A139352 -id:A139352 - cp's OEIS frontend

A139373 Let the binary expansion of n be n = Sum_{k} 2^{r_k}, let e(n) be the number of r_k's that are even, o(n) the number that are odd; sequence lists n such that e(n) > o(n).

1, 4, 5, 7, 13, 16, 17, 19, 20, 21, 22, 23, 25, 28, 29, 31, 37, 49, 52, 53, 55, 61, 64, 65, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 100, 101, 103, 109, 112, 113, 115, 116, 117, 118, 119, 121, 124

Offset: 1

Nadia Heninger and N. J. A. Sloane, Jun 07 2008

nonn

e(n)+o(n) = A000120(n), the binary weight of n. For e(n) = o(n) see A039004.

Cf. A000120, A139351, A139352, A139353, A139354, A139355.

Cf. A039004, A139370, A139371, A139372.

Fortran
```
c See link in A139351
```

aQ[n_] := Module[{d = Reverse[IntegerDigits[n,2]]}, Total@d[[1;;-1;;2]] > Total@d[[2;;-1;;2]]]; Select[Range[180], aQ] (* Amiram Eldar, Dec 15 2018 *)

isok(n) = {my(irb = Vec(select(x->(x%2), Vecrev(binary(n)), 1))); #select(x->(x%2), irb) > #irb/2;} \\ Michel Marcus, Dec 15 2018

A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Oct 05 2023

First differs from A101436 at n = 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050376, A064547, A101436, A139352, A366242, A366243, A366246.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e>3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A139352(e).
a(n) = A064547(n) - A366246(n).
a(n) = A064547(A366245(n)).
a(n) >= 0, with equality if and only if n is in A366242.
a(n) <= A064547(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39310573826635831710..., where f(x) = Sum_{k>=0} (x^(2*4^k)/(1+x^(2*4^k))).

A037863 a(n) = (number of digits <=1) - (number of digits >1) in base 4 representation of n.

Original entry on oeis.org

1, -1, -1, 2, 2, 0, 0, 0, 0, -2, -2, 0, 0, -2, -2, 3, 3, 1, 1, 3, 3, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, -3, -3, -1, -1, -3, -3, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, -3, -3, -1, -1, -3, -3, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 0, 0

Offset: 1

Clark Kimberling

Cf. A110591, A139352.

A037863 := proc(n)
    a := 0 ;
    dgs := convert(n,base,4);
    for i from 1 to nops(dgs) do
        if op(i,dgs)<=1 then
            a := a+1 ;
        else
            a := a-1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

a(n) = my(d=digits(n, 4)); #select(x->(x<=1), d) - #select(x->(x>1), d); \\ Michel Marcus, Jan 27 2025

a(n) = A110591(n)-2*A139352(n). - R. J. Mathar, Jan 27 2025

A366309 The number of infinitary divisors of n that are terms of A366243.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 06 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037445, A139352, A366242, A366243, A366245, A366247, A366246, A366308.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := 2^s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e > 3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 2^A139352(e).
a(n) = 2^A366247(n).
a(n) = A037445(n)/A366308(n).
a(n) = A037445(A366245(n)).
a(n) >= 1, with equality if and only if n is in A366242.
a(n) <= A037445(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} 2^A139352(k)/p^k) = 1.44736831993091923328... .

A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

Original entry on oeis.org

2, 8, 1, 10, 3, 5, 32, 4, 7, 21, 34, 6, 13, 23, 85, 40, 9, 15, 29, 87, 341, 42, 11, 17, 31, 93, 343, 1365, 128, 12, 19, 53, 95, 349, 1367, 5461, 130, 14, 20, 55, 117, 351, 1373, 5463, 21845, 136, 16, 22, 61, 119, 373, 1375, 5469, 21847, 87381, 138, 18, 25, 63

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of even powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For odd powers, see A165275.
For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A062880, (Row 1)=A158705, (Column 1)=A002450, also possibly (Column 2)=A163832.

			Northwest corner:
2....8...10...32...34...40...42...129
1....3....4....6....9...11...12...14
5....7...13...15...17...19...20...22
21..23...29...31...53...55...61...63
Examples:
40 = 32 + 8 = 2^5 + 2^3, so that 40 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 2.

Cf. A139351, A139352, A165275, A165276, A165277, A165278, A165279.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

More terms from Amiram Eldar, Feb 04 2020

A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

Original entry on oeis.org

1, 4, 2, 5, 3, 10, 16, 6, 11, 42, 17, 7, 14, 43, 170, 20, 8, 15, 46, 171, 682, 21, 9, 26, 47, 174, 683, 2730, 64, 12, 27, 58, 175, 686, 2731, 10922, 65, 13, 30, 59, 186, 687, 2734, 10923, 43690, 68, 18, 31, 62, 187, 698, 2735, 10926, 43691, 174762, 69, 19, 34

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of odd powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For even powers, see A165274. For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
It appears that, for n>=3, a(t(n)) = 4*a(t(n-1))+2, where t(n) is the n-th triangular number t(n)=n(n+1)/2 (A000217). [John W. Layman, Sep 15 2009]

			Northwest corner:
  1....4....5...16...17...20...21...64
  2....3....6....7....8....9...12...13
  10..11...14...26...27...30...31...34
  42..43...46...47...58...59...62...63
Examples:
20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.

Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.

Cf. A000217.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

a(27) corrected and a(28)-a(54) added by John W. Layman, Sep 15 2009

More terms from Amiram Eldar, Feb 04 2020

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A139373 Let the binary expansion of n be n = Sum_{k} 2^{r_k}, let e(n) be the number of r_k's that are even, o(n) the number that are odd; sequence lists n such that e(n) > o(n).

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A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

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A037863 a(n) = (number of digits <=1) - (number of digits >1) in base 4 representation of n.

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A366309 The number of infinitary divisors of n that are terms of A366243.

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A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

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A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

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