A089999 -id:A089999 - cp's OEIS frontend

A089998 Smallest square with Hamming weight n (i.e., with exactly n 1's when written in binary).

0, 1, 9, 25, 169, 121, 441, 1521, 2025, 5625, 24025, 47089, 109561, 32761, 393129, 851929, 1540081, 6275025, 15327225, 27258841, 41925625, 127893481, 243204025, 385611769, 998244025, 1979449081, 4823441401, 12870221809, 34324602361

Offset: 0

Reinhard Zumkeller, Nov 20 2003

A000120(a(n)) = n.

Donovan Johnson, Table of n, a(n) for n = 0..60

Cf. A000290, A089999, A061712, A090001, A231897.

a = Table[0, {30}]; Do[c = Count[IntegerDigits[n^2, 2], 1]; If[ a[[c + 1]] == 0, a[[c + 1]] = n^2; Print[c, " = ", n^2]], {n, 1, 360000}] (* Robert G. Wilson v, Dec 03 2003 *)
Join[{0},With[{s=DigitCount[Range[400000]^2,2,1]},Flatten[Table[ Position[ s,?(#==n&),1,1],{n,30}]]]^2] (* _Harvey P. Dale, Mar 03 2013 *)

a(n) = A231897(n)^2. - Hugo Pfoertner, Dec 27 2022

More terms from Robert G. Wilson v, Dec 03 2003

Offset corrected by Donovan Johnson, May 01 2012

A090002 Length of longest contiguous block of 1's in binary expansion of n-th triangular number.

Original entry on oeis.org

0, 1, 2, 2, 1, 4, 1, 3, 1, 2, 3, 1, 3, 2, 2, 4, 1, 2, 2, 5, 2, 3, 6, 1, 2, 1, 5, 4, 2, 2, 3, 5, 1, 2, 2, 3, 2, 6, 3, 2, 2, 3, 3, 3, 4, 2, 3, 2, 2, 2, 5, 3, 2, 3, 3, 2, 4, 3, 4, 3, 3, 3, 4, 6, 1, 2, 2, 3, 1, 4, 2, 7, 1, 2, 3, 2, 3, 3, 2, 2, 2, 5, 2, 4, 5, 3, 3, 4, 4, 5, 12, 2, 2, 2, 3, 3, 2, 1, 4, 2, 3, 5

Offset: 0

Reinhard Zumkeller, Nov 20 2003

a(n) = A038374(A000217(n)).

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000120, A090001, A090003, A089999, A090048.

Join[{0},Max[Length/@Select[Split[IntegerDigits[#,2]],#[[1]]==1&]]&/@ Accumulate[ Range[110]]] (* Harvey P. Dale, Jul 28 2022 *)

A211201 Smallest m such that (sum of binary digits of m*(m+1)/2) = n.

Original entry on oeis.org

0, 1, 2, 6, 5, 10, 19, 22, 37, 77, 108, 165, 90, 313, 461, 620, 1252, 1957, 2610, 3237, 5654, 7797, 9818, 15797, 22245, 34725, 56723, 92634, 122330, 178540, 226838, 507571, 454045, 490426, 1480005, 2284378, 1482842, 2965594, 5931098, 10218573, 11096982, 21793257, 31317157

Offset: 0

Reinhard Zumkeller, Feb 04 2013

Donovan Johnson, Table of n, a(n) for n = 0..71 (terms < 2*10^12)

Cf. A089999, A050493, A000217, A000120.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a211201 = fromJust . (`elemIndex` a050493_list)

a(n) = my(m=0); while (hammingweight(m*(m+1)/2) != n, m++); m; \\ Michel Marcus, Jan 27 2022

A050493(a(n)) = n and A050493(m) <> n for m < a(n).

A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

Original entry on oeis.org

4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167, 12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727, 6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303, 805306367, 1879048191, 3221225471

Offset: 1

Jonathan Vos Post, Jun 23 2007

Semiprime analog of A061712. Extended by Stefan Steinerberger. Includes the subset Mersenne semiprimes A092561.

			a(1) = 4 because the first semiprime A001358(1) is 4 (base 10) which is written 100 in binary, the latter representation having exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 ones.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones and, as it also has no zeros, is the smallest of the Mersenne semiprimes.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.

Join[{4},Table[SelectFirst[Sort[FromDigits[#,2]&/@Permutations[ Join[ PadRight[{}, n,1],{0}]]],PrimeOmega[#]==2&],{n,2,40}]] (* Harvey P. Dale, Feb 06 2015 *)

A358930 a(n) is the smallest n-gonal number with binary weight n.

Original entry on oeis.org

21, 169, 117, 190, 1404, 9976, 3961, 11935, 19966, 113401, 98155, 208879, 261501, 3338221, 916475, 3100671, 9943039, 31457140, 50322871, 100523871, 264240373, 2113871829, 2012739435, 532673535, 7415513007, 33017544153, 17112759966, 50983861215, 59039022015

Offset: 3

Ilya Gutkovskiy, Dec 06 2022

			117 is the smallest pentagonal number with binary weight 5 (117_10 = 1110101_2), so a(5) = 117.

Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to binary expansion of n

Cf. A000120, A089998, A089999, A358931, A358932.

p[n_, k_] := (n - 2)*k*(k - 1)/2 + k; a[n_] := Module[{k = 1, pk}, While[DigitCount[pk = p[n, k], 2, 1] != n, k++]; pk]; Array[a, 25, 3] (* Amiram Eldar, Dec 09 2022 *)

A359315 a(n) is the smallest centered triangular number with binary weight n.

Original entry on oeis.org

1, 10, 19, 46, 31, 235, 631, 1786, 1999, 7669, 7039, 12286, 16381, 180094, 114679, 949231, 2086831, 2883574, 4175839, 12480511, 50329585, 62898151, 132638719, 234618814, 771743710, 2883510271, 4269733885, 8254119871, 17045499901, 33214168831

Offset: 1

Ilya Gutkovskiy, Dec 25 2022

			235 is the smallest centered triangular number with binary weight 6 (235_10 = 11101011_2), so a(6) = 235.

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to binary expansion of n

Cf. A000120, A005448, A089999, A358932, A359316.

seq[len_,nmax_] := Module[{s = Table[0,{len}], n = 1, c = 0, bw, ct}, While[c < len && n < nmax, bw = DigitCount[ct = 3*n*(n-1)/2 + 1, 2, 1]; If[bw <= len && s[[bw]] == 0, c++; s[[bw]] = ct]; n++]; s]; seq[30, 10^6] (* Amiram Eldar, Dec 26 2022 *)

A359317 a(n) is the smallest tetrahedral number with binary weight n.

Original entry on oeis.org

0, 1, 10, 35, 120, 220, 455, 2024, 1771, 4060, 14190, 16215, 129766, 32509, 1414910, 1823471, 5159805, 8171255, 4192244, 24117100, 30865405, 334985911, 192937325, 1610599145, 1048440315, 4261347265, 4244012991, 63828916911, 213588635511, 133110357279

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

			455 is the smallest tetrahedral number with binary weight 6 (455_10 = 111000111_2), so a(6) = 455.

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index entries for sequences related to binary expansion of n

Cf. A000120, A000292, A089999, A358931, A359318.

seq[len_,nmax_] := Module[{s = Table[0,{len}], n = 0, c = 0, bw, t}, While[c < len && n < nmax, bw = DigitCount[t = n*(n+1)*(n+2)/6, 2, 1] + 1; If[bw <= len && s[[bw]] == 0, c++; s[[bw]] = t]; n++]; s]; seq[30, 10^6] (* Amiram Eldar, Dec 26 2022 *)

A354480 a(n) is the smallest decimal palindrome with Hamming weight n (i.e., with exactly n 1's when written in binary).

Original entry on oeis.org

0, 1, 3, 7, 77, 55, 111, 191, 383, 767, 5115, 11711, 15351, 30703, 81918, 97279, 744447, 978879, 1570751, 3665663, 8387838, 66911966, 66322366, 132111231, 199212991, 389545983, 939474939, 3204444023, 3220660223, 11542724511, 34258485243, 33788788733, 34292629243

Offset: 0

Ilya Gutkovskiy, Jun 02 2022

Cf. A000120, A000225, A002113, A061712, A062388, A089226, A089998, A089999, A102029, A114477.

from itertools import count, islice, product
def pals(startd=1): # generator for base-10 palindromes
    for d in count(startd):
        for p in product("0123456789", repeat=d//2):
            if d//2 > 0 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], "0123456789"][d%2]:
                yield int(left + mid + right)
def a(n):
    for p in pals(startd=len(str(2**n-1))):
        if bin(p).count("1") == n:
            return p
print([a(n) for n in range(33)]) # Michael S. Branicky, Jun 02 2022

a(21)-a(32) from Michael S. Branicky, Jun 02 2022

cp's OEIS Frontend

A089998 Smallest square with Hamming weight n (i.e., with exactly n 1's when written in binary).

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A090002 Length of longest contiguous block of 1's in binary expansion of n-th triangular number.

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A211201 Smallest m such that (sum of binary digits of m*(m+1)/2) = n.

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A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

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A358930 a(n) is the smallest n-gonal number with binary weight n.

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A359315 a(n) is the smallest centered triangular number with binary weight n.

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A359317 a(n) is the smallest tetrahedral number with binary weight n.

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A354480 a(n) is the smallest decimal palindrome with Hamming weight n (i.e., with exactly n 1's when written in binary).

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