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

Author

Rémy Sigrist, Nov 30 2019

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Crossrefs

See A327539 for similar sequences.

Cf. A007097, A202259.

Programs

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 *)

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

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.