A056991 Numbers with digital root 1, 4, 7 or 9.

1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• base
• easy

Comments

All squares are members (see A070433).

May also be defined as: possible sums of digits of squares. - Zak Seidov, Feb 11 2008

First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - Zak Seidov, Feb 11 2008

Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.

a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - Gary Detlefs, Mar 19 2010

From Klaus Purath, Feb 20 2023: (Start)

The sum of digits of any term belongs to the sequence. Also the products of any terms belong to the sequence.

This is the union of A017173, A017209, A017245 and A008591.

Positive integers of the forms x^2 + (2*m+1)*x*y + (m^2+m-2)*y^2, for integers m.

This sequence is closed under multiplication. (End)

Links

• R. J. Mathar, Table of n, a(n) for n = 1..22222
• H. I. Okagbue, M. O. Adamu, S. A. Iyase, and A. A. Opanuga, Sequence of Integers Generated by Summing the Digits of their Squares, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
• Eric Weisstein's World of Mathematics, Square Number
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A000290, A008591, A017173, A017209, A017245, A056992, A070433.

For complement see A268226.

Programs

Maple

seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # Gary Detlefs, Mar 19 2010

Mathematica

LinearRecurrence[{1,0,0,1,-1},{1,4,7,9,10},70] (* Harvey P. Dale, Aug 29 2015 *)

PARI

forstep(n=1,1e3,[3,3,2,1],print1(n", ")) \\ Charles R Greathouse IV, Sep 21 2012

Formula

From R. J. Mathar, Feb 14 2008: (Start)

O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).

a(n) = a(n-4) + 9. (End)

a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - Gary Detlefs, Mar 19 2010

a(n) = a(n-1)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 27 2021

a(n) = 3*n - floor(n/4) - 2*floor((n+3)/4). - Ridouane Oudra, Jan 21 2024

E.g.f.: (cos(x) + (9*x - 1)*cosh(x) - 3*sin(x) + (9*x - 2)*sinh(x))/4. - Stefano Spezia, Feb 21 2024

Extensions

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar