cp's OEIS Frontend

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.

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

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 690690, 14804790, 223092870, 8254436190, 200560490130, 8222980095330, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1987938667108592728530, 117288381359406970983270, 7858321551080267055879090
Offset: 0

Views

Author

Bernard Schott, Jan 20 2023

Keywords

Comments

a(11) = 200560490130; a(13) = 304250263527210.
a(n) >= A002110(n) = prime(n)#.
Many terms are primorial numbers, see A360011.

Examples

			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.

Links

Crossrefs

Cf. A001221, A002110, A005349, A113315, A360011.
Similar: A060319 (Fibonacci), A083002 (oblong), A359961 (Zuckerman).

Programs

  • 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

Extensions

a(8)-a(9) from Michel Marcus, Jan 20 2023
a(10)-a(19) from Daniel Suteu, Jan 22 2023

A360301 Smallest exclusionary square (A029783) with exactly n distinct prime factors.

Original entry on oeis.org

2, 18, 84, 858, 31122, 3383898, 188841114, 68588585868, 440400004044, 7722272777722272
Offset: 1

Views

Author

Bernard Schott, Feb 02 2023

Keywords

Comments

There is no 5 in the prime factorization of these terms.
No other terms less than 10^14. - Michael S. Branicky, Feb 02 2023
1.69 * 10^15 < a(10) <= 7722272777722272. - Daniel Suteu, Feb 05 2023

Examples

			84 = 2^2 * 3 * 7 is the smallest integer with 3 distinct prime factors that is also an exclusionary square, because 84^2 = 7056, so a(3) = 84.
858 = 2 * 3 * 11 * 13 is the smallest integer with 4 distinct prime factors that is also an exclusionary square, because 858^2 = 736164, so a(4) = 858.

References

  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Links

Crossrefs

Cf. A029783.
Similar: A060319 (Fibonacci), A083002 (oblong), A359960 (Niven), A359961 (Zuckerman).

Programs

  • PARI
    omega_exclusionary_squares(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q == 5, next); my(v=m*q); while(v <= B, if(j==1, if(v>=A && #setintersect(Set(digits(v)), Set(digits(v^2))) == 0, listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_exclusionary_squares(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 05 2023

Formula

Assuming a(n) exists, a(n) >= A002110(n+1)/5 >> exp((1 + o(1)) * n * log(n)). (The inequality is presumably strict for all n; for n > 34 it seems that all A002110(n) are pandigital.) - Charles R Greathouse IV, Feb 05 2023

Extensions

a(4)-a(7) from Amiram Eldar, Feb 02 2023
a(8)-a(9) from Michael S. Branicky, Feb 02 2023
a(10) from Michael S. Branicky, Feb 07 2023
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