A084561 -id:A084561 - cp's OEIS frontend

A175396 Numbers whose sum of squares of digits is a square.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 34, 40, 43, 50, 60, 68, 70, 80, 86, 90, 100, 122, 148, 184, 200, 212, 221, 236, 244, 263, 269, 296, 300, 304, 326, 340, 362, 366, 400, 403, 418, 424, 430, 442, 447, 474, 481, 488, 500, 600, 608, 623, 629, 632, 636, 663

Offset: 1

Ctibor O. Zizka, Apr 30 2010

Previous name: Numbers n such that Sum_{i=1..r, x(i)^2} is a perfect square, where x(i) = digits of n. r=1+floor(log_10 n).

			34 is a term: 3^2 + 4^2 = 25 = 5^2.
122 is a term: 1^2 + 2^2 + 2^2 = 9 = 3^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A000290, A003132, A084561.

Select[Range[0, 666], IntegerQ[Sqrt[Plus @@ (IntegerDigits[#]^2)]] &] (* Ivan Neretin, Aug 03 2015 *)

isok(n) = {my(digs = digits(n)); issquare(sum(i=1, #digs, digs[i]^2))} \\ Michel Marcus, Jun 02 2013

from math import isqrt
def ok(n): s = sum(int(i)**2 for i in str(n)); return isqrt(s)**2 == s
print(*[k for k in range(664) if ok(k)], sep = ', ')  # Ya-Ping Lu, Jul 07 2025

Corrected and extended by Neven Juric, Jul 12 2010

Simpler definition by Michel Marcus, Jun 02 2013

A348906 Squares with a square number of 1's in their binary expansion.

Original entry on oeis.org

0, 1, 4, 16, 64, 169, 225, 256, 676, 900, 1024, 2209, 2704, 3600, 4096, 5625, 7921, 8836, 10201, 10816, 12321, 13689, 14400, 16384, 19321, 20449, 22201, 22500, 23409, 26569, 27889, 28561, 29929, 30625, 31684, 32041, 35344, 38809, 40401, 40804, 43264, 49284, 52441

Offset: 1

Ctibor O. Zizka, Nov 03 2021

If a number k is of the form 2^(2*r), r >= 0, then k is included in this sequence.

			225 is in the sequence because it is a square and the number of 1's in the binary expansion of 225 is 4 which is a square.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000290 and A084561.

Cf. A000120, A081092, A344603.

q:= n-> issqr(add(i, i=Bits[Split](n))):
select(q, [i^2$i=0..250])[];  # Alois P. Heinz, Nov 03 2021

Select[Range[0, 300]^2, IntegerQ @ Sqrt[DigitCount[#, 2, 1]] &] (* Amiram Eldar, Nov 03 2021 *)

isok(k) = issquare(k) && issquare(hammingweight(k)); \\ Michel Marcus, Nov 03 2021

A381720 Integers whose Hamming weight is a cube.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 256, 383, 447, 479, 495, 503, 507, 509, 510, 512, 639, 703, 735, 751, 759, 763, 765, 766, 831, 863, 879, 887, 891, 893, 894, 927, 943, 951, 955, 957, 958, 975, 983, 987, 989, 990, 999, 1003, 1005, 1006, 1011, 1013, 1014

Offset: 1

Ctibor O. Zizka, Mar 05 2025

The plot of terms of the form k - 2^floor(log_2(k)) shows quasi-periodic structures on the intervals [2^i, 2^(i+1)]. In general, all sequences of the form "Integers whose Hamming weight is f(x)", where f(x) is an integer valued function, are quasi-periodic on intervals [2^i, 2^(i+1)].
The powers of 2 (A000079) are terms.
A023690 is a subsequence.

			For k = 255: A000120(255) = 8 = 2^3 is a cube, thus 255 is a term.

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

Cf. A000079, A000120, A000578, A023690.

Cf. A084561, A344602.

filter:= proc(n) type(surd(convert(convert(n,base,2),`+`),3),integer) end proc:
select(filter, [$0..2000]); # Robert Israel, May 18 2025

Select[Range[0, 1200], IntegerQ[Surd[DigitCount[#, 2, 1], 3]] &] (* Amiram Eldar, Mar 05 2025 *)

isok(k) = ispower(hammingweight(k), 3); \\ Michel Marcus, Mar 05 2025

from itertools import count, islice, combinations
from sympy import integer_nthroot
def A381720_gen(): # generator of terms
    a = []
    yield 0
    for l in count(1):
        b = 1<A381720_list = list(islice(A381720_gen(),53)) # Chai Wah Wu, Mar 06 2025

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A175396 Numbers whose sum of squares of digits is a square.

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A348906 Squares with a square number of 1's in their binary expansion.

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A381720 Integers whose Hamming weight is a cube.

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