A036744 Penholodigital squares: squares containing each of the digits 1..9 exactly once.

139854276, 152843769, 157326849, 215384976, 245893761, 254817369, 326597184, 361874529, 375468129, 382945761, 385297641, 412739856, 523814769, 529874361, 537219684, 549386721, 587432169, 589324176, 597362481, 615387249, 627953481, 653927184, 672935481, 697435281, 714653289, 735982641, 743816529, 842973156, 847159236, 923187456

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• fini
• full

Comments

Improved Mathematica formula provided. Because the range involved is only from Ceiling[Sqrt[123456789]]=11112 and Floor[Sqrt[987654321]]=31427, it only requires analyzing 20,315 numbers, versus 362,880 permutations of nine digits (as in the current formula). - Harvey P. Dale, Apr 17 2002

Since the sum of the digits is 45, the squares are all divisible by 3, so the given Mathematica formula could be sped up by a factor of 3, checking only multiples of 3 rather than all squares. - Joshua Zucker, Nov 28 2005

Eight-digit analog gives 5 squares: 13527684, 34857216, 65318724, 73256481, 81432576. - Zak Seidov, Mar 01 2011

Crossrefs

Cf. A036745, A071519.

Programs

Maple

lim:=floor(sqrt(987654321)): for n from 11112 by 3 to lim do d:=[op(convert(n^2, base, 10))]: pandig:=true: for k from 1 to 9 do if(numboccur(k, d)<>1)then pandig:=false: break: fi: od: if(pandig)then printf("%d, ", n^2): fi: od: # Nathaniel Johnston, Jun 22 2011

Mathematica

Select[Range[11112, 31427]^2, Union[Drop[DigitCount[ # ], -1]] == {1} &]

PARI

A036744 = [n^2 | n <- A071519] \\ or less efficient & more explicit:
A036744 = [n^2 | n <- [1e5\9..1e5\3], vecsort(digits(n^2)) == [1..9]] \\ M. F. Hasler, Jun 28 2023

Formula

a(n) = A071519(n)^2.

Extensions

More terms from Harvey P. Dale, Sep 26 2001

Keyword base added by Reinhard Zumkeller, May 16 2010