A391972 a(n) = 1 if n=1 or a(n-1) is prime; a(n) = a(n-1) + i otherwise, where i = n - r and r is the number of occurrences of 1 among terms a(1) to a(n-1).

1, 2, 1, 3, 1, 4, 8, 13, 1, 7, 1, 8, 16, 25, 35, 46, 58, 71, 1, 15, 30, 46, 63, 81, 100, 120, 141, 163, 1, 24, 48, 73, 1, 27, 54, 82, 111, 141, 172, 204, 237, 271, 1, 36, 72, 109, 1, 39, 78, 118, 159, 201, 244, 288, 333, 379, 1, 48, 96, 145, 195, 246, 298, 351, 405, 460, 516, 573, 631, 1

Offset

1,2

Comments

Runs of growth from 1 up to a prime where each term adds +i for successive i = 1,2,3,... and those increments continue in each run.

During a "reset" from a prime back to 1, i does not increase. Thus, i only increases within runs so that i is smaller than n by the number of occurrences of 1 per the formula below.

The graph forms several near-linear lines. It appears that the distribution of "resets" to 1 doesn't diminish as the sequence progresses.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10015

Formula

a(n) = 1 if n=1 or a(n-1) is prime; a(n) = a(n-1) + i otherwise, where i = n - r and r is the number of occurrences of 1 among terms a(1) to a(n-1).

a(n) = 1 iff n = 1 or n = A087128(k) + k + 1 for some k > 0. - Rémy Sigrist, Feb 13 2026

a(A393398(n)) = A393386(n). - Paolo Xausa, Feb 13 2026

Example

The terms and runs begin
  n    = 1  2  3  4  5  6  7   8  9 10 11 12  13
  a(n) = 1, 2, 1, 3, 1, 4, 8, 13, 1, 7, 1, 8, 16, ...
  step    +1    +2    +3 +4 +5     +6    +7  +8
  runs   \--/  \--/  \---------/  \--/  \-------

Mathematica

Module[{k = 0}, NestList[If[PrimeQ[#], 1, ++k + #] &, 1, 100]] (* Paolo Xausa, Feb 13 2026 *)

Prog

(PARI) lista(nn) = my(list=List(), last=1); listput(list, last); for (n=2, nn, if (isprime(last), last=1, last = last + n - #select(x->(x==1), list)); listput(list, last); ); Vec(list); \\ Michel Marcus, Dec 27 2025

Crossrefs

Cf. A087128, A393386, A393398.

Sequence in context: A197956 A054072 A325642 * A110977 A295785 A069230

Adjacent sequences: A391969 A391970 A391971 * A391973 A391974 A391975

Keyword

nonn

Author

Andrey Mateshko, Dec 24 2025

Extensions

More terms from Michel Marcus, Feb 13 2026

Status

approved