A211201 -id:A211201 - 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

A050493 a(n) = sum of binary digits of n-th triangular number.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 3, 3, 2, 4, 5, 2, 4, 5, 4, 4, 2, 4, 5, 6, 4, 6, 7, 3, 4, 4, 7, 6, 5, 6, 5, 5, 2, 4, 5, 6, 5, 8, 6, 4, 5, 7, 6, 6, 8, 4, 5, 4, 4, 5, 8, 6, 5, 7, 7, 3, 6, 7, 8, 7, 6, 7, 6, 6, 2, 4, 5, 6, 5, 8, 7, 8, 4, 6, 8, 5, 8, 9, 4, 5, 5, 8, 7, 8, 8, 7, 8, 8, 7, 8, 12, 5, 6, 5, 6, 5, 4, 5, 8, 7, 8

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999

See A211201 for smallest numbers m such that a(m) = n. - Reinhard Zumkeller, Feb 04 2013

Cf. A000120, A000217, A004157.

a050493 = a000120 . a000217  -- Reinhard Zumkeller, Feb 04 2013

f[n_]:=Plus@@IntegerDigits[n,2]; lst={};Do[t=n*(n+1)/2;AppendTo[lst,f[t]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2009 *)
Total[IntegerDigits[#,2]]&/@Accumulate[Range[0,100]] (* Harvey P. Dale, Jan 22 2012 *)

a(n)=hammingweight(n*(n+1)) \\ Charles R Greathouse IV, Nov 10 2015

a(n) = Sum_{i=1..floor(log_b(c(n)))+1} (floor(c(n)/b^(i-1)) - floor(c(n)/b^i)*b), b=2, n >= 1, a(0)=0, c(n)=A000217(n).
a(n) = A000120(A000217(n)). - Reinhard Zumkeller, Feb 04 2013
a(n) = [x^(n*(n+1)/2)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A351149 a(n) is the least exponent k such that the Hamming weight of n^(k+1) is not greater than the Hamming weight of n^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 5, 1, 7, 1, 1, 1, 3, 3, 4, 3, 2, 5, 1, 1, 4, 7, 5, 1, 5, 1, 1, 1, 3, 3, 7, 3, 4, 4, 3, 3, 5, 2, 5, 5, 3, 1, 1, 1, 5, 4, 7, 7, 2, 5, 3, 1, 3, 5, 2, 1, 3, 1, 1, 1, 3, 3, 4, 3, 5, 7, 3, 3, 3, 4, 3, 4, 3, 3, 1, 3, 5, 5, 3, 2, 3, 5, 11

Offset: 1

Hugo Pfoertner, Feb 07 2022

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000120, A094694, A211201, A231897.

a[n_] := Module[{k = 1}, While[DigitCount[n^k, 2, 1] < DigitCount[n^(k + 1), 2, 1], k++]; k]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

for(n=1,87, for(k=1,oo, my(hw1=hammingweight(n^k), hw2=hammingweight(n^(k+1))); if(hw2<=hw1, print1(k,", "); break)))

def A351149(n):
    k = 1
    while bin(n**k)[2:].count("1") < bin(n**(k+1))[2:].count("1"): k += 1
    return(k)
print([A351149(n) for n in range(1, 88)]) # Karl-Heinz Hofmann, Feb 07 2022

A359091 a(n) is the index of the smallest n-gonal number with binary weight n.

Original entry on oeis.org

6, 13, 9, 10, 24, 58, 34, 55, 67, 151, 134, 187, 201, 691, 350, 623, 1082, 1870, 2302, 3171, 5017, 13863, 13230, 6663, 24357, 50397, 35604, 60347, 63810, 107019, 181517, 365595, 624858, 1345485, 1002585, 1969415, 1191179, 7651731, 4592173, 7279863, 7403686, 17923182

Offset: 3

Ilya Gutkovskiy, Dec 16 2022

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

Cf. A000120, A211201, A231897, A359092.

p[n_, k_] := (n - 2)*k*(k - 1)/2 + k; a[n_] := Module[{k = 1}, While[DigitCount[p[n, k], 2, 1] != n, k++]; k]; Array[a, 30, 3] (* Amiram Eldar, Dec 17 2022 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

PARI

Python

Formula

Extensions

A050493 a(n) = sum of binary digits of n-th triangular number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A351149 a(n) is the least exponent k such that the Hamming weight of n^(k+1) is not greater than the Hamming weight of n^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

A359091 a(n) is the index of the smallest n-gonal number with binary weight n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica