A358310 Index in A145985 where n-th odd prime p first appears, or -1 if p never appears.

3, 2, 1, 13, -1, 12, -1, 59, 11, -1, -1, 10, -1, 9, 8, 7, -1, -1, 6, -1, -1, 5, 4, -1, 2528242167, -1

Offset

1,1

Comments

Let p = prime(n+1). Certainly if p == 1 (mod 6) then p cannot appear in A145985 (because 10^k-p will be a multiple of 3), so a(n) = -1. In all other cases it appears that a(n) > 0.

a(27) (corresponding to the prime 107) is presently unknown.

Links

Table of n, a(n) for n=1..26.

Example

The 8th odd prime 23 first occurs in A145985 at index 59, so a(8) = 59.
For n=25, the 25th odd prime is 101. The first time when 10^k - 101 is a prime is when k = 12, where 10^12 - 101 = 999999999899. Furthermore, when we look at the numbers t = 10^12 - q for q an increasing prime, t is a prime for q = 11, 41, 101, ..., that is, 101 is the third success.  It follows that 101 is three steps back from the end of the 12th block of descending terms in A145985. The lengths of the blocks are given by A359120. Therefore a(25) = Sum_{j=1..12} A359120(j) - (3-1) = 2528242169 - 2 = 2528242167. - N. J. A. Sloane, Dec 18 2022

Mathematica

With[{prs=Prime[Range[20000000]]}, Table[Position[Select[Table[10^IntegerLength[p]-p, {p, prs}], PrimeQ], n, 1, 1], {n, Prime[Range[24]]}]]/.({}->-1) (* Harvey P. Dale, Dec 17 2022 *)

Crossrefs

Cf. A145985.

Sequence in context: A068440 A246381 A048647 * A180190 A059438 A156628

Adjacent sequences: A358307 A358308 A358309 * A358311 A358312 A358313

Keyword

sign,more

Author

Harvey P. Dale and N. J. A. Sloane, Dec 16 2022.

Extensions

a(25) from N. J. A. Sloane, Dec 18 2022

Status

approved