A307632 Index of first occurrence of n-th prime in A307720.

3, 5, 47, 53, 1374, 1386, 3738, 3756, 6680, 6704, 84626, 84658, 89480, 89522, 91832, 91880, 173092, 173152, 192882, 192950, 524587, 524661, 865301, 865385, 876543, 876641, 890479, 890583, 904273, 904383, 918859, 918987, 1628979, 1629117, 1647107, 1647257, 1666775

Offset: 1

Author

N. J. A. Sloane, Apr 26 2019

Keywords

• nonn

Comments

It follows from the definition of A307720 that if p = k-th prime, k>1 and k odd, and q = (k+1)st prime, then the first time p appears in the sequence is at the start of a subsequence (p,1) [(p-1)/2 times], p, (1,q) [(q+1)/2 times].

For example, the fifth prime (11) first appears in A307720 at step 1374 at the start of the subsequence 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13.

So q appears p+1 steps after p, which explains why the terms of the present sequence appear in pairs.

In fact, it appears that one can make a much stronger statement about what happens immediately after the first occurrence of p. Look at the terms in A307720 following the first 11 at step 1374. It may be that the next O(p^2) terms are forced.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..1229 (taken from Rémy Sigrist's b-file for A307630; first 132 terms from Hans Havermann)

Crossrefs

Cf. A307720, A307630, A307631.

First differences = A348773.

Extensions

More terms from Hans Havermann, Apr 26 2019