4, 5, 6, 7, 8, 10, 21, 23, 27, 31, 43, 45, 60, 67, 82, 91,.... The further terms with index 92, 93, 94, 96, 99 are composite. For the subsequence with prime terms see A211682.
Cf. A213299 for the partial sums.
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LINKS
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Hieronymus Fischer, Table of n, a(n) for n = 1..100
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FORMULA
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a(1+8*k) = 2*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
a(2+8*k) = 3*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
a(3+8*k) = 5*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
a(4+8*k) = 7*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
a(5+8*k) = 23*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
a(6+8*k) = 37*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
a(7+8*k) = 53*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
a(8+8*k) = 73*10^(2k) + 73*sum_{j=0..k-1} 10^(2j), for k>=0.
a(n) = ((2*n+7) mod 8 + d(n+4) - d(n+3))*10^d(n-1) + floor((37+36*(d(n+2)-d(n+1))*10^d(n-1)/99), where d(n)= floor(n/4).
Recursion for n>8:
a(n) = 10*(1+a(n-4)) - a(n-4) mod 10.
G.f.: (2*x*(1+x^10) + 3*x^2*(1 + x^3 + x^5 + x^6) + 5*x^3*(1+x^6) + 7*x^4*(1+x^2))/((1-10*x^4)*(1-x^8)).
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EXAMPLE
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a(11)=537, since all substrings of length <= 2 are primes (5, 3, 7, 53 and 37).
a(21)=237373, the substrings of length <= 2 are 2, 3, 7, 23, 37, 73.
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CROSSREFS
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Cf. A019546, A035232, A039996, A046034, A085823, A211682, A213299.
Sequence in context: A019546 A104179 A096148 * A124674 A177061 A020994
Adjacent sequences: A211678 A211679 A211680 * A211682 A211683 A211684
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KEYWORD
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nonn,base,easy
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AUTHOR
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Hieronymus Fischer, Apr 30 2012
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EXTENSIONS
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Typo in g.f. corrected, Hieronymus Fischer, Sep 03 2012
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STATUS
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approved
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