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.
9 = 3*3 -> 33 = 3*11 -> 311, prime, so a(9) = 311. The trajectory of 8 is more interesting: 8 -> 2 * 2 * 2 -> 2 * 3 * 37 -> 3 * 19 * 41 -> 3 * 3 * 3 * 7 * 13 * 13 -> 3 * 11123771 -> 7 * 149 * 317 * 941 -> 229 * 31219729 -> 11 * 2084656339 -> 3 * 347 * 911 * 118189 -> 11 * 613 * 496501723 -> 97 * 130517 * 917327 -> 53 * 1832651281459 -> 3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107 and 3331113965338635107 is prime, so a(8) = 3331113965338635107.
b:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[])): a:= n-> `if`(isprime(n) or n=1, n, a(b(n))): seq(a(n), n=1..48); # Alois P. Heinz, Jan 09 2021
f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], { #[[2]] }] & /@ FactorInteger@n, 2]; g[n_] := NestWhile[ f@# &, n, !PrimeQ@# &]; g[1] = 1; Array[g, 41] (* Robert G. Wilson v, Sep 22 2007 *)
step(n)=my(f=factor(n),s="");for(i=1,#f~,for(j=1,f[i,2],s=Str(s,f[i,1]))); eval(s) a(n)=if(n<4,return(n)); while(!isprime(n), n=step(n)); n \\ Charles R Greathouse IV, May 14 2015
from sympy import factorint, isprime
def f(n): return int("".join(str(p)*e for p, e in factorint(n).items()))
def a(n):
if n == 1: return 1
fn = n
while not isprime(fn): fn = f(fn)
return fn
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Jul 11 2022
def digitLen(x,n):
r=0
while(x>0):
x//=n
r+=1
return r
def concatPf(x,n):
r=0
f=list(factor(x))
for c in range(len(f)):
for d in range(f[c][1]):
r*=(n**digitLen(f[c][0],n))
r+=f[c][0]
return r
def hp(x,n):
x1=concatPf(x,n)
while(x1!=x):
x=x1
x1=concatPf(x1,n)
return x
#example: prints the home prime of 8 in base 10
print(hp(8,10))
13 is already prime, so a(13) = 0. Starting with 14 we get 14 = 2*7, 27 = 3*3*3, 333 = 3*3*37, 3337 = 47*71, 4771 = 13*367, 13367 is prime; so a(14) = 5.
a037273 1 = -1 a037273 n = length $ takeWhile ((== 0) . a010051) $ iterate (\x -> read $ concatMap show $ a027746_row x :: Integer) n -- Reinhard Zumkeller, Jan 08 2013
nxt[n_] := FromDigits[Flatten[IntegerDigits/@Table[#[[1]], {#[[2]]}]&/@ FactorInteger[n]]]; Table[Length[NestWhileList[nxt, n, !PrimeQ[#]&]] - 1, {n, 48}] (* Harvey P. Dale, Jan 03 2013 *)
row_a027746(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ after M. F. Hasler in A027746 tonum(vec) = my(s=""); for(k=1, #vec, s=concat(s, Str(vec[k]))); eval(s) a(n) = if(n==1, return(-1)); my(x=n, i=0); while(1, if(ispseudoprime(x), return(i)); x=tonum(row_a027746(x)); i++) \\ Felix Fröhlich, May 17 2021
from sympy import factorint
def a(n):
if n < 2: return -1
klst, f = [n], sorted(factorint(n, multiple=True))
while len(f) > 1:
klst.append(int("".join(map(str, f))))
f = sorted(factorint(klst[-1], multiple=True))
return len(klst) - 1
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Aug 02 2021
Starting with 14 (the seventh composite number) we get 14=2*7, 27=3*3*3, 333=3*3*37, 3337=47*71, 4771=13*367, 13367 is prime; so a(7)=5.
a037271 = length . takeWhile ((== 0) . a010051'') .
iterate a037276 . a002808
-- Reinhard Zumkeller, Apr 03 2012
maxComposite = 49; maxIter = 40; concat[n_] := FromDigits[ Flatten[ IntegerDigits /@ Flatten[ Apply[ Table, {#[[1]], {#[[2]]}} & /@ FactorInteger[n], {1}]]]]; composites = Select[ Range[2, maxComposite], ! PrimeQ[#] &]; a[n_] := ( lst = NestWhileList[ concat, composites[[n]], ! PrimeQ[#] &, 1, maxIter]; If[PrimeQ[ Last[lst]], Length[lst] - 1, - 1]); Table[a[n], {n, 1, Length[composites]}] (* Jean-François Alcover, Jul 10 2012 *)
For a(20), the trajectory is 20->39->77->153->305->609->1217, a prime number. That required 6 steps, so a(20)=6.
y=as.bigz(rep(0,500)); ys=rep(0,500);
for(i in 1:500) { n=as.bigz(i); k=0;
while(isprime(n)==0 & ndig(n)<1000 & k<5000) { k=k+1; n=2*n-1 }
if(ndig(n)>=1000 | k>=5000) { ys[i]=-1; y[i]=-1;
} else {ys[i]=k; y[i]=n; }
}
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