A329644 -id:A329644 - cp's OEIS frontend

A329638 Sum of A329644(d) for all such divisors d of n for which that value is positive. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 13, 1, 18, 2, 18, 1, 22, 1, 46, 5, 22, 1, 10, 1, 30, 14, 82, 2, 19, 1, 256, 2, 22, 1, 41, 1, 66, 9, 226, 1, 46, 1, 24, 8, 130, 1, 29, 2, 70, 2, 748, 1, 42, 1, 1362, 22, 30, 10, 42, 1, 214, 254, 44, 1, 43, 1, 3838, 15, 406, 2, 120, 1, 78, 5, 5458, 1, 52, 2, 12250, 2, 70, 1, 26, 2, 934

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A323243, A323244, A329639, A329641, A329644.

Cf. also A318878.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n,0,sigma(A156552(n)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A329644(n) = if(1==n,0, 2^A297113(n) - A324543(n));
A329638(n) = sumdiv(n,d,if((d=A329644(d))>0,d,0));

a(n) = Sum_{d|n} [A329644(d) > 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A323244(n) + A329639(n).

A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, 0, 14, 0, 0, 5, 0, 0, 1, 0, 0, 0, 13, 5, 0, 0, 0, 1, 21, 0, 14, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 5, 0, 12, 14, 0, 0, 17, 0, 0, 26, 74, 0, 350, 40, 0, 14, 53, 0, 0, 0, 70, 0, 0, 13, 18, 7, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A323244, A329638, A329640, A329641, A329644.

Cf. also A318879.

A329639(n) = sumdiv(n,d,if((d=A329644(d))<0,-d,0));

a(n) = -Sum_{d|n} [A329644(d) < 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A329638(n) - A323244(n).

A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Original entry on oeis.org

1, 1, 1, 4, -1, 1, 0, 4, -5, 1, 24, -16, 4, -13, 1, -8, 40, -48, 4, -29, 1, 104, -88, 72, -112, 4, -61, 1, -48, 184, -248, 136, -240, 4, -125, 1, 352, -400, 344, -568, 264, -496, 4, -253, 1, 80, 544, -1104, 664, -1208, 520, -1008, 4, -509, 1, 1424, -784, 928, -2512, 1304, -2488, 1032, -2032, 4, -1021, 1

Offset: 1

Antti Karttunen, Nov 22 2019

			The top left corner of the array:
   n   p_n |k=1,     2, 3,      4,     5,      6,     7,       8,      9,      10
  ---------+----------------------------------------------------------------------
   1 ->  2 |  1,     1, 4,      0,    24,     -8,   104,     -48,    352,      80,
   2 ->  3 |  1,    -1, 4,    -16,    40,    -88,   184,    -400,    544,    -784,
   3 ->  5 |  1,    -5, 4,    -48,    72,   -248,   344,   -1104,    928,   -2512,
   4 ->  7 |  1,   -13, 4,   -112,   136,   -568,   664,   -2512,   1696,   -5968,
   5 -> 11 |  1,   -29, 4,   -240,   264,  -1208,  1304,   -5328,   3232,  -12880,
   6 -> 13 |  1,   -61, 4,   -496,   520,  -2488,  2584,  -10960,   6304,  -26704,
   7 -> 17 |  1,  -125, 4,  -1008,  1032,  -5048,  5144,  -22224,  12448,  -54352,
   8 -> 19 |  1,  -253, 4,  -2032,  2056, -10168, 10264,  -44752,  24736, -109648,
   9 -> 23 |  1,  -509, 4,  -4080,  4104, -20408, 20504,  -89808,  49312, -220240,
  10 -> 29 |  1, -1021, 4,  -8176,  8200, -40888, 40984, -179920,  98464, -441424,
  11 -> 31 |  1, -2045, 4, -16368, 16392, -81848, 81944, -360144, 196768, -883792,

Cf. A000079, A000203, A000225, A075708, A182944, A329610, A329644, A329890.
Rows 1-2: A329891, A329892 (from n>=1).
Column 1: A000012, Column 2: -A036563(n) (from n>=1), Column 3: A010709.

up_to = 105;
A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329637sq(n,k) = ((2^(n+k-1)) - (((2^n)-1) * A329890(k)));
A329637list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A329637sq(col,(a-(col-1))))); (v); };
v329637 = A329637list(up_to);
A329637(n) = v329637[n];

A(n, k) = A329644(A182944(n, k)).

A(n, k) = A000079(n+k-1) - (A000225(n) * A329890(k)).

A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 10, 1, 16, 2, 6, 1, 12, 1, 18, -3, 18, 1, 22, -4, 46, 4, 22, 1, 10, 1, 30, 14, 82, -2, 14, 1, 256, -12, 22, 1, 36, 1, 66, 8, 226, 1, 46, -12, 19, 8, 130, 1, 28, -19, 70, -12, 748, 1, 42, 1, 1362, 16, 22, 10, 42, 1, 214, 254, 40, 1, 38, 1, 3838, 10, 406, -10, 106, 1, 78, -12, 5458, 1, 26, -72, 12250, -348, 30, 1, 12

Offset: 1

Antti Karttunen, Jan 10 2019

sign

After a(1) = 0, the other zeros occur for k >= 1, at A005940(1+A000396(k)), which, provided no odd perfect numbers exist, is equal to A324201(k) = A062457(A000043(k)): 9, 125, 161051, 410338673, ..., etc.

There are 2321 negative terms among the first 10000 terms.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713.
Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732.
Cf. A329644 (Möbius transform).
Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654.

Array[2 # - If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 90] (* Michael De Vlieger, Apr 21 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));

from sympy import divisor_sigma, primepi, factorint
def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

a(n) = 2*A156552(n) - A323243(n).
a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).
a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n).
From Antti Karttunen, Mar 12 2019 & Nov 23 2019: (Start)
a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d).
A002487(a(n)) = A324115(n).
a(n) = A329638(n) - A329639(n).
a(n) = A329645(n) - A329646(n).
(End)

A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

Original entry on oeis.org

0, 1, 3, 3, 7, 2, 15, 4, 9, 5, 31, 3, 63, 2, 8, 16, 127, -1, 255, 4, 21, 16, 511, 8, 21, 20, 12, 27, 1023, 6, 2047, 8, 20, 48, 20, 20, 4095, 2, 78, 32, 8191, -6, 16383, 17, 9, 288, 32767, 8, 45, -3, 122, 45, 65535, 4, 53, 20, 270, 278, 131071, 2, 262143, 688, 12, 72, 56, 23, 524287, 125, 260, -8, 1048575, 20, 2097151, 260, 3, 363, 44, -7, 4194303

Offset: 1

Antti Karttunen, Mar 07 2019

sign

The first four zeros after a(1) occur at n = 192, 288, 3645, 6075.
There are 1562 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
----------------------------------------------------------------------------------
  A324201  divisors                             a(n) applied to each:         Sum
        9: [1, 3, 9]                         -> [0, 3, 9]                      12 = 2*6
      125: [1, 5, 25, 125]                   -> [0, 7, 21, 28]                 56 = 2*28
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 31, 93, 124, 496, 248]    992 = 2*496
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 127, 381, 508, 2032, 1016, 9144, 3048]  16256 = 2*8128
The second term (the first nonzero) of the latter list = A000668(n), and the sum is always twice the corresponding perfect number, which forces either it or at least many of its divisors to be present. For example, in the fourth case, although 8128 = A000396(4) itself is not present, we still have 127, 508, 1016 and 2032 in the list. See also A329644.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644.

Table[DivisorSum[n, MoebiusMu[n/#] If[# == 1, 0, DivisorSigma[1, Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]]] &], {n, 79}] (* Michael De Vlieger, Mar 11 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
memoA323243 = Map();
A323243(n) = if(1==n, 0, my(v); if(mapisdefined(memoA323243,n,&v),v, v=sigma(A156552(n)); mapput(memoA323243,n,v); (v)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));

a(n) = Sum_{d|n} A008683(n/d) * A323243(d).
a(A000040(n)) = A000225(n).
a(A001248(n)) = A173033(n) - A000225(n) = A068156(n) = 3*(2^n - 1).
a(2*A000040(n)) = A324549(n).
a(A002110(n)) = A324547(n).
a(n) = 2*A297112(n) - A329644(n), and for n > 1, a(n) = 2^A297113(n) - A329644(n). - Antti Karttunen, Dec 08 2019

A329641 a(n) = gcd(A329638(n), A329639(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 1, 1, 18, 1, 18, 1, 22, 1, 46, 1, 22, 1, 10, 1, 30, 14, 82, 2, 1, 1, 256, 2, 22, 1, 1, 1, 66, 1, 226, 1, 46, 1, 1, 8, 130, 1, 1, 1, 70, 2, 748, 1, 42, 1, 1362, 2, 2, 10, 42, 1, 214, 254, 4, 1, 1, 1, 3838, 5, 406, 2, 2, 1, 78, 1, 5458, 1, 26, 2, 12250, 2, 10, 1, 2, 1, 934

Offset: 1

Antti Karttunen, Nov 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A323243, A324201, A329610, A329638, A329639, A329640, A329644.

A323243(n) = if(1==n,0,sigma(A156552(n)));
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A329644(n) = sumdiv(n,d,moebius(n/d)*((2*A156552(d))-A323243(d)));
A329641(n) = { my(t=0,u=0); fordiv(n, d, if((d=A329644(d))>0, t +=d, u -= d)); gcd(u,t); };

a(n) = gcd(A329638(n), A329639(n)).

a(A324201(n)) = A329610(n).

A329642 a(n) = Sum_{d|n} [1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 11, 1, 16, 2, 6, 1, 13, 1, 19, -3, 18, 1, 24, -4, 46, 4, 23, 1, 13, 1, 30, 14, 82, -2, 18, 1, 256, -12, 24, 1, 39, 1, 67, 9, 226, 1, 52, -12, 20, 8, 131, 1, 28, -19, 72, -12, 748, 1, 53, 1, 1362, 17, 22, 10, 45, 1, 215, 254, 43, 1, 48, 1, 3838, 11, 407, -10, 109, 1, 84, -12, 5458, 1, 48, -72, 12250, -348, 32, 1, 18

Offset: 1

Antti Karttunen, Nov 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A008683, A156552, A323243, A323244, A329643, A329644.

Cf. A329645 (inverse Möbius transform).

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n,0,sigma(A156552(n)));
A329642(n) = sumdiv(n,d,(1==moebius(n/d))*((2*A156552(d))-A323243(d)));

a(n) = Sum_{d|n} [1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).

a(n) = A329643(n) + A329644(n).

A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 6, 0, 2, 2, 6, 0, 4, 0, 7, 2, 2, 0, 16, 1, 2, 0, 18, 0, 11, 0, 6, 2, 2, 2, 22, 0, 2, 2, 24, 0, 17, 0, 20, 2, 2, 0, 28, 1, 1, 2, 48, 0, 16, 2, 28, 2, 2, 0, 39, 0, 2, -3, 30, 2, 36, 0, 84, 2, 19, 0, 36, 0, 2, -2, 258, 2, 38, 0, 28, 4, 2, 0, 69, 2, 2, 2, 72, 0, 31, 2, 228, 2, 2, 2, 76, 0, 4, 14, 37, 0, 94, 0, 136, -3

Offset: 1

Antti Karttunen, Nov 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A006881, A008683, A156552, A323243, A323244, A329642, A329644.

Cf. A329646 (inverse Möbius transform).

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n,0,sigma(A156552(n)));
A329643(n) = sumdiv(n,d,(-1==moebius(n/d))*((2*A156552(d))-A323243(d)));

a(n) = Sum_{d|n} [-1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).
a(n) = A329642(n) - A329644(n).
For all n, a(A000040(n)) = 0, a(A006881(n)) = 2.

A329891 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^n - (sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, 1, 4, 0, 24, -8, 104, -48, 352, 80, 1424, -2480, 8736, 2048, 16384, -7008, 111456, -80384, 473600, -427008, 1630976, 750592, 5731232, -12709664, 36894720, 12300416, 68246368, -38345568, 459223232, -401666240, 2015330304, -862152384, 5535523520, 4631692288, 21015756800, -61319782400, 165674113600, 46426506688, 279934140416, -484569911296

Offset: 0

Antti Karttunen, Nov 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..255
Index entries for sequences related to sigma(n)

Row 1 of A329637.

Cf. A000203, A000225, A075708, A156552, A323243, A323244, A329644, A329890, A329892.

A329891(n) = if(n<=1,n,(2^n - sigma((2^n)-1)) + sigma((2^(n-1))-1));

a(0) = 0; for n >= 1, a(n) = A323244(2^n) - A323244(2^(n-1)) = 2^n - A329890(n).

a(n) = A329644(2^n).

A329892 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^(n+1) - 3*(sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, -1, 4, -16, 40, -88, 184, -400, 544, -784, 2224, -11536, 18016, -10240, 16384, -86560, 203296, -503296, 896512, -2329600, 2795776, -1942528, 8805088, -54906208, 77129728, -30207616, 70521376, -383472160, 840798784, -2278740544, 3898507264, -6881424448, 8016635968, -3284792320, 28687532032, -252678823936, 359583387328, -135598386880

Offset: 0

Antti Karttunen, Nov 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..255
Index entries for sequences related to sigma(n)

Row 2 of A329637.

Cf. A000203, A000225, A075708, A156552, A323243, A329644, A329890, A329891.

A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329892(n) = if(!n,n,2^(n+1) - 3*A329890(n));

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n, 0, sigma(A156552(n)));
A329644(n) = sumdiv(n, d, moebius(n/d)*((2*A156552(d))-A323243(d)));
A329892(n) = A329644(3^n);

a(n) = A329644(3^n).

a(0) = 0; for n >= 1, a(n) = 2^(n+1) - 3*A329890(n).

cp's OEIS Frontend

A329638 Sum of A329644(d) for all such divisors d of n for which that value is positive. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

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A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

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Python

Formula

A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

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Mathematica

PARI

Formula

A329641 a(n) = gcd(A329638(n), A329639(n)).

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PARI

Formula

A329642 a(n) = Sum_{d|n} [1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

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PARI

Formula

A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

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PARI

Formula

A329891 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^n - (sigma((2^n)-1) - sigma((2^(n-1))-1)).