A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040).

2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, 75, 81, 90, 95, 99, 102, 105, 108, 127, 129, 131, 135, 139, 156, 158, 161, 163, 167, 173, 182, 187, 190, 193, 196, 205, 210, 214, 216, 232, 235, 241, 244, 247, 254, 263, 265, 270

Offset

1,1

Comments

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

Links

Table of n, a(n) for n=1..58.

Formula

For just the odd primes we have a(n) - 1.

Example

The sequence of prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with twos at (A376343):
  2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, ...

Mathematica

Join@@Position[First/@Split[Differences[Select[Range[100], PrimeQ]]], 2]

Crossrefs

Positions of 2's in A037201.

The repeats were at positions A064113 before being omitted.

A variation for squarefree numbers is A376342.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, differences A057820.

A005117 lists squarefree numbers, differences A076259 (ones A375927).

A013929 lists nonsquarefree numbers, differences A078147.

A333254 lists run-lengths of differences between consecutive primes.

Cf. A007921, A030173, A053289, A061398, A373817, A373822, A373947, A376305, A376308.

Sequence in context: A185601 A157795 A294060 * A003066 A075349 A156024

Adjacent sequences: A376340 A376341 A376342 * A376344 A376345 A376346

Keyword

nonn

Author

Gus Wiseman, Sep 26 2024

Status

approved