A070843
Largest n-digit number with exactly n distinct prime divisors. There are no further terms.
Original entry on oeis.org
9, 99, 996, 9996, 99990, 999570, 9999990, 99981420, 999068070, 9592993410
Offset: 1
a(4) = 9996 with the prime divisors 2,3,7 and 17.
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Do[k = 10^n - 1; While[Length[FactorInteger[k]] != n, k-- ]; Print[k], {n, 1, 10}] (* Ryan Propper, Nov 05 2005 *)
lnpd[n_]:=Module[{k=10^n-1},While[PrimeNu[k]!=n,k--];k]; Array[lnpd,10] (* Harvey P. Dale, Sep 30 2021 *)
A342108
Smallest positive integer m with n digits and such that omega(m) = bigomega(m) = n.
Original entry on oeis.org
2, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 223092870, 6469693230
Offset: 1
10010 = 2*5*7*11*13 is the smallest 5-digit number such that omega(10010) = bigomega(10010) = 5, hence a(5) = 10010.
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a={};For[n=1,n<=10,n++,For[m=10^(n-1),m<10^n,m++,If[PrimeOmega[m]==PrimeNu[m]==n,AppendTo[a,m];Break[]]]];a (* Stefano Spezia, Mar 04 2021 *)
A364468
Number of primitive n-bead necklaces (turning over is allowed) comprising elements of two flavors where complements and scalings are equivalent.
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 3, 8, 12, 20, 35, 62, 106, 189, 343, 603, 1130, 2055, 3860, 7154, 13562, 25463, 48607, 92204, 176646, 337587, 649151, 1246819, 2404519, 4636389, 8963497, 17334800, 33585928, 65107935, 126385919, 245492221, 477345359, 928772649, 1808662015, 3524337599, 6872457828, 13409202675, 26179870365
Offset: 0
For a(4) = 1, there is one solution: "1110". The other primitive sequence "1100" can be reduced to "10", which no longer uses 4 elements.
For a(6) = 3, there are three solutions: "111110", "111010", and "110010". The other primitive sequences "111100" and "111000" can be reduced to "110" and "10", respectively, which no longer use 6 elements.
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a11(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n));
a46(n) = {
my(s=0);
fordiv (n, d,
s+=moebius(d)*a11(n/d));
s};
a364468(n) = {
my(s=a46(n));
fordiv (n, k,
s-=if(k!=1&&k!=n, a364468(k), 0));
s};
for (k=1,42, print1(a364468(k),", ")) \\ Hugo Pfoertner, Jul 26 2023
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