A130448 Squares whose decimal representation contains no proper subsequence which is a positive square.

1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876

Offset

1,2

Comments

Is this sequence finite and if so what is the last term?

Yes, the sequence must be finite. This follows from a well-known result: there are no infinite antichains for the subsequence ordering. - Jeffrey Shallit, Mar 05 2014

If it exists, a(29) is greater than 10^18. - Mauro Fiorentini, Jan 24 2017

If it exists, a(51) is greater than 2*10^30. - Giovanni Resta, Jan 08 2018

Links

Martin Raab, Table of n, a(n) for n = 1..55 (terms 1..50 from Giovanni Resta, terms 51..55 from Martin Raab)

Michael S. Branicky, Python program

Wikipedia, Subsequence

Example

576 is in the list because none of its proper subsequences 5, 7, 6, 57, 76 or 56 are squares.

Mathematica

fQ[n_] := Module[{d = IntegerDigits[n], ds, sq}, ds = FromDigits /@ Union[Most[Rest[Subsets[d]]]]; sq = Select[ds, # > 0 && IntegerQ[Sqrt[#]] &, 1]; sq == {}]; Select[Range[0, 100000]^2, fQ] (* T. D. Noe, Mar 05 2014 *)

Prog

(PARI) isok(n) = {my(d = digits(n)); for (k = 1, #d, for (j= 1, #d - k + 1, if (j != #d, sd = vector(j, i, d[k+i-1]); nsd = fromdigits(sd); if (nsd && issquare(nsd), return(0)); ); ); ); return (1); } \\ Michel Marcus, Apr 21 2018
(Python) # see linked program for faster version
from math import isqrt
from itertools import chain, combinations as C, count, islice
def issquare(n): return isqrt(n)**2 == n
def psets(s): # nonempty proper subsets of s
    return chain.from_iterable(C(s, r) for r in range(1, len(s)))
def cond(s):
    ss = ("".join(t) for t in psets(s) if t[0] != "0")
    return not any(issquare(int(u)) for u in ss)
def agen(): yield from (k**2 for k in count(1) if cond(str(k**2)))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 23 2023

Crossrefs

Cf. A131599, A238334 (substrings).

Sequence in context: A046659 A063760 A238334 * A046451 A341790 A350152

Adjacent sequences: A130445 A130446 A130447 * A130449 A130450 A130451

Keyword

base,nonn,fini

Author

Martin Raab, Aug 07 2007

Extensions

Edited by N. J. A. Sloane, Sep 14 2007

a(23)-a(28) from Mauro Fiorentini, Jan 24 2017

Status

approved