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A178613
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The smaller member prime(i) of an emirp pair (prime(i),prime(j)), such that the digit sum of i equals the digit sum of j.
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0
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37, 359, 769, 1409, 7687, 10711, 10853, 11243, 11593, 13441, 13751, 14423, 14551, 14879, 15307, 15661, 16879, 17959, 30853, 31193, 33863, 34589, 37307, 37489, 38449, 73369, 74959, 75239, 78259, 78839, 79669, 90089, 92779, 100267, 101531
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OFFSET
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1,1
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COMMENTS
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We consider base-10 emirp pairs (13,31) = (prime(6),prime(11)), (17,71) = (prime(7),prime(20)), (37,73) = (prime(12),prime(21)), ... (see A006567) and the digit sums of their prime indices (6,2=1+1), (7,2=2+0), (3=1+2,3=2=1),.. (see A156793).
If the digits sums of the two indices are the same, the smaller representative of the emirp pair is entered into the sequence.
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REFERENCES
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W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, 13th edition, Dover Publications, 2010
C. Mauduit, J. Rivat: Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals of Mathematics, Vol. 171, No. 3, 1591-1646, 2010
H Schubart: Einfuehrung in die klassische und moderne Zahlentheorie Vieweg, Braunschweig, 1974
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LINKS
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EXAMPLE
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37 = prime(12) and 73 = prime(21) are an emirp pair with equal digit sums of the indices 1+2 = 3 = 2+1, which puts 37 into the sequence.
359 = prime(72) and 953 = prime(162) are an emirp pair with digit sums 7+2 = 9 = 1+6+2, which puts 359 into the sequence.
The 6th term is from the pair (10711 = prime(1306), 11701 = prime(1405)), see A033548
16th term: (17959 = prime(2059), 95971 = prime(9250)).
21st term: (34589 = prime(3694), 98543 = prime(9463)).
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MATHEMATICA
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f[n_] := Plus @@ IntegerDigits@ PrimePi@n; fQ[n_] := Block[{id = IntegerDigits@n}, rid = Reverse@ id; q = FromDigits@ rid; rid != id && PrimeQ@ FromDigits@ rid && n < q && f@n == f@q]; lst = {}; p = 13; While[p < 102148, If[ fQ@p, AppendTo[lst, p]]; p = NextPrime@p]; lst (* Robert G. Wilson v, Jul 31 2010 *)
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 30 2010
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EXTENSIONS
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STATUS
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approved
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