Huaineng He - cp's OEIS frontend

A384296 Square numbers whose iterative sums of digits are squares.

0, 1, 4, 9, 36, 81, 100, 121, 144, 225, 324, 400, 441, 900, 1521, 2025, 2304, 2601, 3600, 8100, 10000, 10201, 10404, 11025, 12100, 12321, 14400, 22500, 32400, 40000, 40401, 44100, 62001, 69696, 90000, 101124, 103041, 121104, 123201, 149769, 152100, 173889, 178929, 199809, 202500, 230400, 251001

Offset: 1

Huaineng He, May 24 2025

			10201 = 101^2. 1+0+2+0+1 = 4, 4 = 2^2.
178929 = 423^2. 1+7+8+9+2+9 = 36, 36 = 6^2. 3+6 = 9, 9 = 3^2.

Cf. A004159, A117676, A070027.

A subsequence of A053057.

filter:= proc(n) local L,i,t;
  t:= n;
  do
    if not issqr(t) then return false fi;
    if t < 10 then return true fi;
    t:= convert(convert(t,base,10),`+`)
  od
end proc:
select(filter, [seq(i^2,i=0..1000)]); # Robert Israel, May 25 2025

q[n_] := AllTrue[FixedPointList[DigitSum, n], IntegerQ[Sqrt[#]] &]; Select[Range[0, 500]^2, q] (* Amiram Eldar, May 25 2025 *)

A383642 Numbers k = x + y with x and y positive integers such that x*y is a cube.

Original entry on oeis.org

2, 6, 9, 12, 16, 20, 28, 30, 33, 34, 35, 42, 48, 54, 56, 58, 65, 70, 72, 75, 84, 86, 90, 91, 96, 105, 110, 114, 120, 124, 126, 128, 132, 133, 152, 153, 156, 160, 162, 180, 182, 189, 198, 201, 205, 209, 210, 217, 224, 236, 238, 240, 243, 246, 250, 254, 258, 264, 267

Offset: 1

Huaineng He, May 03 2025

nonn

Includes all numbers of the form m*(m + 1).
2 is the only prime member.
If k >= 1 is in the sequence then so is k*m^3 where m >= 1. - David A. Corneth, May 05 2025

			k=12, 12=3+9, 3*9=3^3.
k=65, 65=25+40, 25*40=10^3.

Cf. A000578, A337140.

Supersequence of A003325.

kMax = 300; result = {}; For[k = 2, k <= kMax, k++, For[a = 1, a < k, a++, b = k - a; product = a * b; cubeRoot = Round[product^(1 / 3)]; If[cubeRoot^3 == product, result = Append[result, k];]]]; Sort[Union[result]]

isok(k) = for(i=1, k\2, if(ispower(i*(k-i), 3), return(1))); \\ Michel Marcus, May 04 2025

is(n) = {my(maxc = sqrtnint(((n/2)^2)\1, 3)); for(i = 1, maxc, if(issquare(n^2 - 4*i^3, &sqrtD), P = (n + sqrtD)/2; if(denominator(P) == 1, return(1)))); 0} \\ David A. Corneth, May 05 2025

upto(n) = {my(maxc = sqrtnint(((n/2)^2)\1, 3), res = List(), f); for(i = 1, maxc, f = factor(i); f[,2]*=3; d = divisors(f); forstep(j = (#d+1)\2, 1, -1, c = d[j] + d[#d + 1 - j]; if(c <= n, listput(res, c), next(2)))); Set(res)} \\ David A. Corneth, May 05 2025

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

Original entry on oeis.org

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

Huaineng He, May 08 2025

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.

			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.

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

Cf. A002378, A007953, A028839, A128203.

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

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

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

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

Offset corrected by Robert Israel, Jun 09 2025

cp's OEIS Frontend

User: Huaineng He

A384296 Square numbers whose iterative sums of digits are squares.

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A383642 Numbers k = x + y with x and y positive integers such that x*y is a cube.

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PARI

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PARI

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

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Maple

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cp's OEIS Frontend

User: Huaineng He

A384296 Square numbers whose iterative sums of digits are squares.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A383642 Numbers k = x + y with x and y positive integers such that x*y is a cube.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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