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.

A217973 Niven (or Harshad) numbers not containing the digit 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84, 111, 112, 114, 117, 126, 132, 133, 135, 144, 152, 153, 156, 162, 171, 192, 195, 198, 216, 222, 224, 225, 228, 234, 243, 247, 252, 261, 264, 266, 285, 288, 312, 315, 322, 324, 333, 336
Offset: 1

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Author

Charles R Greathouse IV, Oct 16 2012

Keywords

Comments

Andreescu & Andrica prove that this sequence is infinite.
For each positive integer n, there exists a n-digit Niven (or Harshad) number not containing the digit 0 (see A348318 for more explanations and links). - Bernard Schott, Oct 20 2021

References

  • Titu Andreescu and Dorin Andrica, Number Theory, Structures, Examples, and Problems, Problem 5.2.3 on pages 109-110.

Links

Crossrefs

Intersection of A005349 and A052382.
A216405 is a subsequence.
Cf. A348150, A348316, A348317, A348318.

Programs

  • Maple
    filter:= proc(n) local L;
L:= convert(n,base,10);
not has(L,0) and n mod convert(L,`+`) = 0
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 01 2016
  • Mathematica
    Select[Range[400], IntegerQ[ #/(Plus @@ IntegerDigits[#])] && DigitCount[#, 10, 0] == 0 &]  (* Alonso del Arte, Oct 16 2012 *)
  • PARI
    is(n)=vecsort(digits(n))[1]&&n%sumdigits(n)==0
  • Python
    def ok(n): s = str(n); return '0' not in s and n%sum(map(int, s)) == 0
print([k for k in range(337) if ok(k)]) # Michael S. Branicky, Oct 20 2021

A348316 a(n) is the largest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

Original entry on oeis.org

9, 84, 999, 9963, 99972, 999984, 9999966, 99999966, 999999999, 9999999828, 99999999898, 999999999853, 9999999999936, 99999999999783, 999999999999984, 9999999999999858, 99999999999999939, 999999999999999831, 9999999999999999951, 99999999999999999922, 999999999999999999687
Offset: 1

Views

Author

Bernard Schott, Oct 11 2021

Keywords

Comments

This sequence is inspired by a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition.
The problem asked for a proof that, for each positive integer n, there exists a n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Kalva in English).
This sequence lists the largest such n-digit integer.

Examples

			9963 has 4 digits, does not contain 0 and is divisible by 9+9+6+3 = 27 (9963 = 27*369), while there is no integer k with 9964 <= k <= 9999 that is divisible by sum of its digits, hence a(4) = 9963.

Links

Crossrefs

Cf. A002283, A005349, A014950, A217973, A348150, A348317, A348318.

Programs

  • Mathematica
    hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{k = 10^n}, While[! hQ[k], k--]; k]; Array[a, 20] (* Amiram Eldar, Oct 11 2021 *)
  • Python
    def a(n):
    s, k = "9"*n, int("9"*n)
    while '0' in s or k%sum(map(int, s)): k -= 1; s = str(k)
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Oct 11 2021

Formula

a(n) = A002283(n) = 10^n - 1 iff n is in A014950 (compare with A348150 formula).

Extensions

More terms from Amiram Eldar, Oct 11 2021

A348318 Number of n-digit Niven (or Harshad) numbers not containing the digit 0.

Original entry on oeis.org

9, 14, 108, 710, 4978, 35724, 273032, 2097356, 16674554, 135091242, 1112325268, 9296413365, 77991481271, 654495034497, 5420117473932
Offset: 1

Views

Author

Bernard Schott, Oct 17 2021

Keywords

Comments

This sequence comes from a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition. This problem asked for a proof that, for each positive integer n, there exists an n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Scholes in English).
This sequence gives the number of such n-digit Niven numbers not containing the digit 0.

Examples

			a(2) = 14 because there are 14 integers {12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84} in the set of Niven numbers with 2 digits and not containing the digit 0.

Links

Crossrefs

Cf. A005349, A009994, A140866, A217973, A348150, A348316, A348317.

Programs

  • Mathematica
    q[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{c = 0, k = (10^n - 1)/9}, While[k < 10^n, If[q[k], c++]; k++]; c]; Array[a, 6] (* Amiram Eldar, Oct 17 2021 *)
  • Python
    from itertools import product
def a(n): return sum(1 for p in product("123456789", repeat=n) if int("".join(p))%sum(map(int, p)) == 0)
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Oct 17 2021

Extensions

a(6)-a(10) from Amiram Eldar, Oct 17 2021
a(11)-a(15) from Martin Ehrenstein, Oct 22 2021
Showing 1-3 of 3 results.

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