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.
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).
k | divisors | concatenation ---+----------------+-------------- 4 | (1) 2 (4) | 2 6 | (1) 2, 3 (6) | 23 9 | (1) 3 (9) | 3 21 | (1) 3, 7 (21) | 37 22 | (1) 2, 11 (22) | 211 25 | (1) 5 (25) | 5 33 | (1) 3, 11 (33) | 311 39 | (1) 3, 13 (39) | 313
with(numtheory): for k from 2 to 1000 do: v0:=divisors(k): nn:=nops(v0): if nn > 2 then v1:=[seq(v0[j],j=2..nn-1)]: v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))): v3:=parse(v2): if isprime(v3) = true then lprint(k,v3) fi: fi: od: # Simon Plouffe
fQ[n_] := PrimeQ@ FromDigits@ Most@ Rest@ Divisors@ n; Select[ Range[2, 320], fQ]
from sympy import divisors, isprime
def ok(n):
s = "".join(str(d) for d in divisors(n)[1:-1])
return s != "" and isprime(int(s))
print([k for k in range(319) if ok(k)]) # Michael S. Branicky, Oct 01 2024
Number 214 is in sequence because smallest digit of all divisors of 214 (1, 2, 107, 214) is 0.
Select[Range[400],Min[Flatten[IntegerDigits/@Divisors[#]]]==0&] (* Harvey P. Dale, Dec 03 2021 *)
18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 2.
In the same way all even numbers have the divisor 2 and thus are in this sequence; numbers N in { 20,...,29, 120,...,129, 200,...,299 } have the digit 2 in N which is divisor of itself. - _M. F. Hasler_, Apr 22 2015
[n: n in [1..1000] | [2] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))]
Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 2] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)
is(n)=!bittest(n,0)||setsearch(Set(digits(n)),2)||fordiv(n,d,setsearch(Set(digits(d)),2)&&return(1)) \\ M. F. Hasler, Apr 22 2015
18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 3.
[n: n in [1..1000] | [3] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];
Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 3] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)
is(n)=fordiv(n,d, if(setsearch(Set(digits(d)),3), return(1))); 0 \\ Charles R Greathouse IV, Apr 30 2015
16 is in sequence because the list of divisors of 16: (1, 2, 4, 8, 16) contains digit 4.
[n: n in [1..1000] | [4] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))]
Select[Range@ 141, Part[Plus @@ DigitCount@ Divisors@ #, 4] > 0 &] (* Michael De Vlieger, Apr 20 2015 *) Select[Range[200],Count[Flatten[IntegerDigits/@Divisors[#]],4]>0&] (* Harvey P. Dale, May 05 2022 *)
is(n)=fordiv(n,d, if(setsearch(Set(digits(d)),4), return(1))); 0 \\ Charles R Greathouse IV, Apr 30 2015
20 is in sequence because the list of divisors of 20: (1, 2, 4, 5, 10, 20) contains digit 5.
[n: n in [1..1000] | [5] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];
Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 5] > 0 &] Select[Range[200],Max[DigitCount[Divisors[#],10,5]]>0&] (* Harvey P. Dale, Sep 15 2018 *)
is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 5), return(1))); 0
use ntheory ":all"; for my $n (1..1000) { say $n if scalar(grep {/5/} divisors($n)) } # Dana Jacobsen, May 07 2015
use ntheory ":all"; my @a257222 = grep { scalar(grep {/5/} divisors($)) } 1..1000; # _Dana Jacobsen, May 07 2015
from sympy import divisors A257222_list = [n for n in range(1,10**3) if '5' in set().union(*(set(str(d)) for d in divisors(n,generator=True)))] # Chai Wah Wu, May 06 2015
a(1) = 1: d(1) = {1}.
a(2) = 12: d(12) = {1, 2, 3, 4, 6, 12}.
a(3) = 124: d(124) = {1, 2, 4, 31, 62, 124}.
a(4) = 135: d(135) = {1, 3, 5, 9, 15, 27, 45, 135}.
isok(k) = my(s=""); fordiv(k, d, s=concat(s, Str(d)); if (eval(s)==k, return(1)); if (eval(s)> k, return(0))); \\ Michel Marcus, Sep 22 2022
is(n, {u = 10^5}) = { my(oldu = u, s, d, fe); s = ""; u = min(u, n); fe = factor(n, u); d = divisors(fe); if(#fe~ > 0 && fe[#fe~, 1] > u, d = select(x -> x < fe[#fe~, 1], d); ); for(i = 1, #d, if(d[i] > u, return(is(n, 10*oldu)); ); s=concat(s, Str(d[i])); if(eval(s) == n, return(1)); if(eval(s) > n, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 12 2022, Nov 07 2022
from sympy import divisors
def ok(n):
target, s = str(n), ""
if target[0] != "1": return False
for d in divisors(n):
s += str(d)
if len(s) >= len(target): return s == target
elif not target.startswith(s): return False
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Sep 22 2022
a(6) = 21: the divisors of 21 are 1,3,7,21, and their concatenation 13721 is noncomposite.
Select[Range[10^3], ! CompositeQ@ FromDigits@ Flatten@ IntegerDigits@ Divisors@ # &] (* Michael De Vlieger, Sep 23 2016 *)
is(n)=my(d=divisors(n)); d[1]="1"; isprime(eval(concat(d))) || n==1 \\ Charles R Greathouse IV, Sep 23 2016
from sympy import divisors, isprime
def ok(m): return m==1 or isprime(int("".join(str(d) for d in divisors(m))))
print([m for m in range(1, 900) if ok(m)]) # Michael S. Branicky, Feb 05 2022
a(12) = 6 because digit 6 is largest digit of all divisors of 12: (1, 2, 3, 4, 6, 12).
Flatten[Table[Take[Sort[Flatten[IntegerDigits[Divisors[n]]]], -1], {n, 100}]] (* Alonso del Arte, Mar 23 2012 *)
a(n)=my(t);fordiv(n, d, t=max(t, vecmax(eval(Vec(Str(d))))); if(t>8, return(t)));t \\Charles R Greathouse IV, Mar 20 2012
from sympy import divisors
def a(n): return int(max("".join(map(str, divisors(n)))))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 22 2021
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