A089998 -id:A089998 - cp's OEIS frontend

A089999 Smallest triangular number with Hamming weight n (i.e., with exactly n 1's when written in binary).

0, 1, 3, 21, 15, 55, 190, 253, 703, 3003, 5886, 13695, 4095, 49141, 106491, 192510, 784378, 1915903, 3407355, 5240703, 15986685, 30400503, 48201471, 124780503, 247431135, 602930175, 1608777726, 4290575295, 7482375615, 15938355070

Offset: 0

Reinhard Zumkeller, Nov 20 2003

A000120(a(n)) = n.

a(n) = A000217(A211201(n)). - Reinhard Zumkeller, Mar 18 2013

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

Cf. A211201, A000217, A089998, A061712, A090002.

a089999 = a000217 . a211201  -- Reinhard Zumkeller, Mar 18 2013

a = Table[0, {30}]; Do[t = n(n + 1)/2; c = Count[ IntegerDigits[t, 2], 1]; If[ a[[c + 1]] == 0, a[[c + 1]] = t], {n, 1, 10^8}]; a (* Robert G. Wilson v, Dec 03 2003 *)

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

Offset corrected by Donovan Johnson, May 01 2012

A231897 a(n) = smallest m such that wt(m^2) = n (where wt(i) = A000120(i)), or -1 if no such m exists.

Original entry on oeis.org

0, 1, 3, 5, 13, 11, 21, 39, 45, 75, 155, 217, 331, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 1015143, 908091, 980853, 2960011, 4568757, 2965685, 5931189, 11862197, 20437147

Offset: 0

N. J. A. Sloane, Nov 19 2013

Conjecture: a(n) is never -1. (It seems likely that the arguments of Lindström (1997) could be modified to establish this conjecture.)

a(n) is the smallest m such that A159918(m) = n (or -1 if ...).

Hugo Pfoertner, Table of n, a(n) for n = 0..110 (terms 0..70 from Donovan Johnson, significant extension enabled by programs provided in Code Golf challenge).
Code Golf Stackexchange, Smallest and largest 100-bit square with maximum Hamming weight, fastest code challenge started Dec 15 2022.
Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.

Cf. A000120, A159918, A230097, A211201, A231898, A214560.

A089998 are the corresponding squares.

a231897 n = head [x | x <- [1..], a159918 x == n]
-- Reinhard Zumkeller, Nov 20 2013

a(n)=if(n,my(k); while(hammingweight(k++^2)!=n,); k, 0) \\ Charles R Greathouse IV, Aug 06 2015

def wt(n): return bin(n).count('1')
def a(n):
    m = 2**(n//2) - 1
    while wt(m**2) != n: m += 1
    return m
print([a(n) for n in range(32)]) # Michael S. Branicky, Feb 06 2022

a(n) = 2*A211201(n-1) + 1 for n >= 1. - Hugo Pfoertner, Feb 06 2022

a(26)-a(40) from Reinhard Zumkeller, Nov 20 2013

A090001 Length of longest contiguous block of 1's in binary expansion of n^2.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Nov 20 2003

a(n) = A038374(A000290(n)).

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000120, A090003, A090002, A090000, A089998, A090047.

Join[{0},Table[Max[Length/@Select[Split[IntegerDigits[n^2,2]], MemberQ[ #,1]&]],{n,110}]] (* Harvey P. Dale, Nov 28 2014 *)

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 *)

A359316 a(n) is the smallest centered square number with binary weight n.

Original entry on oeis.org

1, 5, 13, 85, 61, 221, 761, 1013, 2813, 12013, 23545, 54781, 16381, 196565, 425965, 770041, 3137513, 7663613, 13629421, 20962813, 63946741, 121602013, 192805885, 499122013, 989724541, 2411720701, 6435110905, 17162301181, 29929502461, 63753420281

Offset: 1

Ilya Gutkovskiy, Dec 25 2022

			221 is the smallest centered square number with binary weight 6 (221_10 = 11011101_2), so a(6) = 221.

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

Cf. A000120, A001844, A089998, A358932, A359315.

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

A359318 a(n) is the smallest square pyramidal number with binary weight n.

Original entry on oeis.org

0, 1, 5, 14, 30, 55, 819, 506, 1785, 1015, 16206, 51039, 98021, 81375, 1113775, 964535, 2607099, 5494655, 1048061, 6029275, 50331190, 356343295, 534555645, 516941815, 4021378559, 2143222510, 12842950505, 34091142526, 68651299705, 124545644405, 273736383990

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

			819 is the smallest square pyramidal number with binary weight 6 (819_10 = 1100110011_2), so a(6) = 819.

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

Cf. A000120, A000330, A089998, A358931, A359317.

V:= Array(0..40): count:= 0:
for k from 1 while count < 40 do
  m:= k*(k+1)*(2*k+1)/6;
  w:= convert(convert(m,base,2),`+`);
  if w <= 40 and V[w] = 0 then
    count:= count+1; V[w]:= m;
  fi
od:
convert(V,list); # Robert Israel, Dec 26 2022

seq[len_,nmax_] := Module[{s = Table[0,{len}], n = 0, c = 0, bw, sp}, While[c < len && n < nmax, bw = DigitCount[sp = n*(n+1)*(2*n+1)/6, 2, 1] + 1; If[bw <= len && s[[bw]] == 0, c++; s[[bw]] = sp]; n++]; s]; seq[31, 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

A089999 Smallest triangular number with Hamming weight n (i.e., with exactly n 1's when written in binary).

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A231897 a(n) = smallest m such that wt(m^2) = n (where wt(i) = A000120(i)), or -1 if no such m exists.

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A090001 Length of longest contiguous block of 1's in binary expansion of n^2.

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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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Mathematica

A359316 a(n) is the smallest centered square number with binary weight n.

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Mathematica

A359318 a(n) is the smallest square pyramidal 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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