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.
Smallest prime formed from 20 is 1201, by placing 1 on both sides. Smallest prime formed from 33 is 233, by placing a 2 in front.
a068164 n = head (filter isPrime (digitExtensions n)) digitExtensions n = filter (includes n) [0..] includes n k = listIncludes (show n) (show k) listIncludes [] _ = True listIncludes (h:_) [] = False listIncludes l1@(h1:t1) (h2:t2) = if (h1 == h2) then (listIncludes t1 t2) else (listIncludes l1 t2) isPrime 1 = False isPrime n = not (hasDivisorAtLeast 2 n) hasDivisorAtLeast k n = (k*k <= n) && (((n `rem` k) == 0) || (hasDivisorAtLeast (k+1) n))
A068164 := proc(n) local p,pdigs,plen,dmas,dmasdigs,i,j; # test all primes ascending p := 2; while true do pdigs := convert(p,base,10) ; plen := nops(pdigs) ; # binary digit mask over p for dmas from 2^plen-1 to 0 by -1 do dmasdigs := convert(dmas,base,2) ; pdel := [] ; for i from 1 to nops(dmasdigs) do if op(i,dmasdigs) = 1 then pdel := [op(pdel),op(i,pdigs)] ; end if; end do: if n = add(op(j,pdel)*10^(j-1),j=1..nops(pdel)) then return p; end if; end do: p := nextprime(p) ; end do: end proc: seq(A068164(n),n=1..120) ; # R. J. Mathar, Nov 13 2014
from sympy import sieve
def dmo(n, t):
if t < n: return False
while n and t:
if n%10 == t%10:
n //= 10
t //= 10
return n == 0
def a(n):
return next(p for p in sieve if dmo(n, p))
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jan 21 2023
E.g. 34^2 = 1156 is first square that contains the string "11".
With[{e = 2}, Table[Function[p, k = 1; While[Length@ SequencePosition[ IntegerDigits[Set[c, k^e]], p] == 0, k++]; c]@ IntegerDigits@ Prime@ n, {n, 57}] ^(1/e)] (* Michael De Vlieger, May 04 2017, Version 10.1 *)
Table[ i=0; While[ StringPosition[ ToString[ i^2 ], ToString[ n ] ]=={}, i++ ]; i, {n, 0, 80} ]
IsSubList:= proc(T, S)
local i;
if T = [] then return true fi;
if S = [] then return false fi;
i:= ListTools:-Search(T[1],S);
if i = 0 then false else procname(T[2..-1],S[i+1..-1]) fi
end proc:
f:= proc(n) local T,S,k,v;
T:= convert(n,base,10);
for k from 1 do
v:= combinat:-fibonacci(k);
S:= convert(v,base,10);
if IsSubList(T,S) then return v fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 10 2020
a[n_] := Block[{p = RegularExpression[ StringJoin @@ Riffle[ ToString /@ IntegerDigits[ n], ".*"]], f, k=2}, While[! StringContainsQ[ ToString[f = Fibonacci[ k++]], p]]; f]; Array[a, 42] (* Giovanni Resta, Mar 10 2020 *)
def dmo(n, t):
if t < n: return False
while n and t:
if n%10 == t%10:
n //= 10
t //= 10
return n == 0
def fibo(f=1, g=2):
while True: yield f; f, g = g, f+g
def a(n):
return next(f for f in fibo() if dmo(n, f))
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jan 21 2023
from math import isqrt
from itertools import count
def dmo(n, t):
if t < n: return False
while n and t:
if n%10 == t%10:
n //= 10
t //= 10
return n == 0
def a(n):
return next(t for t in (i*(i+1)//2 for i in count(isqrt(2*n))) if dmo(n, t))
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jan 21 2023
a(4) = 2 since 4 = 2^2. Table of the first 20 terms of related sequences: n A068165 A091873 a(n)^2 a(n) 1: 1 1 1 1 2: 25 5 25 5 3: 36 6 36 6 4: 4 2 4 2 5: 25 5 25 5 6: 16 4 16 4 7: 576 24 576 24 8: 81 9 81 9 9: 9 3 9 3 10: 100 10 100 10 11: 121 11 121 11 12: 121 11 121 11 13: 1369 37 361 19 14: 144 12 144 12 15: 1156 34 1156 34 16: 16 4 16 4 17: 1764 42 1764 42 18: 1089 33 81 9 19: 169 13 169 13 20: 2025 45 1024 32 ...
Table[If[IntegerQ@ Sqrt@ n, Sqrt@ n, k = Floor@ Sqrt@ n; Function[t, While[Function[w, Times @@ Boole@ Map[w[[#1]] >= #2 & @@ # &, #] < 1]@ DigitCount[k^2] &@ Apply[Join, Map[Lookup[t, #] /. d_ /; IntegerQ@ d :> If[d > 0, {d, #}, {10, #}] &, Keys@ t]], k++]]@ KeyDrop[PositionIndex@ DigitCount@ n, 0]; k], {n, 69}] (* Michael De Vlieger, May 05 2017, Version 10.1 *)
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