A073642 -id:A073642 - cp's OEIS frontend

A272011 Irregular triangle read by rows: strictly decreasing sequences of nonnegative numbers given in lexicographic order.

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

Offset: 0

Peter Kagey, Apr 17 2016

Length of n-th row given by A000120(n);
Maximum of n-th row given by A000523(n);
Minimum of n-th row given by A007814(n);
GCD of n-th row given by A064894(n);
Sum of n-th row given by A073642(n + 1).
n-th row begins at index A000788(n - 1) for n > 0.
The first appearance of n is at A001787(n).
a(A001787(n) + 1) = a(A001787(n)) for all n > 0.
a(A001787(n) + 2) = 0 for all n > 0.
a(A001787(n) + 3) = a(A001787(n)) for all n > 1.
a(A001787(n) + 4) = 1 for all n > 1.
a(A001787(n) + 5) = a(A001787(n)) for all n > 1.
Row n < 1024 lists the digits of A262557(n). - M. F. Hasler, Dec 11 2019

			Row n is given by the exponents in the binary expansion of n. For example, row 5 = [2, 0] because 5 = 2^2 + 2^0.
Row 0: []
Row 1: [0]
Row 2: [1]
Row 3: [1, 0]
Row 4: [2]
Row 5: [2, 0]
Row 6: [2, 1]
Row 7: [2, 1, 0]

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000120, A000523, A001787, A007814, A064894, A073642.
Cf. A133457 gives the rows in reverse order.
Cf. A272020, A262557.

Map[Length[#] - Flatten[Position[#, 1]] &, IntegerDigits[Range[50], 2]] (* Paolo Xausa, Feb 13 2024 *)

apply( A272011_row(n)=Vecrev(vecextract([0..exponent(n+!n)],n)), [0..39]) \\ For n < 2^10: row(n)=digits(A262557[n]). There are 2^k rows starting with k, they start at row 2^k. - M. F. Hasler, Dec 11 2019

A372474 Least k such that the k-th prime number has exactly n zeros in its binary expansion.

Original entry on oeis.org

2, 1, 8, 7, 19, 32, 99, 55, 174, 310, 565, 1029, 1902, 3513, 6544, 6543, 23001, 43395, 82029, 155612, 295957, 564164, 1077901, 3957811, 3965052, 7605342, 14630844, 28194383, 54400029, 105097568, 393615809, 393615807, 762939128, 1480206930, 2874398838, 5586502349

Offset: 0

Gus Wiseman, May 11 2024

			The prime numbers A000040(a(n)) together with their binary expansions and binary indices begin:
         3:                          11 ~ {1,2}
         2:                          10 ~ {2}
        19:                       10011 ~ {1,2,5}
        17:                       10001 ~ {1,5}
        67:                     1000011 ~ {1,2,7}
       131:                    10000011 ~ {1,2,8}
       523:                  1000001011 ~ {1,2,4,10}
       257:                   100000001 ~ {1,9}
      1033:                 10000001001 ~ {1,4,11}
      2053:                100000000101 ~ {1,3,12}
      4099:               1000000000011 ~ {1,2,13}
      8209:              10000000010001 ~ {1,5,14}
     16417:             100000000100001 ~ {1,6,15}
     32771:            1000000000000011 ~ {1,2,16}
     65539:           10000000000000011 ~ {1,2,17}
     65537:           10000000000000001 ~ {1,17}
    262147:         1000000000000000011 ~ {1,2,19}
    524353:        10000000000001000001 ~ {1,7,20}
   1048609:       100000000000000100001 ~ {1,6,21}
   2097169:      1000000000000000010001 ~ {1,5,22}
   4194433:     10000000000000010000001 ~ {1,8,23}
   8388617:    100000000000000000001001 ~ {1,4,24}
  16777729:   1000000000000001000000001 ~ {1,10,25}
  67108913: 100000000000000000000110001 ~ {1,5,6,27}
  67239937: 100000000100000000000000001 ~ {1,18,27}

Positions of first appearances in A035103.
For squarefree instead of prime we have A372473, firsts of A372472.
Counting ones (weight) gives A372517, firsts of A014499.
Counting squarefree bits gives A372540, firsts of A372475, runs A077643.
Counting squarefree ones gives A372541, firsts of A372433.
Counting bits (length) gives A372684, firsts of A035100.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A070939 gives length of binary expansion (number of bits).
Cf. A059015, A066195, A069010, A073642, A145037, A211997, A230877, A359359, A359400, A371571, A372516.

nn=10000;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
dcs=DigitCount[Select[Range[nn],PrimeQ],2,0];
Table[Position[dcs,i][[1,1]],{i,0,spnm[dcs]}]

from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372474(n):
    for l in count(n):
        m = 1<Chai Wah Wu, May 13 2024

a(n) = A000720(A066195(n)). - Robert Israel, May 13 2024

a(22)-a(35) from and offset corrected by Chai Wah Wu, May 13 2024

A372472 Number of zeros in the binary expansion of the n-th squarefree number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 09 2024

			The 12th squarefree number is 17, with binary expansion (1,0,0,0,1), so a(12) = 3.

Positions of first appearances are A372473.
Restriction of A023416 to A005117.
For prime instead of squarefree we have A035103, ones A014499, bits A035100.
Counting 1's instead of 0's (so restrict A000120 to A005117) gives A372433.
For binary length we have A372475, run-lengths A077643.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A003714, A039004, A049093, A049094, A059015, A069010, A070939, A073642, A145037, A211997, A368494, A372474, A372516.

A372583 := proc(n)
    A023416(A005117(n)) ;
end proc:
seq(A372583(n),n=1..200) ; # R. J. Mathar, May 20 2024

DigitCount[Select[Range[100],SquareFreeQ],2,0]

a(n) = A023416(A005117(n)).

a(n) + A372433(n) = A070939(A005117(n)) = A372475(n).

A059867 Number of irreducible representations of the symmetric group S_n that have odd degree.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 8, 8, 16, 16, 32, 32, 64, 64, 16, 16, 32, 32, 64, 64, 128, 128, 128, 128, 256, 256, 512, 512, 1024, 1024, 32, 32, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 4096, 4096

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Feb 28 2001

Ayyer et al. (2016, 2016) obtain this sequence (which they call "odd partitions") as the number of partitions of n such that the dimension of the corresponding irreducible representation of S_n is odd.

			a(3) = 2 because S_3 the degrees of the irreducible representations of S_3 are 1,1,2.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Arvind Ayyer, Amritanshu Prasad, Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016.
Arvind Ayyer, A. Prasad, S. Spallone, Representations of symmetric groups with non-trivial determinant, arXiv preprint arXiv:1604.08837 [math.RT], 2016. See Eq. (14).
I. G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bulletin of the London Mathematical Society, 3(2):189-192, 1971.
John McKay, Irreducible representations of odd degree, Journal of Algebra 20, 1972 pages 416-418.
Igor Pak, Greta Panova, Bounds on Kronecker coefficients via contingency tables, Linear Algebra and its Applications (2020), Vol. 602, 157-178.

Cf. A000120, A029930; A029931: the bisection of log_2(a(n)); A073642, A089248.

a[n_] := 2^Total[Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1];
Array[a, 60] (* Jean-François Alcover, Jul 21 2018 *)

A059867(n)={my(d=binary(n));prod(k=1,#d,if(d[#d+1-k],2^(k-1),1));} \\ Joerg Arndt, Apr 29 2013

a(n) = {my(b = Vecrev(binary(n))); 2^sum(k=1, #b, (k-1)*b[k]);} \\ Michel Marcus, Jan 11 2016

def A059867(n) : dig = n.digits(2); return prod(2^n for n in range(len(dig)) if dig[n]==1) # Eric M. Schmidt, Apr 27 2013

If n = sum 2^e[i] in binary, then the number of odd degree irreducible complex representations of S_n is 2^sum e[i]. In words: write n in binary and take the product of the powers of 2 that appear.
G.f.: prod(k>=0, 1 + 2^k * x^2^k). a(n) = 2^A073642(n). - Ralf Stephan, Jun 02 2003
a(1)=1, a(2n) = 2^e1(n)*a(n), a(2n+1) = a(2n), where e1(n) = A000120(n). - Ralf Stephan, Jun 19 2003

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A271410 LCM of exponents in binary expansion of 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 4, 12, 12, 12, 12, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 20, 20, 60, 60, 60, 60, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 12, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 7, 7, 14, 14, 21, 21, 42

Offset: 0

Peter Kagey, Apr 11 2016

			a(2) = lcm(2) = 2 because 2*2 = 2^2;
a(3) = lcm(1, 2) = 2 because 2*3 = 2^1 + 2^2;
a(7) = lcm(1, 2, 3) = 6 because 2*7 = 2^3 + 2^2 + 2^1.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A029931, A064894, A073642, A096111, A116417.

lcm[n_]:=Module[{idn2=IntegerDigits[n,2]},LCM@@Pick[Reverse[Range[ Length[ idn2]]], idn2,1]]; Join[{1},Array[lcm,100]] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = my(ve = select(x->x==1, Vecrev(binary(2*n)), 1)); lcm(vector(#ve, k, ve[k]-1)); \\ Michel Marcus, Apr 12 2016

a(n)=lcm(Vec(select(x->x, Vecrev(binary(n)), 1))) \\ Charles R Greathouse IV, Apr 12 2016

from math import lcm
def A271410(n): return lcm(*(i for i, b in enumerate(bin(n)[:1:-1],1) if b == '1')) # Chai Wah Wu, Dec 12 2022

A029930 If 2n = Sum 2^e_i, a(n) = Product 2^e_i.

Original entry on oeis.org

1, 2, 4, 8, 8, 16, 32, 64, 16, 32, 64, 128, 128, 256, 512, 1024, 32, 64, 128, 256, 256, 512, 1024, 2048, 512, 1024, 2048, 4096, 4096, 8192, 16384, 32768, 64, 128, 256, 512, 512, 1024, 2048, 4096, 1024, 2048, 4096, 8192, 8192, 16384, 32768, 65536, 2048

Offset: 0

N. J. A. Sloane

			14 = 8+4+2 so a(7) = 8*4*2 = 64.

Seiichi Manyama, Table of n, a(n) for n = 0..8191
Arvind Ayyer, A. Prasad and S. Spallone, Representations of symmetric groups with non-trivial determinant, arXiv preprint arXiv:1604.08837 [math.RT], 2016. See Eq. (14).

Cf. A000120, A029931, A073642.

A bisection of A059867.

HammingWeight := n -> add(i, i = convert(n, base, 2)):
a := proc(n) option remember; `if`(n = 0, 1,
ifelse(n::even, 2^HammingWeight(n/2)*a(n/2), 2*a(n-1))) end:
seq(a(n), n = 0..48); # Peter Luschny, Oct 30 2021

e1[n_] := Total[IntegerDigits[n, 2]]; a[0] = 1; a[n_] := a[n] = If[EvenQ[ n], 2^e1[n/2] a[n/2], 2 a[n-1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 07 2016 *)

a(n) = {my(bd = Vecrev(binary(n))); prod(k=1, #bd, if (bd[k], 2^k, 1));} \\ Michel Marcus, Mar 07 2016

From Ralf Stephan, Jun 19 2003: (Start)
G.f.: Prod_{k>=0} 1+2^(k+1)x^2^k.
a(0) = 1, a(2n) = 2^e1(n)*a(n), a(2n+1) = 2a(2n), where e1(n) = A000120(n).
a(n) = 2^A029931(n). (End)

A372515 Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 3, 1, 3, 3, 1, 2, 2, 1, 1, 2, 3, 4, 2, 3, 4, 1, 3, 4, 3, 4, 1, 2, 4, 2, 4, 1, 4, 4, 1, 2, 3, 2, 3, 1, 3, 3, 1, 2, 2, 1, 1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 3, 4, 5, 3, 4, 5, 1, 2, 4, 5, 2, 4, 5, 1, 4, 5, 4, 5, 1, 2, 3, 5, 2, 3, 5, 1, 3, 5

Offset: 1

Gus Wiseman, May 26 2024

			The reversed binary expansion of 100 is (0,0,1,0,0,1,1), with zeros at positions {1,2,4,5}, so row 100 is (1,2,4,5).
Triangle begins:
   1:
   2: 1
   3:
   4: 1 2
   5: 2
   6: 1
   7:
   8: 1 2 3
   9: 2 3
  10: 1 3
  11: 3
  12: 1 2
  13: 2
  14: 1
  15:
  16: 1 2 3 4

Row lengths are A023416, partial sums A059015.
For ones instead of zeros we have A048793, lengths A000120, sums A029931.
Row sums are A359400, non-reversed A359359.
Same as A368494 but with empty rows () instead of (0).
A003714 lists numbers with no successive binary indices.
A030190 gives binary expansion, reverse A030308.
A039004 lists the positions of zeros in A345927.
Cf. A069010, A070939, A073642, A230877, A344618, A359402, A359495.

Table[Join@@Position[Reverse[IntegerDigits[n,2]],0],{n,30}]

A198192 Replace 2^k in the binary representation of n with n-k (i.e. if n = 2^a + 2^b + 2^c + ... then a(n) = (n-a) + (n-b) + (n-c) + ...).

Original entry on oeis.org

0, 1, 1, 5, 2, 8, 9, 18, 5, 15, 16, 29, 19, 34, 36, 54, 12, 30, 31, 52, 34, 57, 59, 85, 41, 68, 70, 100, 75, 107, 110, 145, 27, 61, 62, 99, 65, 104, 106, 148, 72, 115, 117, 163, 122, 170, 173, 224, 87, 138, 140, 194, 145, 201, 204, 263, 156, 216, 219, 282, 226

Offset: 0

Brian Reed, Oct 21 2011

			a(5) = (5-2) + (5-0) = 8 because 5 = 2^2 + 2^0.
a(7) = (7-2) + (7-1) + (7-0) = 18 because 7 = 2^2 + 2^1 + 2^0.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000120, A073642.

% n is number of terms to be computed:
function [B] = predAddition(n)
   for i = 0:n
      k = i;
      c = 0;
      s = 0;
      while(k ~= 0)
         if ((i - c) >= 0)
            s = s + mod(k,2)*(i-c);
         end
         c = c + 1;
         k = (k - mod(k,2))/2;
      end
      B(i+1) = s;
   end
end

b:= (n, k)-> `if`(n=0, 0, k*(n mod 2)+b(floor(n/2), k-1)):
a:= n-> b(n, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 25 2011

a(n) = n*A000120(n) - A073642(n). - Franklin T. Adams-Watters, Oct 22 2011

a(n) = b(n,n) with b(0,k) = 0, b(n,k) = k*(n mod 2) + b(floor(n/2),k-1) for n>0. - Alois P. Heinz, Oct 25 2011

A198193 Replace 2^k in the binary representation of n with n+(k-L) where L = floor(log(n)/log(2)).

Original entry on oeis.org

0, 1, 2, 5, 4, 8, 11, 18, 8, 15, 18, 28, 23, 35, 39, 54, 16, 30, 33, 50, 38, 57, 61, 83, 47, 70, 74, 100, 81, 109, 114, 145, 32, 61, 64, 96, 69, 103, 107, 144, 78, 116, 120, 161, 127, 170, 175, 221, 95, 141, 145, 194, 152, 203, 208, 262, 165, 220, 225, 283

Offset: 0

Brian Reed, Oct 26 2011

That is, if n = 2^a + 2^b + 2^c + ... then a(n) = (n+(a-L)) + (n+(b-L)) + (n+(c-L)) + ...).

			a(4) = (4+(2-2)) = 4 because int(log(4)/log(2)) = 2 and 4 = 2^2.
a(6) = (6+(2-2)) + (6+(1-2)) = 11 because int(log(6)/log(2)) = 2 and 6 = 2^2 + 2^1.

Cf. A000120, A073642, A198192.

% n is number of terms to be computed, b is the base. The examples all use b=2:
function [V] = revAddition(n,b)
   for i = 0:n
      k = i;
      if (i > 0)
         l = floor(log(i)/log(b));
      end
      s = 0;
      while(k ~= 0)
         if ((i-l) >= 0)
            s = s + mod(k,b)*(i-l);
         end
         l = l - 1;
         k = (k - mod(k,b))/b;
      end
      V(i+1) = s;
   end
end

read("transforms") :
A198193 := proc(n)
        (n-A000523(n))*wt(n)+A073642(n) ;
end proc:
seq(A198193(n),n=0..20) ; # R. J. Mathar, Nov 17 2011

Table[b = Reverse[IntegerDigits[n, 2]]; L = Length[b] - 1; Sum[b[[k]] (n + k - 1 - L), {k, Length[b]}], {n, 0, 59}] (* T. D. Noe, Nov 01 2011 *)

def A198193(n): return sum((n-i)*int(j) for i,j in enumerate(bin(n)[2:])) # Chai Wah Wu, Mar 13 2021

Let L = A000523(n), then a(n) = (n-L)*A000120(n) + A073642(n).

A309983 Numbers n resulting from adding the exponents of 2 associated with the "1" terms of their binary representation and subtracting the exponents of 2 associated with the "0" terms of their binary representation.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 0, 0, 2, 2, 4, 4, 6, 6, -2, -2, 0, 0, 2, 2, 4, 4, 4, 4, 6, 6, 8, 8, 10, 10, -5, -5, -3, -3, -1, -1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 3, 3, 5, 5, 7, 7, 9, 9, 9, 9, 11, 11, 13, 13, 15, 15, -9, -9, -7, -7, -5, -5, -3, -3, -3, -3, -1

Offset: 1

Mark Povich, Aug 26 2019

			When n=18, a(n) = 0. Convert 18 to binary (=10010). The 1s are in the 2^4 place and the 2^1 place. Take those exponents and add them (=5). The 0's are in the 2^3, 2^2, and 2^0 places. Subtract those exponents (=5) from the previous sum to get 0.
When n=26, a(n) = 6. Convert 26 to binary (=11010). The 1s are in the 2^4, 2^3, and 2^1 places. Take those exponents and add them (=8). The 0's are in the 2^2 and 2^0 places. Subtract those exponents (=2) from the previous sum to get 6.

Mark Povich, Table of n, a(n) for n = 1..9999
noyonict, Number conversion in Python 3

Cf. A073642 (when only 1 digits are considered).

a[n_] := Block[{d = Reverse@IntegerDigits[n, 2]}, Total@ Flatten@ {Position[d, 1]-1, 1-Position[d, 0]}]; Array[a, 74] (* Giovanni Resta, Aug 26 2019 *)

a(n) = {my(b=Vecrev(binary(n))); my(v1 = Vec(select(x->(x==1), b, 1))); my(v0 = Vec(select(x->(x==0), b, 1))); (vecsum(v1) - #v1) - (vecsum(v0) - #v0);} \\ Michel Marcus, Aug 26 2019

def dec_to_bin(dec_num): #define a function that converts decimal to binary.
    bin_num = 0
    power = 0
    while dec_num > 0:
        bin_num += 10 ** power * (dec_num % 2)
        dec_num //= 2
        power += 1
    return bin_num
def rev_bin(n):
    return list(reversed(str(dec_to_bin(n))))
n = 18
neg = [pos for pos, num in enumerate(rev_bin(n)) if num == "0"]
posi = [pos for pos, num in enumerate(rev_bin(n)) if num == "1"]
print(sum(posi)-sum(neg))

def A309983(n):
    r, i = 0, 0
    while n > 0:
        d, n = n%2, n//2
        if d == 1:
            r = r+i
        else:
            r = r-i
        i = i+1
    return r # A.H.M. Smeets, Oct 07 2019

cp's OEIS Frontend

A272011 Irregular triangle read by rows: strictly decreasing sequences of nonnegative numbers given in lexicographic order.

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A372474 Least k such that the k-th prime number has exactly n zeros in its binary expansion.

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A372472 Number of zeros in the binary expansion of the n-th squarefree number.

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A059867 Number of irreducible representations of the symmetric group S_n that have odd degree.

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A271410 LCM of exponents in binary expansion of 2n.

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A029930 If 2n = Sum 2^e_i, a(n) = Product 2^e_i.

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A372515 Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.

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