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 1-4 of 4 results.

A190220 Numbers all of whose divisors are numbers whose decimal digits are in nonincreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 31, 33, 40, 41, 43, 44, 53, 55, 61, 62, 63, 66, 71, 73, 77, 82, 83, 86, 88, 93, 97, 99, 110, 211, 220, 311, 331, 421, 422, 431, 433, 440, 443, 511, 521, 541, 622, 631, 633, 641, 643, 653, 661, 662, 733, 743, 751
Offset: 1

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Author

Jaroslav Krizek, May 06 2011

Keywords

Comments

Subset of A009996. Superset of A028867, A190219 and A190217.

Examples

			Number 110 is in sequence because all divisors of 110 (1, 2, 5, 10, 11, 22, 55, 110) are numbers whose decimal digits are in nonincreasing order.

Links

Programs

  • Maple
    with(numtheory): A190220 := proc(n) option remember: local d, dd, i, j, k, m, poten: if(n=1)then return 1: fi: for k from procname(n-1)+1 do d:=divisors(k): poten:=1: for i from 1 to nops(d) do m:=-1: dd:=convert(d[i], base, 10): for j from 1 to nops(dd) do if(m<=dd[j])then m:=dd[j]: else poten:=0: break: fi: od: if(poten=0)then break:fi: od: if(poten=1)then return k: fi: od: end: seq(A190220(n), n=1..64); # Nathaniel Johnston, May 14 2011

A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

Original entry on oeis.org

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

Views

Author

Bernard Schott, Oct 29 2022

Keywords

Comments

As A009995 is finite with 1023 terms, a(n) is bounded with a(n) <= 1022 and not 1023, since A009995(1) = 0.

Examples

			22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly decreasing order: {11, 22}, hence a(22) = 4-2 = 2.
52 has 6 divisors {1, 2, 4, 13, 26, 52} of which four have decimal digits that are in strictly decreasing order {1, 2, 4, 52}, hence a(52) = 4.

Links

Crossrefs

Cf. A009995, A190219, A358100, A358101.
Similar: A086971 (semiprimes), A087990 (palindromic), A355593 (alternating), A357171 (increasing order).

Programs

  • Maple
    f:= proc(n) local L;
   if n < 10 then return true fi;
   L:= convert(n,base,10);
   andmap(type,L[2..-1]-L[1..-2],positive)
end proc:
g:= n -> nops(select(f,numtheory:-divisors(n))):
map(g, [$1..100]); # Robert Israel, Oct 31 2022
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* Amiram Eldar, Oct 29 2022 *)
  • PARI
    a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, ,12) == dd); \\ Michel Marcus, Oct 30 2022
  • Python
    from sympy import divisors
def c(n): s = str(n); return all(s[i+1] < s[i] for i in range(len(s)-1))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Feb 12 2024

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..1023} 1/A009995(n) = 3.89840673699905364734... (this is a rational number whose numerator and denominator have 1292 and 1291 digits, respectively). - Amiram Eldar, Jan 06 2024

A358100 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly decreasing order.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 80, 60, 252, 120, 240, 540, 360, 630, 420, 960, 1440, 840, 1260, 2880, 3360, 4320, 2520, 6720, 5040, 8640, 10080, 15120, 50400, 20160, 40320, 30240, 171360, 90720, 383040, 60480, 120960, 181440, 362880, 544320, 937440, 786240, 2056320
Offset: 1

Views

Author

Bernard Schott, Nov 01 2022

Keywords

Comments

This sequence is finite since A009995 is finite with 1022 nonzero terms, hence the last term is a(1022) = lcm of the 1022 positive terms of A009995.

Examples

			For n=7, the divisors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} of which 7 have their decimal digits in strictly decreasing order (all except 15). No integer < 30 has 7 such divisors, so a(7) = 30.

Crossrefs

Cf. A009995, A190219, A358099, A358101.
Similar: A087997 (palindromic), A355303 (undulating), A357172 (increasing order).

Programs

  • Mathematica
    s[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[45, 3*10^6] (* Amiram Eldar, Nov 01 2022 *)
  • PARI
    f(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, , 12) == dd); \\ A358099
a(n) = my(k=1); while(f(k)!=n, k++); k; \\ Michel Marcus, Nov 01 2022

Extensions

More terms from Amiram Eldar, Nov 01 2022

A358101 Positions of records in A358099, i.e., integers whose number of divisors whose decimal digits are in strictly decreasing order sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 60, 120, 240, 360, 420, 840, 1260, 2520, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 181440, 362880, 544320, 786240, 1572480, 1874880, 3749760, 5624640, 7862400, 14938560, 23587200, 24373440, 31872960, 63745920, 95618880
Offset: 1

Views

Author

Bernard Schott, Nov 03 2022

Keywords

Comments

As A009995 is finite, this sequence is necessarily finite.
Corresponding records are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, ...

Examples

			a(9) = 60 is in the sequence because A358099(60) = 10 is larger than any earlier value in A358099.

Crossrefs

Cf. A009995, A190219, A358099, A358100.
Similar sequences: A093036, A340548, A357173.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; fm = 0; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Nov 03 2022 *)

Extensions

More terms from Amiram Eldar, Nov 03 2022
Showing 1-4 of 4 results.

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