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.

Previous Showing 11-17 of 17 results.

A256824 Reverse concatenation of distinct digits of all divisors of n in base 10.

Original entry on oeis.org

1, 21, 31, 421, 51, 6321, 71, 8421, 931, 5210, 1, 64321, 31, 7421, 531, 86421, 71, 986321, 91, 54210, 7321, 21, 321, 864321, 521, 6321, 97321, 87421, 921, 653210, 31, 864321, 31, 74321, 7531, 9864321, 731, 98321, 931, 854210, 41, 764321, 431, 421, 95431, 64321
Offset: 1

Views

Author

Jaroslav Krizek, Apr 10 2015

Keywords

Comments

Concatenation of elements of set of all digits of all divisors of n in decreasing order in base 10.
There are precisely 512 distinct terms of this sequence - see A256825 (possible values of a(n) in increasing order).
Minimal term is 1, maximal term is 9876543210.
Numbers n such that a(n) = 1 are in A243534, numbers n such that a(n) = 9876543210 are in A095050.
See A256826 - the smallest numbers k such that a(k) = A256825(n).

Examples

			For n = 12; list of divisors of 12 in base 10: 1, 2, 3, 4, 6, 12 contains five distinct digits (1, 2, 3, 4, 6) whose reverse concatenation is 64321.

Links

Crossrefs

Cf. A009995, A095050, A243534, A256825, A256826.

Programs

  • Magma
    [Seqint(Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))): n in [1..100]];
  • Mathematica
    Table[FromDigits[Reverse[Union[Flatten[IntegerDigits[Divisors[n]]]]]],{n,100}] (* Ivan N. Ianakiev, Apr 14 2015 *)
  • PARI
    a(n) = {my(v = []); fordiv(n, d, v = vecsort(concat(v, digits(d)),,8)); subst(Polrev(v), x, 10);} \\ Michel Marcus, Apr 11 2015

A243543 Smallest number whose list of divisors contains n distinct digits (in base 10).

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 72, 54, 108
Offset: 1

Views

Author

Jaroslav Krizek, Jun 19 2014

Keywords

Comments

Finite sequence with 10 terms.

Examples

			a(9) = 54 because 54 is the smallest number whose list of divisors contains 9 distinct digits; the list of divisors of 54: (1, 2, 3, 6, 9, 18, 27, 54) contains 9 distinct digits (1, 2, 3, 4, 5, 6, 7, 8, 9).

Crossrefs

Cf. Sequences of numbers n such that list of divisors of n contains k distinct digits: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050.
Cf. also A095048, A037278, A176558, A243360.

A059436 Smallest number whose set of divisors contains each digit 0-9 at least n times.

Original entry on oeis.org

108, 540, 1140, 1890, 3420, 5460, 7980, 11760, 16380, 23520, 23520, 23760, 38220, 38220, 41580, 41580, 41580, 71820, 71820, 71820, 83160, 83160, 83160, 124740, 124740, 143640, 166320, 166320, 249480, 249480, 249480, 249480, 311220, 335160, 415800, 415800, 415800, 415800, 415800, 415800
Offset: 1

Views

Author

Erich Friedman, Feb 01 2001

Keywords

Examples

			The divisors of 540 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540 and every digit appears at least twice in this list.

Links

Crossrefs

Cf. A027750; subsequence of A095050, A095048.

Programs

  • Haskell
    import Data.List (group, sort)
a059436 n = head [x | x <- [1..],
   let dds = map length $ group $ sort $ concatMap show $ a027750_row x,
   minimum dds == n, length dds == 10]
-- Reinhard Zumkeller, Feb 04 2012
  • Mathematica
    T = 0*Range[25]; Do[d = Last /@ Tally@ Flatten[ IntegerDigits /@ Divisors@ n]; If[Length@d == 10, m = Min[25, d]; While[m > 0 && (T[[m]] == 0 || n < T[[m]]), T[[m--]] = n]], {n, 125000}]; T (* Giovanni Resta, May 15 2016 *)
sded[n_]:=With[{fid=Flatten[IntegerDigits/@Divisors[n]]},If[Length[Union[fid]]==10,{n,Min[ Tally[fid][[;;,2]]]},Nothing]]; Table[SelectFirst[sded/@Range[500000],#[[2]]>k&],{k,0,39}][[;;,1]] (* Harvey P. Dale, Mar 27 2024 *)

Extensions

More terms from David W. Wilson, Aug 31 2001
Offset corrected by R. J. Mathar, Jun 02 2010
a(10)-a(36) corrected by Giovanni Resta, May 15 2016
More terms from Harvey P. Dale, Mar 27 2024

A256825 Numbers with digits in strictly decreasing order containing digit 1.

Original entry on oeis.org

1, 10, 21, 31, 41, 51, 61, 71, 81, 91, 210, 310, 321, 410, 421, 431, 510, 521, 531, 541, 610, 621, 631, 641, 651, 710, 721, 731, 741, 751, 761, 810, 821, 831, 841, 851, 861, 871, 910, 921, 931, 941, 951, 961, 971, 981, 3210, 4210, 4310, 4321, 5210, 5310, 5321
Offset: 1

Views

Author

Jaroslav Krizek, Apr 10 2015

Keywords

Comments

a(n) = possible values of A256824(m) in increasing order where A256824(m) = reverse concatenation of distinct digits of all divisors of m in base 10.
There are precisely 512 terms. Maximal term is 9876543210.
Subsequence of A009995 (numbers with digits in strictly decreasing order).
See A256826 - the smallest number k such that A256824(k) = a(n).

Examples

			21 is in sequence because there are numbers m such that A256824(m) = 21 (for m = 2, 22, 121, 211, 2111, ...).

Links

Crossrefs

Cf. A009995, A095050, A243534, A256824, A256826.

Programs

  • Excel
    [Row n = 1 … 1023; Column A: A(n) = A009995(n); Column B: B(n) =  =IF(FIND("1";A(n);1)>0;A(n)); Arrangement of column B]
  • PARI
    lista(nn) = for (n=1, nn, if ((d = digits(n)) && ((sd=vecsort(d,,8))==Vecrev(d)) && vecsearch(sd, 1), print1(n, ", "))); \\ Michel Marcus, Apr 11 2015

A256826 a(n) = the smallest number k such that A256824(k) = A256825(n).

Original entry on oeis.org

1, 101, 2, 3, 41, 5, 61, 7, 181, 19, 202, 103, 23, 401, 4, 43, 505, 25, 15, 451, 601, 122, 163, 461, 1661, 107, 127, 37, 47, 157, 67, 1801, 281, 83, 1481, 5581, 1861, 187, 109, 29, 9, 149, 59, 619, 79, 89, 2003, 404, 403, 123, 10, 503, 115, 4051, 12451, 453
Offset: 1

Views

Author

Jaroslav Krizek, Apr 13 2015

Keywords

Comments

A256824(n) = reverse concatenation of distinct digits of all divisors of n in base 10, A256825(n) = possible values of A256824(m) in increasing order.
Finite sequence with 512 terms. Maximal term is a(185) = 88511.

Examples

			a(11) = 202 because 202 is the smallest number k such that reverse concatenation of distinct digits of all divisors of k (i.e. 1, 2, 101, 202) in base 10 = A256824(k) = A256824(202) = A256825(11) = 210.

Links

Crossrefs

Cf. A009995, A095050, A243534, A256824, A256825.

Programs

A345390 Numbers whose set of divisors contains every digit at least twice.

Original entry on oeis.org

540, 720, 760, 810, 918, 1080, 1140, 1170, 1260, 1404, 1440, 1512, 1520, 1530, 1560, 1620, 1740, 1800, 1820, 1824, 1836, 1872, 1890, 1908, 1960, 2016, 2028, 2052, 2070, 2072, 2088, 2106, 2112, 2124, 2142, 2156, 2160, 2184, 2208, 2280, 2340, 2380, 2430, 2508, 2520
Offset: 1

Views

Author

Tanya Khovanova, Jun 17 2021

Keywords

Comments

Compare to A095050: Numbers such that all ten digits are needed to write all positive divisors in decimal representation.

Examples

			The divisors of 918 are 1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, and 918. Every digit appears at least twice. Thus, 918 is in this sequence.

Links

Crossrefs

Cf. A059436, A095050.

Programs

  • Maple
    q:= n-> (p-> is(min(seq(coeff(p, x, j), j=0..9))>1))(add(x^i, i=
     map(d-> convert(d, base, 10)[], [numtheory[divisors](n)[]]))):
select(q, [$10..2600])[];  # Alois P. Heinz, Apr 21 2022
  • Mathematica
    Select[Range[3000], Length[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] == 10 && Min[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] > 1 &]
  • Python
    from sympy import divisors
def ok(n):
    digits_used = {d:0 for d in "0123456789"}
    for div in divisors(n, generator=True):
        for d in str(div): digits_used[d] += 1
        if all(digits_used[d] > 1 for d in "0123456789"): return True
    return False
print([k for k in range(2521) if ok(k)]) # Michael S. Branicky, Jun 25 2022

A353066 Numbers whose set of divisors contains every digit at least three times.

Original entry on oeis.org

1140, 1890, 2280, 2340, 2610, 2660, 2700, 2808, 2880, 2940, 2970, 3420, 3480, 3510, 3540, 3600, 3654, 3672, 3780, 3870, 3920, 3952, 3990, 4032, 4140, 4320, 4368, 4380, 4410, 4560, 4590, 4680, 4740, 4760, 4770, 4860, 4896, 4940, 4950, 5130, 5220, 5320, 5400, 5454
Offset: 1

Views

Author

Tanya Khovanova, Apr 21 2022

Keywords

Comments

Every multiple of a term is also a term.

Examples

			The divisors of 1140 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 19, 20, 30, 38, 57, 60, 76, 95, 114, 190, 228, 285, 380, 570, 1140. Digits tally from 0 to 9: 8, 10, 6, 4, 3, 6, 3, 3, 4, 3. The minimum is 3, thus, 1140 is in this sequence.

Links

Crossrefs

Cf. A059436 (at least n times).
Subsequence of A095050 (at least once) and of A345390 (at least twice).

Programs

  • Maple
    q:= n-> (p-> is(min(seq(coeff(p, x, j), j=0..9))>2))(add(x^i, i=
     map(d-> convert(d, base, 10)[], [numtheory[divisors](n)[]]))):
select(q, [$10..5555])[];  # Alois P. Heinz, Apr 21 2022
  • Mathematica
    Select[Range[10000], Length[Tally[Flatten[IntegerDigits[Divisors[#]]]]] == 10 && Min[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] >= 3 &]
  • PARI
    upto(n) = { my(v = vector(n, i, -1)); for(i = 1, n, if(v[i] == -1, c = is(i); if(c == 1, v[i] = 1; for(j = 1, n\i, v[i*j] = 1; ) , v[i] = 0 ) ) ); Vec(select(x->x==1,v,1)) }
is(n) = { my(v = vector(10, i, 3), d = divisors(n), todo = 30, i, j); for(i = 1, #d, dd = digits(d[i]); for(j = 1, #dd, if(v[dd[j]+1] > 0, v[dd[j]+1]--; todo--; if(todo <= 0, return(1) ) ) ) ); 0 } \\ David A. Corneth, Jul 11 2022
  • Python
    from sympy import divisors
def ok(n):
    counts = [0]*10
    for d in divisors(n, generator=True):
        for di in str(d): counts[int(di)] += 1
        if min(counts) > 2: return True
    return False
print([k for k in range(5455) if ok(k)]) # Michael S. Branicky, Apr 21 2022
Previous Showing 11-17 of 17 results.

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