OFFSET
1,3
COMMENTS
Grundy numbers of the game in which you decrease n by a number prime to n, and the game ends when 1 is reached. - Eric M. Schmidt, Jul 21 2013
a(n) = the smallest part of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(21) = 2; indeed, the partition having Heinz number 21 = 3*7 is [2,4]. - Emeric Deutsch, Jun 04 2015
a(n) is the number of numbers whose largest proper divisor is n, i.e., for n>1, number of occurrences of n in A032742. - Stanislav Sykora, Nov 04 2016
For n > 1, a(n) gives the number of row where n occurs in arrays A083221 and A246278. - Antti Karttunen, Mar 07 2017
REFERENCES
John H. Conway, On Numbers and Games, 2nd Edition, p. 129.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
Wikipedia, Nimber (explains the term Grundy number).
FORMULA
EXAMPLE
a(15) = 2 because 15=3*5, 3<5 and 3 is the 2nd prime.
MAPLE
with(numtheory):
a:= n-> `if`(n=1, 0, pi(min(factorset(n)[]))):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 03 2013
MATHEMATICA
a[1] = 0; a[n_] := PrimePi[ FactorInteger[n][[1, 1]] ]; Table[a[n], {n, 1, 96}](* Jean-François Alcover, Jun 11 2012 *)
PROG
(Haskell)
a055396 = a049084 . a020639 -- Reinhard Zumkeller, Apr 05 2012
(PARI) a(n)=if(n==1, 0, primepi(factor(n)[1, 1])) \\ Charles R Greathouse IV, Apr 23 2015
(Python)
from sympy import primepi, isprime, primefactors
def a049084(n): return primepi(n)*(1*isprime(n))
def a(n): return 0 if n==1 else a049084(min(primefactors(n))) # Indranil Ghosh, May 05 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, May 15 2000
STATUS
approved