A367451 - OEIS

%I #22 Mar 27 2024 08:58:26

%S 1,4,9,324,1444,26244,114244,116964,191844,213444,311364,412164,

%T 777924,2196324,3392964,6441444,11142244,13264164,16499844,21734244,

%U 23213124,23444964,24621444,26193924,27436644,36216324,44116164,47361924,61121124,61496964,63393444

%N Squares in A353729.

%C The sequence is infinite because the squares of numbers of the form (10^n + 14)/3, n >= 2, 38, 338, 3338, ... are 1444, 114244, 11142244, ... and have only the digits 1, 2, 4. Deleting any digit results in a number divisible by 4.

%C a(1) = 1 and a(3) = 9 are the only terms with an odd last digit. There are no other terms with the last digit 1, 5, 9 because the squares are odd, the tens digit must be even, and by deleting it the remaining number is odd.

%C The last digit of the terms cannot be 6 because the last two digits of the squares would be 16, 36, 56, 76, 96, so the penultimate digit is odd, and removing the 6 would leave an odd number. Thus, the terms a(n) with n >= 4, have the last digit 4.

%H Jason Yuen, <a href="/A367451/b367451.txt">Table of n, a(n) for n = 1..10000</a>

%e 324 = 18^2 is a term, since 3 divides 24, 2 divides 34 and 4 divides 32.

%t A353729Q[n_] := Block[{d = IntegerDigits[n], i=0}, FreeQ[d, 0] && (While[++i <= Length[d] && Divisible[FromDigits[Drop[d, {i}]], d[[i]]]]; i > Length[d])];

%t Select[Range[15000]^2, A353729Q] (* _Paolo Xausa_, Feb 27 2024 *)

%o (Magma) ints:=func<n| n eq 0 select[0] else Intseq(n)>;f:=func<n,i|Seqint([ints(n)[j]: j in [1..#ints(n)]|j ne (#ints(n)-i+1)])>; [p:p in [s*s:s in [1..8000]] |not 0 in ints(p) and forall{i:i in [1..#ints(p)]|IsIntegral(f(p,i)/ints(p)[(#Intseq(p)-i+1)])}];

%o (PARI) isok(k) = if (issquare(k), my(d=digits(k)); if (vecmin(d), for(i=1, #d, my(list=List(d)); listpop(list, i); if (fromdigits(Vec(list)) % d[i], return(0));); return(1););); \\ _Michel Marcus_, Dec 19 2023

%Y Intersection of A000290 and A353729.

%K nonn,base

%O 1,2

%A _Marius A. Burtea_, Dec 16 2023