1,2

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 (asher.auel(AT)reed.edu), May 17 2001 and Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 15 2005. Thanks to Andrew Plewe for suggesting that these sequences might be identical.

T. D. Noe, Table of n, a(n) for n=1..5000

Further properties of skinny numbers, from David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Jun 14 2007:

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. 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}.

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 (rpropper(AT)stanford.edu), Sep 09 2005

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 (deutsch(AT)duke.poly.edu), 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

r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[If[r[n]^2 == r[n^2], Print[n]], {n, 1, 10^4}] (Propper)

Cf. A007953, A004159, A061903, A061910.

Cf. A129967, A129968, A129969, A129970, A129971.

Numbers n such that A067552(n) = 0.

Sequence in context: A102859 A123977 A069967 this_sequence A159952 A007961 A060811

Adjacent sequences: A061906 A061907 A061908 this_sequence A061910 A061911 A061912

base,easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com), Jun 14 2007