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

1,2

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

(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)))

Crossrefs

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

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

Sequence in context: A325106 A309687 A268084 * A017876 A356860 A017865

Adjacent sequences: A261082 A261083 A261084 * A261086 A261087 A261088

Keyword

nonn

Author

Antti Karttunen, Sep 23 2015

Status

approved