A256065 Zeroless numbers that when incremented or decremented by the product of their digits produce a square.

2, 8, 46692, 58896, 59949, 186633, 186673, 949968, 1587616, 2989584, 58988961, 245878784, 914457625, 2439577764, 2754991369, 4161798288, 4161798468, 4629457984, 4897936656, 29859851664, 34828536976, 41664977536, 59998484736, 96745892625, 134994579556

Offset: 1

Author

Derek Orr, Mar 13 2015

Keywords

• nonn
• base

Comments

If a term has a zero in it, its digit product is 0. Thus it is trivial to include cubes with one or more zeros.

Intersection of A066567, A228187, and A052382.

Is this sequence finite?

Replacing "squares" with "cubes", this sequence would only consist of {4} for n < 10^8. 4 is believed to be the only number to satisfy this property with cubes.

If it exists, a(20) > 10^10.

a(80) > 10^27. - Hiroaki Yamanouchi, Mar 16 2015

Examples

			46692 + 4*6*6*9*2 = 49284 = 222^2 and 46692 - 4*6*6*9*2 = 210^2. So 46692 is a member of this sequence.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..79

Crossrefs

Cf. A066567 (when incremented), A228187 (when decremented), A052382 (zeroless).

Programs

Mathematica

pdsQ[n_]:=With[{p=Times@@IntegerDigits[n]},p>0&&AllTrue[Sqrt[n+{p,-p}],IntegerQ]]; Select[Range[3*10^6],pdsQ] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Jun 06 2025 *)

PARI

for(n=0,10^7,d=digits(n);p=prod(i=1,#d,d[i]);if(p&&issquare(n-p)&&issquare(n+p),print1(n,", ")))

Extensions

a(12)-a(19) from Michel Marcus, Mar 14 2015

a(20)-a(25) from Hiroaki Yamanouchi, Mar 16 2015