A066591 Primes which can be expressed as a concatenation of nonnegative squares.

11, 19, 41, 101, 109, 149, 181, 191, 199, 251, 401, 409, 419, 449, 491, 499, 641, 811, 911, 919, 941, 991, 1009, 1019, 1049, 1091, 1109, 1181, 1259, 1289, 1361, 1409, 1481, 1499, 1601, 1609, 1619, 1699, 1811, 1901, 1949, 1999, 2251, 2549, 2591, 3691

Offset

1,1

Comments

All terms are == {1,9} mod 10. - Zak Seidov, Jul 16 2015

The surprising prime 162536496481 is the concatenation of the 6 double-digit squares in increasing order (see Prime Curios! link). - Bernard Schott, Nov 19 2020

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2103 from Robert Israel)

Chris K. Caldwell and G. L. Honaker, Jr., 1625364981, Prime Curios!

Example

96181 is a term as it is a concatenation of 961 and 81 both of which are squares. 100169 is a term as it is a concatenation of 100 and 169 in one way and also that of 1, 0, 0, 16 and 9 in another way.

Maple

N:= 10^4: # to get all terms <= N
catn:= proc(x, y) if y=0 then 10*x else x*10^(ilog10(y)+1)+y fi end proc:
Sq:= {seq(i^2, i=0..floor(sqrt(N)))}: Agenda:= Sq: S:= Sq:
while Agenda <> {} do
Agenda:= select(`<=`, {seq(seq(catn(f, g), f=Agenda), g=Sq)}, N) minus S;
S:= S union Agenda;
od:
sort(convert(select(isprime, S), list)); # Robert Israel, Jul 16 2015

Prog

(Python)
from sympy import sieve
from itertools import count, islice
def iscat(w, A):
    return False if len(w) < 2 else any(w[:i] in A and (w[i:] in A or iscat(w[i:], A)) for i in range(1, len(w)))
def agen():
    S = {"0"}
    for d in count(2):
        S |= {str(i*i) for i in range(10**(d-2), 10**(d-1))}
        for p in sieve.primerange(10**(d-1), 10**d):
            if iscat(str(p), S):
                yield p
print(list(islice(agen(), 50))) # Michael S. Branicky, Feb 20 2024

Crossrefs

A061246 and A167535 are subsequences. - Zak Seidov, Jul 16 2015

Cf. A000290, A068493.

Sequence in context: A259190 A165944 A100557 * A061246 A353102 A068493

Adjacent sequences: A066588 A066589 A066590 * A066592 A066593 A066594

Keyword

base,easy,nonn

Author

Amarnath Murthy, Dec 21 2001

Extensions

Corrected and extended by Christopher Lund (clund(AT)san.rr.com), Apr 11 2002

Status

approved