A329428 Starting from n: as long as the decimal representation starts with a prime number, replace the largest such prefix with the index of the corresponding prime number; a(n) corresponds to the final value.

0, 1, 1, 1, 4, 1, 6, 4, 8, 9, 10, 1, 12, 6, 14, 15, 16, 4, 18, 8, 10, 1, 12, 9, 14, 15, 16, 4, 18, 10, 10, 1, 12, 9, 14, 15, 16, 12, 18, 10, 40, 6, 42, 14, 44, 45, 46, 15, 48, 49, 10, 1, 12, 16, 14, 15, 16, 12, 18, 4, 60, 18, 62, 63, 64, 65, 66, 8, 68, 69, 40

Offset

0,5

Comments

As long as we have a number whose decimal representation is the concatenation of a prime number, say the k-th prime number, and a minimal string possibly empty or with leading zeros, say v, we replace this number with the concatenation of k and v; eventually none of the prefixes will be a prime number.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Formula

a(n) <= n with equality iff n belongs to A202259.

a(prime(k)) = a(k) for any k > 0 where prime(k) denotes the k-th prime number.

a(10*k + v) = 10*a(k) + v for any k > 0 and v in {0, 2, 4, 5, 6, 8}.

a(A007097(k)) = 1 for any k >= 0.

Example

For n = 991:
- let pi denote A000720,
- 991 gives pi(991) = 167,
- 167 gives pi(167) = 39,
- 39 gives pi(3) followed by 9 = 29,
- 29 gives pi(29) = 10,
- neither 1 nor 10 is a prime number, so a(991) = 10.

Mathematica

Array[Which[# == 0, 0, # == 1, 1, True, FixedPoint[If[! IntegerQ@ #1, FromDigits[#2], FromDigits[Join @@ {IntegerDigits@ PrimePi@ #1, Drop[#2, IntegerLength@ #1]}]] & @@ {SelectFirst[Table[FromDigits[#[[1 ;; i]]], {i, Length@ #, 1, -1}], PrimeQ], #} &@ IntegerDigits[#] &, #]] &, 71, 0] (* Michael De Vlieger, Dec 01 2019 *)
f[n_]:=Block[{p, r, d = IntegerDigits@ n, v=n}, Do[{p, r}= FromDigits/@ TakeDrop[d, k]; If[PrimeQ@ p, v=PrimePi[p] 10^(Length[d]-k)+r; Break[]], {k, Length@d, 1, -1}]; v]; a[n_]:= FixedPoint[f, n]; Array[a, 71, 0] (* Giovanni Resta, Dec 02 2019 *)

Prog

(PARI) t(n) = if (n==0, 0, isprime(n), primepi(n), 10*t(n\10)+(n%10))
a(n) = while (n!=n=t(n), ); n

Crossrefs

See A327539 for similar sequences.

Cf. A007097, A202259.

Sequence in context: A115607 A318876 A330356 * A288469 A076717 A200360

Adjacent sequences: A329425 A329426 A329427 * A329429 A329430 A329431

Keyword

nonn,base

Author

Rémy Sigrist, Nov 30 2019

Status

approved