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.
lst={}; Do[AppendTo[lst, CarmichaelLambda[n]], {n, 6*7!}]; lst; Take[Union[lst], 123] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
(* warning: there seems to be no guarantee that no terms near the end are omitted! - Joerg Arndt, Dec 23 2014 *)
TakeWhile[Union@ Table[CarmichaelLambda@ n, {n, 10^6}], # <= 168 &] (* Michael De Vlieger, Mar 19 2016 *)
list(lim)=my(v=List([1]),u,t); forprime(p=3,lim\3+1, u=List(); listput(u,p-1); while((t=u[#u]*p)<=lim, listput(u,t)); for(j=1,#v, for(i=1,#u, t=lcm(u[i],v[j]); if(t<=lim && t!=v[j], listput(v,t)))); v=List(Set(v))); forprime(p=lim\3+2,lim+1, listput(v,p-1)); v=List(Set(v)); for(i=1,#v, t=2*v[i]; if(t>lim, break); listput(v,t); while((t*=2)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 23 2017
is(n)=if(n%2, return(n==1)); my(f=factor(n),pe); for(i=1,#f~, if(n%(f[i,1]-1)==0, next); pe=f[i,1]^f[i,2]; forstep(q=2*pe+1,n+1,2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV, Jun 25 2017
a(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);} \\ Michel Marcus, May 12 2018
Because A002174(5)=8 and psi(k)=8 for k=32,96,160,480, we have a(5)=4.
a079612(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);}
nb(n) = sumdiv(n, d, moebius(n/d)*numdiv(a079612(d)));
lista(nn) = for (n=1, nn, if (nbs = nb(n), print1(nbs, ", "))); \\ Michel Marcus, May 12 2018
a[n_] := Module[{f, fsz, g = 1, h = 1, p, e}, Which[n <= 0, Return[0], n == 1, Return[2], OddQ[n], Return[0]]; f = FactorInteger[n][[All, 1]]; fsz = Length[f]; For[k = 1, k <= fsz, k++, p = f[[k]]; e = 1; While[Mod[n, CarmichaelLambda[p^e]] == 0, e++]; g *= p^(e-1)]; Do[If[PrimeQ[d+1] && Mod[g, d+1] != 0, h *= (d+1)], {d, Divisors[n]}]; g *= h; If[CarmichaelLambda[g] != n, 0, g]];
a /@ Range[100] (* Jean-François Alcover, Oct 18 2019, after Gheorghe Coserea *)
lambda(n) = { \\ A002322
my(f=factor(n), fsz=matsize(f)[1]);
lcm(vector(fsz, k, my(p=f[k,1], e=f[k,2]);
if (p != 2, p^(e-1)*(p-1), e > 2, 2^(e-2), 2^(e-1))));
};
a(n) = {
if (n <= 0, return(0), n==1, return(2), n%2, return(0));
my(f=factor(n), fsz=matsize(f)[1], g=1, h=1);
for (k=1, fsz, my(p=f[k,1], e=1);
while (n % lambda(p^e) == 0, e++); g *= p^(e-1));
fordiv(n, d, if (isprime(d+1) && g % (d+1) != 0, h *= (d+1)));
g *= h; if (lambda(g) != n, 0, g);
};
vector(64, n, a(n)) \\ Gheorghe Coserea, Feb 21 2019
minvl(n, v) = {vgt = select(x->(x>=n), v, 1); first = vgt[1]; vgtd = vector(#vgt-1, k, vgt[k+1] - vgt[k]); vgtdr = Vecrev(vgtd); vokdiff = select(x->x!=1, vgtdr, 1); if (#vokdiff, #v - vokdiff[1]+1, first);}
lista(nn) = {v = read("v002322.txt"); for (n=1, nn, print1(minvl(n, v), ", "););}
a(8) = 32 because lambda(32) = 8.
with(numtheory):for k from 1 to 100 do:id:=0:for n from 1 to 1000 while(id=0) do: if lambda(n) = k then id:=1:printf(`%d, `,n):else fi:od:if id=0 then printf(`%d, `,0):else fi:od:
nn = 100; t = Table[0, {nn}]; Do[c = CarmichaelLambda[k]; If[c <= nn && t[[c]] == 0, t[[c]] = k], {k, 1000}]; t
1,2; 3,4,6,8,12,24; 5,10,15,16,20,30,40,48,60,80,120,240
a(6) = 3 because phi(6) = lambda(3) = 2.
with(numtheory): for n from 1 to 100 do: ii:=0:for k from 1 to 10^6 while(ii=0) do:if phi(n)=lambda(k) then ii:=1: printf(`%d, `,k):else fi:od:if ii=0 then printf(`%d, `,0): else fi:od:
Table[k=0; While[!EulerPhi[n] == CarmichaelLambda[k], k++]; k, {n, 100}] (* program will go into an infinite loop at n = 2047 *)
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