A178613 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.

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

Offset

1,1

Comments

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.

References

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

Links

Table of n, a(n) for n=1..35.

Example

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)).

Mathematica

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 *)

Crossrefs

Cf. A006567, A156793.

Sequence in context: A133554 A177513 A137834 * A124337 A027944 A283629

Adjacent sequences: A178610 A178611 A178612 * A178614 A178615 A178616

Keyword

base,nonn

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 30 2010

Extensions

More terms from Robert G. Wilson v, Jul 31 2010

Status

approved