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.
Table of values at powers of 2, from _Rémy Sigrist_, Apr 25 2020: k a(2^k) -- ------ 0 1 1 2 2 6 3 13 4 27 5 51 6 119 7 248 8 535 9 1311 10 2994 11 6838 12 15945 13 36740 14 83716 15 186184 16 427070 17 971213 18 2203026 19 4964477
a(1) = 1 as '1' || '2' = '12', which is divisible by 3 (where || denotes decimal concatenation). a(2) = 80: the concatenation 2 || 3 || ... || 82 is 23456789101112131415161718192021222324252627282930313233343536373839\ 40414243444546474849505152535455565758596061626364656667686970717273747\ 576777879808182, which is divisible by 83. a(7) = 13 as '7' || '8' || '9' || '10' || '11' || '12' || ... || '20' = 7891011121314151617181920, which is divisible by 21. a(8) = 2 as '8' || '9' || '10' = 8910, which is divisible by 11.
grow := proc(n,M) # searches out to a limit of M, returns [n,n+k] or [n,-1] if no k was found
local R,i;
R:=n;
for i from n+1 to M do
R:=R*10^length(i)+i;
if (i mod 2) = 0 then
if (R mod (i+1)) = 0 then return([n, i]); fi;
fi;
od:
[n, -1];
end;
for n from 1 to 100 do lprint(grow(n,20000)); od;
apply( {a(n,L=10^logint(n*10,10),c=n)= n%2||c=c*L+n+1; for(k=n+++n%2,oo, kM. F. Hasler, Feb 20 2020
f:= proc(n) local k,p;
p:= n;
for k from 1 do
p:= p*(n+k);
if (p/(n+k+1))::integer then return k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020
a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 04 2020, after Maple *)
a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } \\ Jinyuan Wang, Feb 25 2020
\\ See Corneth link
def a(n):
k, p = 1, n*(n+1)
while p%(n+k+1): k += 1; p *= (n+k)
return k
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 06 2021
f:= proc(n) local k,p;
p:= n;
for k from 1 do
p:= p*(n+k);
if (p/(n+k+1))::integer then return n+k+1 fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020
a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[n+k+1]]]];
Array[a, 100] (* Jean-François Alcover, Jul 18 2020, after Maple *)
def a(n):
k, p = 1, n*(n+1)
while p%(n+k+1): k += 1; p *= (n+k)
return n + k + 1
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jun 06 2021
a(1) = 2 as '1' || '2' = '12', which is divisible by 3 (where || denotes decimal concatenation). a(7) = 20 as '7' || '8' || '9' || '10' || '11' || '12' || ... || '20' = 7891011121314151617181920, which is divisible by 21. a(8) = 10 as '8' || '9' || '10' = 8910, which is divisible by 11. a(2) = 82: the concatenation 2 || 3 || ... || 82 is 23456789101112131415161718192021222324252627282930313233343536373839\ 40414243444546474849505152535455565758596061626364656667686970717273747\ 576777879808182, which is divisible by 83.
grow := proc(n,M) # searches out to a limit of M, returns [n,n+k] or [n,-1] if no k was found local R,i; R:=n; for i from n+1 to M do R:=R*10^length(i)+i; if (i mod 2) = 0 then if (R mod (i+1)) = 0 then return([n, i]); fi; fi; od: [n, -1]; end; for n from 1 to 100 do lprint(grow(n,20000)); od;
apply( {A332584(n,L=10^#Str(n),c=n)= until((c=c*L+n)%(n+1)==0, n++M. F. Hasler, Feb 20 2020
def A332584(n): r, m = n, n + 1 while True: r = r*10**(len(str(m))) + m if m % 2 == 0 and r % (m+1) == 0: return m m += 1 # Chai Wah Wu, Jun 12 2020
A332558(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } a(n) = {my(s=A332558(n)+n); s!/(n-1)!/(s+1); } \\ Jinyuan Wang, Feb 25 2020
a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[ p, n+k+1], Return[FactorInteger[n+k+1][[-1, 1]]]]]];
Array[a, 1000] (* Jean-François Alcover, Aug 17 2020 *)
a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return((n+k)!/(n-1)!))); } \\ Jinyuan Wang, Feb 25 2020
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