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 A308725 Number of steps to reach 6 or 7 when iterating x -> A227215(x) starting at x=n, where A227215(n) gives the smallest such sum a+b+c of three positive integers for which a*b*c = n. 1
 3, 3, 2, 2, 1, 0, 0, 1, 1, 2, 4, 1, 3, 3, 2, 2, 7, 2, 6, 2, 5, 4, 6, 2, 5, 3, 2, 5, 5, 3, 4, 3, 3, 3, 4, 3, 9, 5, 8, 5, 7, 2, 6, 3, 5, 4, 4, 5, 3, 2, 6, 8, 9, 2, 8, 4, 7, 4, 6, 2, 5, 4, 4, 2, 7, 3, 4, 6, 3, 4, 6, 4, 5, 6, 4, 7, 7, 3, 4, 4, 3, 4, 8, 4, 7, 5, 4, 8, 7, 4, 6, 3, 5, 3, 6, 4, 9, 3, 8, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Starting from n, choose factorization n = m1*m2*m3 so that the sum x = m1+m2+m3 is minimal, then set n = x and repeat. a(n) gives the number of steps needed to reach either 6 or 7. The process is guaranteed to reach either term, because we only use factorization n = n*1*1 when n is either 1 or a prime number, that are the only cases (apart from A227215(4)=5) for which A227215(n) > n as then A227215(n) = n+2. Moreover, for n > 3, at least one of n, n+2, n+4 is composite, leading to a further significant drop in the trajectory after at most two consecutive +2 steps. - Comment clarified by Antti Karttunen, Jul 12 2019 Records: 3, 4, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, ..., occur at: n = 1, 11, 17, 37, 107, 233, 307, 1289, 3986, 6637, 14347, 69029, .... - Antti Karttunen, Jul 12 2019 LINKS Antti Karttunen, Table of n, a(n) for n = 1..10000 FORMULA If n is 6 or 7, a(n) = 0, otherwise a(n) = 1 + a(A227215(n)). - Antti Karttunen, Jul 11 2019 EXAMPLE 1 = 1*1*1  --> 1 + 1 + 1  = 3     3 = 1*1*3  --> 1 + 1 + 3  = 5     5 = 1*1*5  --> 1 + 1 + 5  = 7, thus a(1) = 3. .     4 = 1*2*2  --> 1 + 2 + 2  = 5,     5 = 1*1*5  --> 1 + 1 + 5  = 7, thus a(4) = 2. .   560 = 7*8*10 --> 7 + 8 + 10 = 25    25 = 1*5*5  --> 1 + 5 +  5 = 11    11 = 1*1*11 --> 1 + 1 + 11 = 13    13 = 1*1*13 --> 1 + 1 + 13 = 15    15 = 1*3*5  --> 1 + 3 +  5 =  9     9 = 1*3*3  --> 3 + 3 +  1 =  7, thus a(560) = 6. .    84 = 3*4*7  --> 3 + 4 + 7 = 14    14 = 1*2*7  --> 1 + 2 + 7 = 10    10 = 1*2*5  --> 1 + 2 + 5 =  8     8 = 2*2*2  --> 2 + 2 + 2 =  6, thus a(84) = 4. MATHEMATICA maxTerm = 99 (* Should be increased if output -1 appears. *); f[m_] := Module[{m1, m2, m3, factors}, factors = {m1, m2, m3} /. {ToRules[ Reduce[1 <= m1 <= m2 <= m3 && m == m1 m2 m3, {m1, m2, m3}, Integers]]}; SortBy[factors, Total] // First]; a[n_] := Module[{cnt = 0, m = n, fm, step}, While[!(m == 6 || m == 7), step = {fm = f[m], m = Total[fm]}; (* Print[n, " ", step]; *) cnt++; If[cnt > maxTerm, Return[-1]]]; cnt]; Array[a, 100] (* Jean-François Alcover, Jul 03 2019 *) PROG (PARI) A227215(n) = { my(ms=3*n); fordiv(n, i, for(j=i, (n/i), if(!(n%j), for(k=j, n/(i*j), if(i*j*k==n, ms = min(ms, (i+j+k))))))); (ms); }; \\ Like code in A227215. A308725(n) = if((6==n)||(7==n), 0, 1+A308725(A227215(n))); \\ Memoized implementation: memoA308725 = Map(); A308725(n) = if((6==n)||(7==n), 0, my(v); if(mapisdefined(memoA308725, n, &v), v, v = 1+A308725(A227215(n)); mapput(memoA308725, n, v); (v))); \\ Antti Karttunen, Jul 12 2019 CROSSREFS Cf. A227215, A308190. Sequence in context: A131589 A338113 A309119 * A202691 A328143 A278402 Adjacent sequences:  A308722 A308723 A308724 * A308726 A308727 A308728 KEYWORD nonn AUTHOR Ali Sada, Jun 20 2019 STATUS approved

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Last modified August 11 07:25 EDT 2022. Contains 356053 sequences. (Running on oeis4.)