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.
Sqrt[#]&/@Select[Range[0,3800]^2,SubsetQ[{0,1,4,9},IntegerDigits[#]]&] (* Harvey P. Dale, Jun 14 2022 *)
maxk=10000; L=List(); for(k=0, maxk, if(setminus(vecsort(digits(k^2),,8), [0,1,4,9]) == [], listput(L, k))); Vec(L)
1049^2 = 1100401, therefore 1049 is a term. A046030(7)=14 is not a term, as 14^2=196 and 6 is not a square digit. 104900499999000^2 = 11004114900040199000001000000.
M:= 15: # for terms of up to M digits
f:= proc(n,d) n >= 10^(d-1) and convert(convert(n^2,base,10),set) subset {0,1,4,9} end proc:
g:= proc(n,d) convert(convert(n^2 mod 10^d,base,10),set) subset {0,1,4,9} end proc:
R:= 0, 1:
C:= [0,1,9]:
for d from 2 to M do
C:= select(g,map(t -> (t, t+10^(d-1), t+4*10^(d-1), t+9*10^(d-1)), C),d);
V:= select(f, C,d);
R:= R, op(V);
od:
sort([R]); # Robert Israel, Dec 18 2023
a = {}; Do[d = FromDigits[ ReplaceAll[ IntegerDigits[n, 4], {3 -> 9, 2 -> 4}]]; If[ Union[ Join[ IntegerDigits[d^2], {0, 1, 4, 9}]] == {0, 1, 4, 9}, a = Append[a, d]], {n, 0, 3*10^5}]; a
With[{c={0,1,4,9}},Select[FromDigits/@Tuples[c,11],SubsetQ[c, IntegerDigits[ #^2]]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2017 *)
See PARI link
from itertools import count, islice def A077429_gen(): # generator of terms for m in count(0): s = bin(m)[2:] if len(s)&1: s='0'+s n = int(''.join({'00':'0','01':'1','10':'4','11':'9'}[s[i:i+2]] for i in range(0,len(s),2))) if set(str(n**2)) <= {'0','1','4','9'}: yield n A077429_list = list(islice(A077429_gen(),20)) # Chai Wah Wu, Dec 19 2023
816 is in the sequence because 81 and 16 are squares.
with(numtheory):
for n from 10 to 20000 do:
x:=convert(n, base, 10):n1:=nops(x):
s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):
s2:=(n-irem(n,10))/10:ss1:=sqrt(s1):ss2:=sqrt(s2):
if ss1=floor(ss1) and ss2=floor(ss2)
then
printf(`%d, `, n):
else
fi:
od:
isok(n) = (left = n\10) && issquare(left) && (pt = 10^(#Str(n)-1)) && issquare(n - (n\pt)*pt); \\ Michel Marcus, Jun 25 2014
Flatten[Table[FromDigits/@Tuples[{1,4,9},n],{n,4}]] (* Harvey P. Dale, May 18 2015 *)
isok(n) = {digs = digits(n); for (i=1, #digs, if (! digs[i] || ! issquare(digs[i]), return (0));); return (1);} \\ Michel Marcus, Aug 31 2013
a(6) = 1100401 = 1049^2. A019544(8)=441 is not a term, as 441=21^2 and 2 is not a square digit.
N:= 30: # for terms of up to 2*N digits
R:= {1}: T:= {1,9}:
for d from 2 to N do
T:= select(t -> convert(convert(t^2 mod 10^d, base,10),set) subset {0,1,4,9}, map(t -> (t, t + 10^(d-1), t + 4*10^(d-1), t + 9*10^(d-1)), T));
R:= R union select(t -> convert(convert(t^2,base,10),set) subset {0,1,4,9},T);
od:
R2:= map(t -> t^2, R):
Res:= map(t -> seq(t*10^(2*i), i=0..(2*N-ilog10(t)-1)/2), R2) union {0}:
sort(convert(Res,list)); # Robert Israel, Aug 26 2024
a = {}; Do[d = FromDigits[ ReplaceAll[ IntegerDigits[n, 4], {3 -> 9, 2 -> 4}]]; If[ Union[ Join[ IntegerDigits[d^2], {0, 1, 4, 9}]] == {0, 1, 4, 9}, a = Append[a, d^2]], {n, 0, 3*10^4}]; a
997 is a term because 997 is prime and 997^2 = 994009 with 0, 4, 9 that are all squares.
p=primes(10000000); m=1;
for u=1:length(p) digit=dec2base(p(u).^2,10)-'0';
if (mod(sqrt(digit), 1) == 0) sol(m)=p(u); m=m+1; end
end
sol % Marius A. Burtea, Jun 30 2019
ok[n_] := AllTrue[IntegerDigits[n], MemberQ[{0, 1, 4, 9}, #] &]; mo = Select[Range[1, 10^6, 2], ok@Mod[#^2, 10^6] &]; Reap[Sow@2; Do[x = 10^6 t + m; If[PrimeQ[x] && ok[x^2], Sow[x]], {t, 0, 202}, {m, mo}]][[2, 1]] (* Giovanni Resta, Jul 02 2019 *)
Select[Prime[Range[11211000]],AllTrue[Sqrt[IntegerDigits[#^2]],IntegerQ]&] (* Harvey P. Dale, Aug 17 2021 *)
isok(p) = isprime(p) && (d=digits(p^2)) && (#select(x->issquare(x), d) == #d); \\ Michel Marcus, Jun 30 2019
a:= proc(n) option remember; local d, i, k;
for k from 1+`if`(n=1, 1, a(n-1))
do for i to length(k)-1 do
if issqr(iquo(k, 10^i, 'd')) and
issqr(d) then return k fi
od
od
end:
seq(a(n), n=1..60); # Alois P. Heinz, Aug 29 2015
qQ[n_] := Floor[ Sqrt@ n]^2 == n; ok[n_] := Catch[ Do[ If[ qQ@ Floor[n / 10^k] && qQ@ Mod[n, 10^k],Throw@ True], {k, IntegerLength[n] -1}]; False]; Select[Range@ 1000, ok] (* Giovanni Resta, Aug 29 2015 *)
The a(36) = 11 distinct ways to represent 36 using only square digits are [1,0,0,1,0,0] in base 2, [1,1,0,0] in base 3, [1,0,0] in base 6, [4,4] in base 8, [4,0] in base 9, [1,16] in base 20, [1,9] in base 27, [1,4] in base 32, [1,1] in base 35, [1,0] in base 36 and [36] in bases >= 37.
A384436:=proc(n) local a,b,c; a:=0; for b from 2 to n+1 do c:=convert(n,'base',b); if select(issqr,c)=c then a:=a+1 fi od; return max(1,a) end proc; seq(A384436(n),n=0..81);
a[n_] := Sum[Boole[AllTrue[IntegerDigits[n, b], IntegerQ[Sqrt[#]] &]], {b, 2, n+1}]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, May 29 2025 *)
a(n) = sum(b=2, n+1, my(d=digits(n,b)); #select(issquare, d) == #d); \\ Michel Marcus, May 29 2025
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