A174926 Smallest prime which has a decimal representation which shows n^2 embedded in otherwise only decimal square digits 0, 1, 4 and 9.

101, 11, 41, 19, 1601, 251, 1361, 149, 641, 181, 1009, 12101, 14401, 1699, 11969, 2251, 12569, 1289, 13241, 1361, 4001, 4441, 48409, 10529, 15761, 62501, 6761, 4729, 7841, 8419, 9001, 9619, 102409, 10891, 115601, 12251, 129641, 11369, 14449

Offset

0,1

Comments

There are four decimal square digits: 0 = 0^2 = 0, 1 = 1^2, 4 = 2^2, 9 = 3^2.

It is conjectured that sequence is infinite.

Some primes of the form n^2//1 = 10 * n^2 + 1 are in this sequence: for n = 1, 2, 5, ...

Note this curiosity of a double appearance of 1361 as 1//6^2//1 = p(6^2) = 1361 = p(19^2) = 1//19^2 or of 13691 = prime(1618) = 37^2//1 > 11369 = prime(1373) = 1//37^2 = p(37^2), 38th term of sequence

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

Let // denote concatenation of digits. Then:
101 = prime(26) = 1//0^2//1.
11 = prime(5) = 1^2//1.
41 = prime(13) = 2^2//1.
19 = prime(8) = 1//3^2.
1601 = prime(252) = 4^2//0//1.
251 = prime(54) = 5^2//1.
1361 = prime(218) = 1//6^2//1.
149 = prime(35) = 1//7^2.
641 = prime(116) = 8^2//1.
181 = prime(42) = 1//9^2.
1009 = prime(169) = 10^2//9.
12101 = prime(1448) = 11^2//0//1.

Maple

U[1]:= [0, 1, 4, 9]: V[1]:= [1, 4, 9]: W[1]:= [1, 9]:
for d from 2 to 5 do U[d]:= map(t -> (10*t, 10*t+1, 10*t+4, 10*t+9), U[d-1]);
V[d]:= select(t -> ilog10(t) = d-1, U[d]);
W[d]:= select(t -> member(t mod 10, [1, 9]), U[d]) od:
f:= proc(k) local d, i, R, m, v, w, x;
   if isprime(k) then return k fi;
   m:= 1 + ilog10(k);
   for d from 1 to 5 do
     R:= sort([seq(v*10^m+k, v=V[d]), seq(k*10^d+w, w=W[d]), seq(seq(seq(v*10^(m+i)+k*10^i+w, v=V[d-i]), w=W[i]), i=1..d-1)]);
     for x in R do if isprime(x) then return x fi od
   od;
 FAIL;
end proc:
f(0):= 101:
map(t -> f(t^2), [$0..50]); # Robert Israel, Jul 22 2026

Crossrefs

Cf. A174884, A062584.

Sequence in context: A385512 A381606 A029948 * A029949 A083152 A073053

Adjacent sequences: A174923 A174924 A174925 * A174927 A174928 A174929

Keyword

base,nonn,changed

Author

_Ulrich Krug_, Apr 02 2010

Extensions

Corrected by Robert Israel, Jul 22 2026

Status

approved