A328032 If there are m primes between 10^(n-1) and 10^n, a(n) is the middle prime if m is odd, otherwise the larger of the two middle primes.

5, 47, 509, 5273, 53047, 532907, 5356259, 53765519, 539119753, 5402600081, 54118210441, 541947386821, 5425907665571, 54313871643797, 543611236251491, 5440228524355381, 54438462600610513, 544705097744731559, 5449909581264135103

Offset: 1

Author

Robert G. Wilson v, Oct 02 2019

Keywords

• nonn
• hard

Comments

This sequence, unlike A309329, only contains primes.

For n > 2, a(n) > 10*a(n-1) for the terms shown. Does this continue?

The prime index of a(n): 3, 15, 97, 699, 5411, 44046, 371539, 3213018, 28304495, 252950023, 2286553663, 20862983416, 191836724429, 1775503643821, 16524756086736, 154541455728298, 1451397749344080, 13681755722697547, 129398810782042734, 1227438634918631724, 11674044544289825385, 111297278087667319110, 1063393839148059937607, 10180460079478002418395, 97640954583246485139774, 938046530135790455369642, 9025853588857058793877502, ..., .

Examples

			a(1) is 5 since, among the single-digit primes, i.e., {2, 3, 5, 7}, the two middle primes are {3, 5}, of which the larger one is 5;
a(2) is 47 since it is the middle prime of the two-digit primes, i.e., {11, 13, 17, ..., 47, ..., 83, 89, 97};
a(3) is 509 since it is the middle prime of the three-digit primes, i.e., {101, 103, 107, ..., 509, ..., 983, 991, 997}.
		

Crossrefs

Cf. A006879, A006880, A309329, A309359.

Programs

Mathematica

f[n_] := Block[{p = PrimePi[ 10^(n -1)], q = PrimePi[ 10^n]}, Prime[ Ceiling[(q +p +1)/2]]]; Array[f, 13]

Formula

a(n) is the next prime after A309329(n) - 1.