This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
75 -> 7^5 = 16807 -> 1^6*8^0*7 = 7.
33 -> 3^3 = 27 -> 2^7 = 128 -> 1^2*8 = 8.
25 -> 2^5 = 32 -> 3^2 = 9.
def powertrain(n):
p, s = 1, str(n)
if len(s)%2 == 1: s += '1'
for b, e in zip(s[0::2], s[1::2]): p *= int(b)**int(e)
return p
def aupto(limit, target=0):
alst = []
for n in range(1, limit+1):
m, ptm = n, powertrain(n)
while m != ptm: m, ptm = ptm, powertrain(ptm)
if m == target: alst.append(n)
return alst
print(aupto(1191, target=9)) # Michael S. Branicky, Sep 25 2021
E.g. 240 -> (0)42 -> 4^2 = 16; 12345 -> 54321 -> 5^4*3^2*1 = 5625.
a133048 0 = 0 a133048 n = train $ dropWhile (== 0) $ a031298_row n where train [] = 1 train [x] = x train (u:v:ws) = u ^ v * (train ws) -- Reinhard Zumkeller, May 27 2013
powerback:=proc(n) local a,i,j,t1,t2,t3; if n = 0 then RETURN(0); fi; t1:=convert(n, base, 10); t2:=nops(t1); for i from 1 to t2 do if t1[i] > 0 then break; fi; od: a:=1; t3:=t2-i+1; for j from 0 to floor(t3/2)-1 do a := a*t1[i+2*j]^t1[i+2*j+1]; od: if t3 mod 2 = 1 then a:=a*t1[t2]; fi; RETURN(a); end;
ptm[n_]:=Module[{idn=IntegerDigits[IntegerReverse[n]]},If[ EvenQ[ Length[idn]],Times@@ (#[[1]]^#[[2]]&/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]]&/@Partition[ Most[ idn],2]))Last[idn]]];Array[ptm,70,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2020 *)
The trajectory of 39 is 39 -> 19683 -> 5038848 -> 214990848 -> 17179869184 -> 1735247072139264 -> 19999187712 -> 102372436321763328 -> 8813365017182208 -> 0, so a(39) = 102372436321763328.
maxtraj := proc(n) local h,p,M,t1,t2,i; M:=100; t1:=n; h:=n; for i from 1 to M do t2:=powertrain(t1); if t2 = t1 then RETURN(n,h); fi; if t2 > t1 then h:=t2; fi; t1:=t2; od; RETURN(n,-1); end;
a222493 = a133500 . a221221
Factored forms are 0, 1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9, 3^6, 3^7, 3^8, 3^9, 2^16, 2^18, 5^8, 5^9, 2^9*3^9, 7^9, 2^27, 3^18, 2^27*3, 2^29, 2^27*5, 2^28*3, 2^27*7, 2^30, 2^27*3^2, 2^2*3^18, 3^18*5, 2*3^19, 3^18*7, 2^3*3^18, 3^20, 2^4*3^18, 2^5*3^18, 2^6*3^18, 2^7*3^18, 2^8*3^18, 2^9*3^18, 3^24, 2^27*3^7, 3^25, 2^27*3^8, 3^26, 2^27*3^9, 3^27, 2^43, 2^16*3^18, 2^45, 2^18*3^18, 3^18*5^8, 2^27*5^9, ...
M:=100000; t1:=[]; t2:=[]; rec:=-1; for n from 0 to M do m:=powertrain(n); if m > rec then rec:=m; t1:=[op(t1),m]; t2:=[op(t2),n]; fi; od: lprint(t1); lprint(t2);
The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0. The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0. The number 31395559595973 takes 16 steps to converge and so the next term is >= 16. The trajectory is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0. The smallest number that takes 16 steps to converge is 555959597395, for which the trajectory starts 555959597395 -> 471570692025125026702880859375 and then continues as above. - _Michael S. Branicky_, Jan 24 2022
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