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

Author

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

Keywords

• base
• nonn

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.

Examples

			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)).
		

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

Crossrefs

Cf. A006567, A156793.

Programs

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 *)

Extensions

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