A156977 Numbers n such that n^2 contains every decimal digit exactly once.

32043, 32286, 33144, 35172, 35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 40545, 42744, 43902, 44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55446, 55524, 55581, 55626, 56532, 57321, 58413, 58455, 58554, 59403, 60984

Offset: 1

Author

Zak Seidov, Feb 20 2009

Keywords

• fini
• full
• nonn
• base

Comments

There are exactly 87 such numbers, none of them being prime.

Since 0 + 1 +...+ 9 = 5*9, every pandigital number is divisible by 9, hence every term of this sequence is divisible by 3 and so cannot be a prime. - Giovanni Resta, Mar 19 2013 [Comment expanded by N. J. A. Sloane, Jan 15 2022]

Links

• Giovanni Resta, Table of n, a(n) for n = 1..87 (full sequence)
• S. C. Gould, Question 15734, The Educational Times, and Journal of the College of Preceptors 58 (1905), nr. 527 (March 1), p. 157; Solution 15734, Ibid., nr. 529 (May 1), p. 235.

Crossrefs

Cf. A036745, A054037, A054038.

Programs

Magma

[n: n in [Floor(Sqrt(1023456789))..Ceiling(Sqrt(9876543210))] | Set(Intseq(n^2)) eq {0..9}]; // Bruno Berselli, Mar 19 2013 (after Giovanni Resta)

Maple

lim:=floor(sqrt(9876543210)): for n from floor(sqrt(1023456789)) to lim do d:=[op(convert(n^2, base, 10))]: pandig:=true: for k from 0 to 9 do if(numboccur(k, d)<>1)then pandig:=false: break: fi: od: if(pandig)then printf("%d, ",n): fi: od: # Nathaniel Johnston, Jun 22 2011

Mathematica

Select[Range[Floor@Sqrt@1023456789, Ceiling@Sqrt@9876543210], Sort@IntegerDigits[#^2] == Range[0, 9] &] (* Giovanni Resta, Mar 19 2013 *)
Select[Range[31992,99381,3],Union[DigitCount[#^2]]=={1}&] (* Harvey P. Dale, Jan 17 2022 *)

Formula

a(n) = sqrt(A036745(n)).