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 A243977 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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. LINKS EXAMPLE 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. PROG (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 CROSSREFS Cf. A243912, A243911, A244187. Sequence in context: A086249 A176784 A176511 * A328569 A016569 A072801 Adjacent sequences: A243974 A243975 A243976 * A243978 A243979 A243980 KEYWORD nonn,base AUTHOR Derek Orr, Jun 16 2014 EXTENSIONS Programs corrected and improved by Derek Orr, Aug 18 2014 STATUS approved

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Last modified March 27 22:14 EDT 2023. Contains 361575 sequences. (Running on oeis4.)