cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-1 of 1 results.

A061909 Skinny numbers: numbers n such that there are no carries when n is squared by "long multiplication".

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 30, 31, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 130, 200, 201, 202, 210, 211, 212, 220, 221, 300, 301, 310, 311, 1000, 1001, 1002, 1003, 1010, 1011, 1012, 1013, 1020, 1021, 1022, 1030, 1031, 1100, 1101, 1102
Offset: 1

Views

Author

N. J. A. Sloane, Jun 14 2007

Keywords

Comments

There are several equivalent formulations. Suppose the decimal expansion of n is n = Sum_{i = 0..k } d_i 10^i, where 0 <= d_i <= 9 for i = 0..k.
Then n is skinny if and only if:
(i) e_i = Sum_{ j = 0..i } d_j d_{i-j} <= 9 for i = 0 .. 2k-1;
(ii) if P_n(X) = Sum_{i = 0..k } d_i X^i (so P_n(10) = n) then P_{n^2}(X) = P_n(X)^2;
(iii) R(n^2) = R(n)^2, where R(n) means the digit reversal of n;
(iv) (sum of digits of n)^2 = sum of digits of n^2.
This entry is a merging and reworking of earlier entries from Asher Auel, May 17 2001 and Amarnath Murthy, Aug 15 2005. Thanks to Andrew S. Plewe for suggesting that these sequences might be identical.
Also, numbers n in base 10 whose base 10 expansion of n^2 is the same as if n were interpreted in some base b>10 and n^2 also calculated in that base. - Andrew Silberman (sandrew(AT)math.upenn.edu), Oct 30 2006
From David Applegate and N. J. A. Sloane, Jun 14 2007: (Start)
The decimal expansion of a skinny number n may contain only 0's, 1's, 2's and 3's.
There may be at most one 3 and if there is a 3 then there can be no 2's. (Hence of course if there are any 2's then there can be no 3's.)
There is no limit to the number of 1's and 2's - consider for example Sum_{i=0..m} 10^{2^i} and 2*Sum_{i=0..m} 10^{2^i}.
These are necessary conditions, but are not sufficient (e.g., 131 is not skinny). (End)
There are fifty-five skinny numbers without a 0 digit, the greatest being a(5203) = 111111111. - Jason Zimba, Jul 05 2020

Examples

			12 is a member as 12^2 = 144, digit reversal of 144 is 441 = 21^2.
13 is a member as 13 squared is 169 and sqrt(961) = 31.
113 is a member as 113^2 = 12769, reversal(12769) = 96721 = 311^2.
(Sum of digits of 13)^2 = 4^2 = 16 and sum of digits of 13^2 = sum of digits of 169 = 16.
10^k is in the sequence for all k >= 0, since reversal((10^k)^2) = 1 = (reversal(10^k))^2. - _Ryan Propper_, Sep 09 2005

Links

Crossrefs

A085305 is a subsequence.
Cf. A007953, A004159, A061903, A061910, A129967, A129968, A129969, A129970, A129971, A123977, A159953, A169939.
The primes in this sequence are given by A085306.
Numbers n such that A067552(n) = 0.

Programs

  • Haskell
    a061909 n = a061909_list !! (n-1)
a061909_list = filter (\x -> a004086 (x^2) == (a004086 x)^2) [0..]
-- Reinhard Zumkeller, Jul 08 2011
  • Maple
    rev:=proc(n) local nn, nnn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n) if sqrt(rev(n^2))=rev(n) then n else fi end: seq(a(n),n=1..1200); # Emeric Deutsch, Mar 31 2007
f := []: for n from 1 to 1000 do if (convert(convert(n,base,10),`+`))^2 = convert(convert(n^2,base,10),`+`) then f := [op(f), n] fi; od; f; # Asher Auel
  • Mathematica
    r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[If[r[n]^2 == r[n^2], Print[n]], {n, 1, 10^4}] (* Ryan Propper, Sep 09 2005 *)
Select[Range[0,1200],IntegerReverse[#^2]==IntegerReverse[#]^2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)
  • PARI
    is(n)=sumdigits(n)^2==sumdigits(n^2) \\ Charles R Greathouse IV, Jun 21 2017
  • Python
    from itertools import count, islice, product
def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n**2) == sd(n)**2
def agen(): # generator of terms
    yield from [0, 1, 2, 3]
    for d in count(2):
        for f in "123":
            rset = "01" if f == "3" else "012" if f == "2" else "0123"
            for r in product(rset, repeat=d-1):
                t = int(f+"".join(r))
                if ok(t): yield t
print(list(islice(agen(), 53))) # Michael S. Branicky, Dec 23 2022

Formula

a(n) >> n^2.0959..., where the exponent is log 10/log 3. - Charles R Greathouse IV, Sep 21 2012
Showing 1-1 of 1 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.