A261085 Number of steps needed to reach zero when starting from the n-th prime [i.e., setting k to A000040(n)] and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

1, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 15, 17, 19, 20, 22, 23, 24, 26, 16, 18, 20, 21, 22, 22, 23, 24, 23, 24, 25, 26, 28, 29, 31, 33, 33, 34, 36, 37, 39, 40, 40, 41, 44, 47, 34, 35, 49, 51, 52, 54, 54, 55, 57, 58, 59, 58, 59, 62, 48, 49, 50, 66, 69, 71, 73, 74, 74, 76, 55, 57, 59, 60, 61, 63, 63, 65, 68, 69, 71, 72, 74, 62, 64, 65, 66, 67, 67, 70, 72, 73, 75, 76, 77, 80, 81, 75, 77, 79, 79, 81

Offset: 1

Author

Antti Karttunen, Sep 23 2015

Keywords

• nonn

Examples

			For n=4 we have prime(4) = 7, from which we start subtracting the number of divisors, to get the following path: 7 - 2 = 5, 5 - 2 = 3, 3 - 2 = 1, 1 - 1 = 0, and we have reached zero in four steps, thus a(4) = 4.
For n=5 we have prime(5) = 11, for which the similar process results: 11 - 2 = 9, 9 - 3 = 6, 6 - 4 = 2, 2 - 2 = 0, and again we have reached zero in four steps, thus also a(5) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..6543

Crossrefs

Cf. A000005, A000040, A049820, A155043, A259934, A261088.

Cf. A261086 (gives the positions of drops, i.e., where nonmonotonic) and A261087 (the corresponding primes).

Programs

Mathematica

mpr[p_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,p,#>0&]]-1; mpr/@Prime[ Range[ 120]] (* Harvey P. Dale, Aug 18 2022 *)

PARI

uplim = 65537;
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
n=0; forprime(p=2, uplim, n++; write("b261085.txt", n, " ", A155043(p)));

Scheme

(define (A261085 n) (A155043 (A000040 n)))

Formula

a(n) = A155043(A000040(n)).