A129985 Multiplicative persistence of the prime numbers.

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 1, 2, 1, 2, 3, 2, 3, 3, 1, 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 2, 2, 2, 3, 1, 1, 3, 3, 2, 1, 2, 3, 3, 2, 3, 1, 2, 2, 3, 2, 2, 4, 2, 3, 3, 1, 1, 1, 2, 1, 3, 3, 2, 3, 3, 3, 3, 4, 3, 3, 4, 1, 1, 3, 1, 2, 3, 2, 3, 3, 2, 2, 3, 4, 3, 3, 3, 3, 1, 1, 2, 2, 2

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 14 2007

Keywords

• easy
• nonn
• base

Examples

			229 = prime(50) -> 2*2*9 = 36 -> 3*6 = 18 -> 1*8 = 8 -> persistence(229) = 3 = a(50).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A031346.

Programs

Maple

P:=proc(n) local i,k,w,ok,cont,x; for i from 1 by 1 to n do k:=ithprime(i); w:=1; ok:=1; x:=k; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);

Mathematica

s={};Do[i=0;m=Prime[n];While[m!=Times@@IntegerDigits[m],m=Times@@IntegerDigits[m];i++];AppendTo[s,i],{n,100}];s (* James C. McMahon, Feb 03 2025 *)

Formula

a(n) = A031346(A000040(n)). - Alois P. Heinz, Feb 03 2025

Extensions

Offset corrected by Alois P. Heinz, Feb 03 2025