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.

Showing 1-3 of 3 results.

A359961 Smallest Zuckerman number (A007602) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 132, 3276, 27132, 1117116, 111914712, 6111417312, 1113117121116, 1112712811322112, 11171121131111172
Offset: 0

Views

Author

Bernard Schott, Jan 21 2023

Keywords

Examples

			3276 = 2^2*3^2*7*13 is the smallest integer with 4 distinct prime factors that is also Zuckerman number as 3276 / (3*2*7*6) = 13, so a(4) = 3276.

Links

Crossrefs

Cf. A007602, A288069.
Similar: A060319 (Fibonacci), A083002 (oblong), A359960 (Niven).

Programs

  • PARI
    a(n) = my(k=1); while (!(p=vecprod(digits(k))) || (k % p) || (omega(k) != n), k++); k; \\ Michel Marcus, Jan 21 2023

Extensions

a(6)-a(7) from Michel Marcus, Jan 21 2023
a(8)-a(9) from Daniel Suteu, Jan 21 2023
a(10)-a(11) from Bert Dobbelaere, Jan 29 2023

A360011 Integers k such that the product of the first k primes is a Niven number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 16, 18, 19, 21, 22, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 66, 69, 70, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Michel Marcus, Jan 21 2023

Keywords

Comments

Integers k such that A002110(k) belongs to A005349.
So, the sequence A002110(a(n)) is a subsequence of A359960. - Bernard Schott, Jan 21 2023

Examples

			A002110(5) = 2310 and 2310 is divisible by 2+3+1+0=6, so 5 is a term.

Crossrefs

Cf. A002110, A005349, A359960.

Programs

  • Mathematica
    a={}; For[k=0, k<=100, k++, p=Product[Prime[i],{i,k}]; If[Mod[p,Total[IntegerDigits[p]]]==0, AppendTo[a,k]]]; a (* Stefano Spezia, Jan 21 2023 *)
  • PARI
    isok(k) = my(p=factorback(primes(k))); !(p % sumdigits(p));

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
Showing 1-3 of 3 results.

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