A055396 Smallest prime dividing n is a(n)-th prime (a(1)=0).

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 1, 14, 1, 2, 1, 15, 1, 4, 1, 2, 1, 16, 1, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 2, 1, 4, 1, 22, 1, 2, 1, 23, 1, 3, 1, 2, 1, 24, 1, 4, 1, 2, 1, 3, 1

Offset: 1

Author

Henry Bottomley, May 15 2000

Keywords

• nonn

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

Examples

			a(15) = 2 because 15=3*5, 3<5 and 3 is the 2nd prime.
		

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).
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences generated by sieves

Crossrefs

Cf. A004280, A020639, A032742, A038179, A049084, A055399, A061395, A215366, A243055, A257993, A276086.

Cf. also A078898, A246277, A250469 and arrays A083221 and A246278.

Programs

Haskell

a055396 = a049084 . a020639  -- Reinhard Zumkeller, Apr 05 2012

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 *)

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

Formula

From Reinhard Zumkeller, May 22 2003: (Start)

a(n) = A049084(A020639(n)).

A000040(a(n)) = A020639(n); a(n) <= A061395(n).

(End)

From Antti Karttunen, Mar 07 2017: (Start)

A243055(n) = A061395(n) - a(n).

a(A276086(n)) = A257993(n).

(End)