A383745 Numbers k of the form x*(x+1) whose sum of digits is of the form y*(y+1).

0, 2, 6, 20, 42, 110, 132, 156, 240, 420, 462, 552, 600, 930, 992, 1056, 1122, 1560, 1722, 1892, 2352, 2550, 2756, 3306, 3540, 3782, 4422, 4556, 4970, 5700, 5852, 6006, 6806, 7140, 7832, 8372, 8930, 9120, 9506, 10100, 10302, 10506, 10920, 11130, 11990, 12210, 12432

Offset

1,2

Comments

4*a(n)+1 is the square of an odd number.

All members are congruent to 0 or 2 mod 3.

The sum of the digits of 10^s * (10^s+1) is 2 so there are infinitely many a(n) of the form 3*m + 2.

The sum of the digits of (10^t-1) * 10^t is 9*t. Given that t = z*(9*z + 1), it can be proved that there are infinitely many a(n) in the form of 3*m.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) >= digsum(a(n)).

Example

132 is in the sequence because 132 = 11*12 and 1+3+2 = 6 = 2 *3.
2756 is in the sequence because 2756 = 52*53 and 2+7+5+6 = 20 = 4 * 5.

Maple

select(t -> issqr(1+4*convert(convert(t, base, 10), `+`)), [seq(i*(i+1), i=0..120)]); # Robert Israel, Jun 09 2025

Mathematica

Select[2 * Accumulate[Range[0, 150]], IntegerQ[Sqrt[4 * DigitSum[#] + 1]] &] (* Amiram Eldar, May 08 2025 *)

Prog

(PARI) apply(x->(x*(x+1)), select(x->issquare(4*sumdigits(x*(x+1))+1), [0..100])) \\ Michel Marcus, May 08 2025

Crossrefs

Cf. A002378, A007953, A028839, A128203.

Sequence in context: A214307 A087134 A378027 * A370494 A036689 A355390

Adjacent sequences: A383742 A383743 A383744 * A383746 A383747 A383748

Keyword

nonn,base

Author

Huaineng He, May 08 2025

Extensions

Offset corrected by Robert Israel, Jun 09 2025

Status

approved