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a(n) = A076649(n) - A055642(n).
3

%I #19 Jan 11 2017 03:15:36

%S -1,0,0,1,0,1,0,2,1,0,0,1,0,0,0,2,0,1,0,1,0,1,0,2,0,1,1,1,0,1,0,3,1,1,

%T 0,2,0,1,1,2,0,1,0,2,1,1,0,3,0,1,1,2,0,2,1,2,1,1,0,2,0,1,1,4,1,2,0,2,

%U 1,1,0,3,0,1,1,2,1,2,0,3,2,1,0,2,1,1,1,3,0,2,1,2,1,1,1,4,0,1,2,1

%N a(n) = A076649(n) - A055642(n).

%C a(1) is the only negative term in this sequence. - _Ely Golden_, Jan 10 2017

%C a(n) = 0 if and only if n is a member of A109608. - _Ely Golden_, Jan 10 2017

%H Ely Golden, <a href="/A280827/b280827.txt">Table of n, a(n) for n = 1..10000</a>

%H Ely Golden, <a href="/A280827/a280827.txt">Proof that a(n)>=0 for all n>1</a>

%e a(10) = 0, as 2*5 have 2 digits total, and 10 has 2 digits. Thus a(10) = 2-2 = 0.

%e a(1) is defined to be -1, as the empty product has 0 digits, and 1 has 1 digit. Thus a(1) = 0-1 = -1.

%o (SageMath)

%o def digits(x, n):

%o if(x<=0|n<2):

%o return []

%o li=[]

%o while(x>0):

%o d=divmod(x, n)

%o li.insert(0,d[1])

%o x=d[0]

%o return li;

%o def factorDigits(x, n):

%o if(x<=0|n<2):

%o return []

%o li=[]

%o f=list(factor(x))

%o for c in range(len(f)):

%o for d in range(f[c][1]):

%o ld=digits(f[c][0], n)

%o li+=ld

%o return li;

%o def digitDiff(x,n):

%o return len(factorDigits(x,n))-len(digits(x,n))

%o radix=10

%o index=1

%o while(index<=10000):

%o print(str(index)+" "+str(digitDiff(index,radix)))

%o index+=1

%Y Cf. A109608, A076649.

%K sign,base,easy

%O 1,8

%A _Ely Golden_, Jan 08 2017