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.

A332268 a(n) is the number of divisors of n that are Niven numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 4, 2, 1, 8, 2, 2, 4, 4, 1, 7, 1, 4, 2, 2, 3, 9, 1, 2, 2, 8, 1, 7, 1, 3, 5, 2, 1, 9, 2, 5, 2, 3, 1, 8, 2, 5, 2, 2, 1, 11, 1, 2, 6, 4, 2, 4, 1, 3, 2, 6, 1, 12, 1, 2, 3, 3, 2, 4, 1, 9, 5, 2, 1, 10, 2, 2
Offset: 1

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Author

Marius A. Burtea, May 04 2020

Keywords

Comments

If p is a prime number, p >= 11, then a(p) = 1.
Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 36, 40, 54, 63, 72, 81, 108, 162, 216, 243, 324, 486, 648, 972, 1944, have all divisors Niven numbers. There are only finitely many numbers all of whose divisors are Niven numbers. (A337741).
A333456(n) is the least number k such that a(k) = n. - Bernard Schott, Jul 30 2022

Examples

			For n = 4 the divisors are 1, 2, 4 and they are all Niven numbers, so a(4) = 3.
For n = 14 the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Niven numbers, so a(14) = 3.

Links

Crossrefs

Cf. A000005, A005349, A333456, A337741, A340637.

Programs

  • Magma
    [#[d:d in Divisors(k)|d mod &+Intseq(d) eq 0]:k  in [1..100]];
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; Array[a, 100] (* Amiram Eldar, May 04 2020 *)
  • PARI
    a(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ Michel Marcus, May 04 2020

Formula

a(A333456(n)) = n. - Bernard Schott, Jul 30 2022

A340797 Integers whose number of divisors that are Brazilian sets a new record.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 120, 168, 240, 336, 360, 420, 672, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 43680, 45360, 50400, 55440, 65520, 83160, 98280, 110880, 131040, 166320, 196560, 221760, 262080, 277200, 327600
Offset: 1

Views

Author

Bernard Schott, Jan 24 2021

Keywords

Comments

Corresponding number of Brazilian divisors: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 15, 17, 18, 19, 26, ...
Observation: the 58 consecutive highly composite numbers from A002182(12) = 240 to A002182(69) = 2095133040 (maybe more, according to conjectured terms) are also terms of this sequence.

Examples

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 4 are Brazilian: {8, 10, 20, 40}. No positive integer smaller than 40 has as many as four Brazilian divisors; hence 40 is a term.

Links

Crossrefs

Cf. A002182, A125134, A220570, A308851, A340795, A340796.
Similar with: A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A340637 (Niven), A340638 (Zuckerman).

Programs

  • Mathematica
    brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; dm = -1; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 1000}]; s (* Amiram Eldar, Jan 24 2021 *)
  • PARI
    isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
nbd(n) = sumdiv(n, d, isb(d)); \\ A340795
lista(nn) = {my(m=-1); for (n=1, nn, my(x = nbd(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 24 2021

Extensions

a(20)-a(36) from Michel Marcus, Jan 24 2021
a(37)-a(44) from Amiram Eldar, Jan 24 2021

A340638 Integers whose number of divisors that are Zuckerman numbers sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 24192, 48384, 88704, 120960, 241920, 266112, 532224, 1064448, 1862784, 2661120, 3725568, 5322240, 7451136, 10450944, 19160064, 20901888, 28740096, 38320128, 57480192, 99283968, 114960384
Offset: 1

Views

Author

Bernard Schott, Jan 14 2021

Keywords

Comments

A Zuckerman number is a number that is divisible by the product of its digits (A007602).
The terms in this sequence are not necessarily Zuckerman numbers. For example a(7) = 72 has product of digits = 14 and 72/14 = 36/7 = 5.142...
The first seven terms are the first seven terms of A087997, then A087997(8) = 66 while a(8) = 144.

Examples

			The 8 divisors of 24 are all Zuckerman numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Zuckerman numbers, hence 24 is a term.

Links

Crossrefs

Cf. A007602, A335037, A337941.
Subsequence of A335038.
Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Niven numbers (A340637).

Programs

  • Mathematica
    zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; s[n_] := DivisorSum[n, 1 &, zuckQ[#] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 14 2021 *)
  • PARI
    isokz(n) = iferr(!(n % vecprod(digits(n))), E, 0); \\ A007602
lista(nn) = {my(m=0); for (n=1, nn, my(x = sumdiv(n, d, isokz(d));); if (x > m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 15 2021

Extensions

More terms from David A. Corneth and Amiram Eldar, Jan 15 2021
Showing 1-3 of 3 results.

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