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.
a(6)=223 because 6th composite number is 12: A002808(6)=12, then 12=2*2*3 and merging prime factors we get a(6)=223.
Reap[Do[If[!PrimeQ[n],Sow[FromDigits@Flatten[Table[IntegerDigits[ #[[1]]],{#[[2]]}]&/@FactorInteger@(n)]]],{n,2,115}]][[2,1]]
a(5) = 22 as a(4) = 4 which has a factorization 4 = 2*2, so the concatenation of factors is '22'. a(7) = 5 as a(6) = 211 which is prime, and 5 is the smallest number not yet appearing in the sequence. a(14) = 31941 as a(13) = 2337 which has a factorization 2337 = 3*19*41, so the concatenation of factors is '31941'.
nn = 43; c[] = 0; a[1] = c[1] = u = 1; While[c[u] > 0, u++]; Do[If[CompositeQ[#], k = FromDigits@ Flatten@ Map[IntegerDigits[#] &, ConstantArray[##] & @@@ FactorInteger[#]], k = u] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 12 2022 *)
95 = 5*19 -> 519 = 3*173 -> 3173 = 19*167 -> 19167 = 3*6389 and 36389 is prime.
filter:= proc(n) local k,F,x,j;
x:= n;
for k from 1 to 4 do
if isprime(x) then return false fi;
F:= sort(ifactors(x)[2],(a,b) -> a[1] t[1] $ t[2], F);
x:= F[1];
for j from 2 to nops(F) do
x:= x*10^(1+ilog10(F[j]))+F[j]
od;
od;
isprime(x)
end proc:
select(filter, [$2..1000]);# Robert Israel, Jun 25 2019
\\ See Links section.
698 is in the sequence as 698 -> 2349 -> 333329 -> 2571297 -> 3857099 -> 31312323 -> 33771937101 -> 379437170413 -> 73124171910091 -> 374148203145623. Only after the ninth iteration we reach a prime. - _David A. Corneth_, Oct 15 2019
is(n, k) = if(isprime(n), return(0)); for(i = 1, k - 1, n = concatelements(primesvector(n)); if(isprime(n), return(0))); n = concatelements(primesvector(n)); isprime(n) concatelements(v) = my(s = ""); for(i = 1, #v, s = concat(s, v[i])); eval(s) primesvector(n) = my(f = factor(n), res = vector(vecsum(f[,2])), t = 0); for(i = 1, #f~, for(j = 1, f[i, 2], t++; res[t] = f[i, 1])); res \\ David A. Corneth, Oct 15 2019
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; }
}
24 = 2*2*2*3, 2223 = 3*3*13*19, 331319 is prime.
69 = 3*23 -> 323 = 17*19 -> 1719 = 3*3*191 and 33191 is prime.
14 = 2*7 -> 27 = 3*3*3 -> 333 = 3*3*37 -> 3337 = 47*71 -> 4771 = 13*367 and 13367 is prime.
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