A066567 -id:A066567 - cp's OEIS frontend

A228187 Positive numbers which when decremented by the product of their digits produce a square.

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 37, 45, 48, 67, 71, 79, 100, 228, 231, 256, 259, 292, 388, 400, 524, 575, 624, 661, 832, 865, 868, 900, 928, 949, 973, 985, 1024, 1089, 1231, 1317, 1344, 1399, 1541, 1549, 1564, 1600, 1612, 1629, 1723, 1759, 2025, 2164, 2209

Offset: 1

Derek Orr, Aug 15 2013

			388 is a term because 388-(3*8*8) = 196 = 14^2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066567.

a:= proc(n) option remember; local d, k, m;
      for k from 1+`if`(n=1, 0, a(n-1)) do
        d, m:= 1, k;
        while m>0 do d:=d*irem(m,10,'m') od;
        if issqr(k-d) then return k fi
      od
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 19 2013

Select[Range[2500],IntegerQ[Sqrt[#-Times@@IntegerDigits[#]]]&] (* Harvey P. Dale, Aug 16 2025 *)

for(n=0,10^4,d=digits(n);p=prod(i=1,#d,d[i]);if(issquare(n-p),print(n,", "))) \\ Derek Orr, Mar 13 2015

A229185 Numbers incremented by their digit product produce a cube.

Original entry on oeis.org

0, 4, 85, 168, 184, 212, 368, 549, 681, 919, 999, 1000, 1283, 1593, 2181, 3664, 4096, 4717, 6811, 7497, 8000, 9919, 10648, 12143, 15275, 15425, 21664, 21728, 21824, 27000, 39304, 42427, 42811, 47629, 50653, 63424, 64000, 74088, 79507, 84416, 103823, 110592

Offset: 1

Derek Orr, Sep 15 2013

			681 + 6*8*1 = 729 = 9^3.

Cf. A007954, A066567, A229184.

Select[Range[0,100000], IntegerQ[(# + Times @@ IntegerDigits[#])^(1/3)] &] (* T. D. Noe, Sep 16 2013 *)

for(n=0, 10^5, d=digits(n); P=n+prod(i=1, #d, d[i]); if(ispower(P, 3), print1(n, ", "))) \\ Derek Orr, Mar 12 2015

def DP(n):
  p = 1
  for i in str(n):
    p *= int(i)
  return p
for n in range(10**4):
  k = 0
  P = n + DP(n)
  while P >= k**3:
    if P == k**3:
      print(n, end=', ')
      break
    k += 1
# Simplified by Derek Orr, Mar 12 2015

Prepended a(1) = 0 from Derek Orr, Mar 12 2015

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

Original entry on oeis.org

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

Derek Orr, Mar 13 2015

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

			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.

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

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

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 *)

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,", ")))

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

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

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A228187 Positive numbers which when decremented by the product of their digits produce a square.

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A229185 Numbers incremented by their digit product produce a cube.

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

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Mathematica

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