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

1, 2, 6, 30, 210, 2310, 30030, 690690, 14804790, 223092870, 8254436190, 200560490130, 8222980095330, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1987938667108592728530, 117288381359406970983270, 7858321551080267055879090

Offset

0,2

Comments

a(11) = 200560490130; a(13) = 304250263527210.

a(n) >= A002110(n) = prime(n)#.

Many terms are primorial numbers, see A360011.

Links

Table of n, a(n) for n=0..19.

Giovanni Resta, Harshad numbers.

Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Example

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.

Prog

(PARI) a(n) = my(k=1); while ((k % sumdigits(k)) || (omega(k) != n), k++); k; \\ Michel Marcus, Jan 20 2023
(PARI)
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)));
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

Crossrefs

Cf. A001221, A002110, A005349, A113315, A360011.

Similar: A060319 (Fibonacci), A083002 (oblong), A359961 (Zuckerman).

Sequence in context: A002110 A118491 A376237 * A088257 A058694 A336672

Adjacent sequences: A359957 A359958 A359959 * A359961 A359962 A359963

Keyword

nonn,base

Author

Bernard Schott, Jan 20 2023

Extensions

a(8)-a(9) from Michel Marcus, Jan 20 2023

a(10)-a(19) from Daniel Suteu, Jan 22 2023

Status

approved