A055396 - OEIS

%I #63 Jan 20 2021 07:22:40

%S 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,

%T 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,

%U 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

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

%C 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

%C 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

%C 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

%C For n > 1, a(n) gives the number of row where n occurs in arrays A083221 and A246278. - _Antti Karttunen_, Mar 07 2017

%D John H. Conway, On Numbers and Games, 2nd Edition, p. 129.

%H Reinhard Zumkeller, <a href="/A055396/b055396.txt">Table of n, a(n) for n = 1..10000</a>

%H Douglas E. Iannucci and Urban Larsson, <a href="https://arxiv.org/abs/2101.07608">Game values of arithmetic functions</a>, arXiv:2101.07608 [math.NT], 2021.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Nimber">Nimber</a> (explains the term Grundy number).

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>

%F From _Reinhard Zumkeller_, May 22 2003: (Start)

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

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

%F (End)

%F From _Antti Karttunen_, Mar 07 2017: (Start)

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

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

%F (End)

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

%p with(numtheory):

%p a:= n-> `if`(n=1, 0, pi(min(factorset(n)[]))):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 03 2013

%t a[1] = 0; a[n_] := PrimePi[ FactorInteger[n][[1, 1]] ]; Table[a[n], {n, 1, 96}](* _Jean-François Alcover_, Jun 11 2012 *)

%o (Haskell)

%o a055396 = a049084 . a020639  -- _Reinhard Zumkeller_, Apr 05 2012

%o (PARI) a(n)=if(n==1,0,primepi(factor(n)[1,1])) \\ _Charles R Greathouse IV_, Apr 23 2015

%o (Python)

%o from sympy import primepi, isprime, primefactors

%o def a049084(n): return primepi(n)*(1*isprime(n))

%o def a(n): return 0 if n==1 else a049084(min(primefactors(n))) # _Indranil Ghosh_, May 05 2017

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

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

%K nonn

%O 1,3

%A _Henry Bottomley_, May 15 2000