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A243977
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a(n) is the largest run of identical digits that n^k can end with for some k, or 0 if there is no limit to such runs.
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3
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3, 1, 2, 1, 1, 1, 3, 1, 0, 2, 3, 3, 2, 1, 1, 5, 1, 2, 0, 1, 3, 1, 1, 1, 1, 1, 3, 1, 0, 3, 1, 4, 2, 1, 1, 3, 3, 3, 0, 1, 3, 1, 2, 1, 1, 1, 3, 1, 0, 1, 3
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OFFSET
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2,1
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COMMENTS
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a(10*n) = 0 for all n > 0.
a(54) > 30 or 0. The sequence continues for 54 < n < 100 {2, 2, 1, 1, 3, 2, 0, 1, 3, 1, 2, 1, 2, 1, 1, 1, 0, ?, 3, 3, 1, 1, 1, ?, 3, 4, 0, 1, 1, 1, 2, 1, 1, 1, 3, 1, 0, 2, 3, 1, 2, 1, 1, 4, 3, 2} where the question marks represent a(71) and a(77). a(71) > 44 or 0 and a(77) > 16 or 0.
a(53), a(71) and a(77) are conjectured to be infinite. See A244187.
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LINKS
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Table of n, a(n) for n=2..52.
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EXAMPLE
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For a(1), 2^k ends in 1 identical digit when k = 1, 2 identical digits when k = 18, and 3 identical digits when k = 39. 2^k doesn't end in 4 identical digits for any k. Thus a(1) = 3.
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PROG
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(PARI) a(n, p)=lst=[]; for(c=0, 10^p, m=n^c%10^p; if(vecsearch(vecsort(lst), m), for(i=1, #lst, if(vecextract(lst, 2^(i-1), )==[m], return([c, c-i+1])))); if(!vecsearch(vecsort(lst), m), lst=concat(lst, m)))
hup(n)=if(n%10==0, return(0)); ww=[]; p=2; for(ii=1, a(n, p)[1], ww=concat(ww, ii)); while(p<100, v=ww; w=[]; for(q=1, #v, h=digits(n^v[q]%10^p); if(#h==p&&(vecmin(h)==vecmax(h)), w=concat(w, v[q]))); if(w, ww=[]; for(k=1, #w, j=w[k]; while(j<=a(n, p+1)[1], ww=concat(ww, j); j+=a(n, p)[2])); ww=vecsort(ww, , 8); p++); if(!w, return(p-1)))
n=2; while(n<100, print1(hup(n), ", "); n++)
(Python)
def a(n, p):
..lst = []
..for c in range(10**p+1):
....m = n**c%10**p
....if m in lst:
......return [c, c-lst.index(m)]
....else:
......lst.append(m)
def cou(n):
..if n % 10 == 0:
....return 0
..ww = []
..p = 2
..aa = a(n, p)[0]
..ww.extend(range(aa))
..while p < 100:
....newlst = ww
....w = []
....for i in newlst:
......m = n**i%10**p
......if len(str(m))==p and m%int('1'*p)==0:
........w.append(i)
....if w:
......ww = []
......for k in w:
........j = k
........while j <= a(n, p+1)[0]:
..........ww.append(j)
..........j += a(n, p)[1]
......ww.sort()
......p += 1
....else:
......return p-1
n = 2
while n < 100:
..if cou(n):
....print(cou(n), end=', ')
..else:
....print(0, end=', ')
..n += 1
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CROSSREFS
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Cf. A243912, A243911, A244187.
Sequence in context: A086249 A176784 A176511 * A328569 A016569 A072801
Adjacent sequences: A243974 A243975 A243976 * A243978 A243979 A243980
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KEYWORD
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nonn,base
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AUTHOR
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Derek Orr, Jun 16 2014
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EXTENSIONS
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Programs corrected and improved by Derek Orr, Aug 18 2014
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STATUS
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approved
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