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A359960 Smallest Niven (or Harshad) number (A005349) with exactly n distinct prime factors.

%I A359960 #33 Feb 16 2025 08:34:04
%S A359960 1,2,6,30,210,2310,30030,690690,14804790,223092870,8254436190,
%T A359960 200560490130,8222980095330,304250263527210,13082761331670030,
%U A359960 614889782588491410,32589158477190044730,1987938667108592728530,117288381359406970983270,7858321551080267055879090
%N A359960 Smallest Niven (or Harshad) number (A005349) with exactly n distinct prime factors.
%C A359960 a(11) = 200560490130; a(13) = 304250263527210.
%C A359960 a(n) >= A002110(n) = prime(n)#.
%C A359960 Many terms are primorial numbers, see A360011.
%H A359960 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/Harshad_number/">Harshad numbers</a>.
%H A359960 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.
%e A359960 2310 = 2*3*5*7*11 is the smallest integer with 5 prime factors because it is a primorial number, as 2310 / (2+3+1+0) = 385, 2310 is a Niven number: a(5) = 2310.
%o A359960 (PARI) a(n) = my(k=1); while ((k % sumdigits(k)) || (omega(k) != n), k++); k; \\ _Michel Marcus_, Jan 20 2023
%o A359960 (PARI)
%o A359960 omega_niven(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && v%sumdigits(v) == 0, listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
%o A359960 a(n) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_niven(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Jan 22 2023
%Y A359960 Cf. A001221, A002110, A005349, A113315, A360011.
%Y A359960 Similar: A060319 (Fibonacci), A083002 (oblong), A359961 (Zuckerman).
%K A359960 nonn,base
%O A359960 0,2
%A A359960 _Bernard Schott_, Jan 20 2023
%E A359960 a(8)-a(9) from _Michel Marcus_, Jan 20 2023
%E A359960 a(10)-a(19) from _Daniel Suteu_, Jan 22 2023