A213980 Let n = prime(1)^c_1 * prime(2)^c_2 * ... * prime(k)^c_k be the prime factorization of n. Set f(n) = n - 1 + c_1 + c_2 + ... + c_k and f_i, i>=0 (f_0(n) = n, f_1=f) is i-th iteration of f. a(n) is the minimal i such that f_i(n) is prime.

0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 4, 0, 3, 2, 1, 0, 3, 0, 2, 2, 1, 0, 2, 3, 2, 1, 6, 0, 5, 0, 4, 6, 5, 4, 3, 0, 3, 2, 1, 0, 3, 0, 2, 1, 1, 0, 5, 6, 5, 5, 4, 0, 3, 2, 1, 2, 1, 0, 18, 0, 18, 17, 15, 16, 15, 0, 14, 14, 13, 0, 12, 0, 13, 12, 11, 11, 10, 0, 9, 9, 1, 0, 8, 9, 8, 7, 6, 0, 5, 5, 4, 4, 3, 2, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 1, 0, 16, 0

Offset

2,7

Comments

Conjecture: a(n) exists for every n >= 2.

Links

Amiram Eldar, Table of n, a(n) for n = 2..10000

Example

f_1(12) = 12+2+1-1 = 14, f_1(14) = 14+1+1-1 = 15, f_1(15) = 15+1+1-1 = 16, f_1(16) = 16+4-1 = 19.
Since to get to a prime we used 4 iterations, a(12)=4.

Mathematica

a[n_] := Block[{x = n, c = 0}, While[! PrimeQ[x], x = x-1 + Total[Last /@ FactorInteger[x]]; c++]; c]; a/@Range[2, 109] (* Giovanni Resta, Feb 16 2013 *)

Crossrefs

f_1 is A222312.

Sequence in context: A292086 A065177 A064044 * A349373 A144912 A306708

Adjacent sequences: A213977 A213978 A213979 * A213981 A213982 A213983

Keyword

nonn

Author

Vladimir Shevelev, Feb 15 2013

Extensions

a(81) corrected by Giovanni Resta, Feb 16 2013

Status

approved