A129527 -id:A129527 - cp's OEIS frontend

A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 29, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 58, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 116, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 116, 84, 144

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A004709, A138302, A353900.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).

f[p_, e_] := p^e + Sum[p^(e - 2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}

Multiplicative with a(p^e) = p^e + Sum_{k=0..floor(log_2(e))} p^(e-2^k).		
a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .

A328407 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + x) / (1 - x)^3.

Original entry on oeis.org

1, 5, 9, 21, 25, 45, 49, 85, 81, 125, 121, 189, 169, 245, 225, 341, 289, 405, 361, 525, 441, 605, 529, 765, 625, 845, 729, 1029, 841, 1125, 961, 1365, 1089, 1445, 1225, 1701, 1369, 1805, 1521, 2125, 1681, 2205, 1849, 2541, 2025, 2645, 2209, 3069, 2401, 3125, 2601, 3549, 2809, 3645, 3025

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

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

Cf. A000290, A001511, A007434, A016754, A023871, A129527, A209229, A328408.

[n eq 1 select 1 else IsOdd(n) select n^2 else Self(n div 2)+n^2 :n in [1..55]]; // Marius A. Burtea, Oct 15 2019

nmax = 55; CoefficientList[Series[Sum[x^(2^k) (1 + x^(2^k))/(1 - x^(2^k))^3, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n^2, n^2]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^2 &], {n, 1, 55}]
f[p_, e_] := If[p == 2, (4^(e + 1) - 1)/3, p^(2*e)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

G.f.: Sum_{k>=0} x^(2^k) * (1 + x^(2^k)) / (1 - x^(2^k))^3.
G.f.: (1/3) * Sum_{k>=1} J_2(2*k) * x^k / (1 - x^k), where J_2() is the Jordan function (A007434).
Dirichlet g.f.: zeta(s-2) / (1 - 2^(-s)).
a(2*n) = a(n) + 4*n^2, a(2*n+1) = (2*n + 1)^2.
a(n) = Sum_{d|n} A209229(n/d) * d^2.
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023871.
Sum_{k=1..n} a(k) ~ 8*n^3/21. - Vaclav Kotesovec, Oct 15 2019
Multiplicative with a(2^e) = (4^(e+1)-1)/3, and a(p^e) = p^(2*e) for an odd prime p. - Amiram Eldar, Oct 25 2020

A373188 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^2.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 10, 9, 10, 11, 15, 13, 14, 15, 21, 17, 18, 19, 25, 21, 22, 23, 30, 25, 26, 27, 35, 29, 30, 31, 42, 33, 34, 35, 45, 37, 38, 39, 50, 41, 42, 43, 55, 45, 46, 47, 63, 49, 50, 51, 65, 53, 54, 55, 70, 57, 58, 59, 75, 61, 62, 63, 85, 65, 66, 67, 85, 69, 70, 71, 90, 73, 74, 75, 95, 77, 78, 79, 105, 81, 82, 83, 105, 85, 86

Offset: 1

Seiichi Manyama, May 27 2024

nonn

Cf. A373187, A373189.

Cf. A129527, A327625, A359100.

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^4).

a(4*n+1) = 4*n+1, a(4*n+2) = 4*n+2, a(4*n+3) = 4*n+3 and a(4*n+4) = 4*n+4 + a(n+1) for n >= 0.

A331739 a(n) is n minus its largest odd divisor.

Original entry on oeis.org

0, 1, 0, 3, 0, 3, 0, 7, 0, 5, 0, 9, 0, 7, 0, 15, 0, 9, 0, 15, 0, 11, 0, 21, 0, 13, 0, 21, 0, 15, 0, 31, 0, 17, 0, 27, 0, 19, 0, 35, 0, 21, 0, 33, 0, 23, 0, 45, 0, 25, 0, 39, 0, 27, 0, 49, 0, 29, 0, 45, 0, 31, 0, 63, 0, 33, 0, 51, 0, 35, 0, 63, 0, 37, 0, 57, 0, 39, 0, 75, 0, 41, 0, 63, 0, 43, 0, 77, 0, 45, 0, 69, 0, 47

Offset: 1

Antti Karttunen, Feb 02 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000265, A006519, A007814, A135481.

Cf. A129527 (even bisection).

A331739 := proc(n)
    n-A000265(n) ;
end proc:
seq(A331739(n),n=1..40) ; # R. J. Mathar, Jan 24 2022

a[n_] := n - n/2^IntegerExponent[n, 2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 19 2023 *)

PARI
```
A331739(n) = (n-(n>>valuation(n,2)));
```

def A331739(n): return n-(n>>(~n & n-1).bit_length()) # Chai Wah Wu, Jul 01 2022

a(n) = n - A000265(n).

Sum_{k=1..n} a(k) ~ n^2/6. - Amiram Eldar, Sep 13 2024

A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 8, 4, 2, 1, 9, 10, 5, 11, 12, 6, 3, 13, 14, 7, 15, 16, 8, 4, 2, 1, 17, 18, 9, 19, 20, 10, 5, 21, 22, 11, 23, 24, 12, 6, 3, 25, 26, 13, 27, 28, 14, 7, 29, 30, 15, 31, 32, 16, 8, 4, 2, 1, 33, 34, 17, 35, 36, 18, 9, 37, 38, 19, 39, 40, 20, 10, 5, 41, 42, 21

Offset: 1

Franklin T. Adams-Watters, Oct 13 2006

A fractal sequence, generated by the rule a(n) is a new maximum when a(n-1) is odd and a repetition of an earlier value when a(n-1) is even.
From Flávio V. Fernandes, Mar 13 2025: (Start)
a(n) is given by A003602(n) at A001511(n) diagram
  1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9
  . 1 . . . 2 . . . 3 . . . 4 . . .
  . . . 1 . . . . . . . 2 . . . . .
  . . . . . . . 1 . . . . . . . . .
  . . . . . . . . . . . . . . . 1 .
read by backwards 2^n, which is given by A118319(n) at A001511(n) diagram
  1 . 2 . 4 . 5 . 8 . 9 .11 .12 .16
  . 3 . . . 6 . . .10 . . .13 . . .
  . . . 7 . . . . . . .14 . . . . .
  . . . . . . .15 . . . . . . . . .
  . . . . . . . . . . . . . . .31 . - see formula. (End)

			Triangle starts
  1;
  2, 1;
  3;
  4, 2, 1;
  5;
  6, 3;
  7;
  8, 4, 2, 1;
  9;
  10, 5;
  11;
  12, 6, 3;
  13;

Alois P. Heinz, Rows n = 1..10000, flattened

Row lengths are A001511.
Row sums give A129527.
Cf. A120385.
Cf. A003602, A108918, A118319.

T:= proc(n) local m,l; m:=n; l:= m;
      while irem(m, 2, 'm')=0 do l:=l,m od: l
    end:
seq(T(n), n=1..40);  # Alois P. Heinz, Oct 09 2015

Flatten[Function[n, NestWhile[Append[#, Last[#]/2] &, {n}, EvenQ[Last[#]] &]][#] & /@ Range[20]] (* Birkas Gyorgy, Apr 13 2011 *)

a(1) = 1, for n > 1, if a(n-1) is even, a(n) = a(n-1)/2, otherwise a(n) = (max_{k

Ordinal transform of A082850.

a(n) = A003602(A108918(n)). - Flávio V. Fernandes, Mar 13 2025

A129265 Triangle read by rows: T(n,k) is the number of power of two divisors of n that are less than or equal to n/k.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 06 2007

Equals A115361 * A000012 as infinite lower triangular matrices (cf. A129264).

			First few rows of the triangle are:
  1;
  2, 1;
  1, 1, 1;
  3, 2, 1, 1;
  1, 1, 1, 1, 1;
  2, 2, 2, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A129527.
Column 1 is A001511.
Cf. A115361, A000012, A129264.

T(n, k)={sumdiv(n, d, d <= n/k && d==1<Andrew Howroyd, Aug 07 2018

T(n,k) = 1 for n odd.

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 07 2018

A129559 A054523 * A115361.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 4, 2, 0, 1, 4, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 8, 4, 0, 0, 2, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 4, 4, 2, 0, 2, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

row sums = A129527: (1, 3, 3, 7, 5, 9, 7, 15, ...). Left column = phi(2*n), A062570: (1, 2, 2, 4, 4, 4, 6, 8, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 0, 1;
  4, 2, 0, 1;
  4, 0, 0, 0, 1;
  4, 2, 2, 0, 0, 1;
  6, 0, 0, 0, 0, 0, 1;
  8, 4, 0, 2, 0, 0, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A062570.
Row sums are A129527 (inverse Moebius transform of A062570).
Cf. A054523, A115361.

T(n, k)=if(n%k, 0, eulerphi(2*n/k)) \\ Andrew Howroyd, Aug 07 2018

Equals A054523 * A115361 as infinite lower triangular matrices.

T(n,k) = phi(2*n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 07 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 07 2018

A373187 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^4.

Original entry on oeis.org

1, 4, 10, 21, 35, 56, 84, 124, 165, 220, 286, 374, 455, 560, 680, 837, 969, 1140, 1330, 1575, 1771, 2024, 2300, 2656, 2925, 3276, 3654, 4144, 4495, 4960, 5456, 6108, 6545, 7140, 7770, 8601, 9139, 9880, 10660, 11700, 12341, 13244, 14190, 15466, 16215, 17296, 18424

Offset: 1

Seiichi Manyama, May 27 2024

nonn

Cf. A129527, A373186.
Cf. A373188, A373189.
Cf. A000292.

G.f. A(x) satisfies A(x) = x/(1 - x)^4 + A(x^4).

a(4*n+1) = A000292(4*n+1), a(4*n+2) = A000292(4*n+2), a(4*n+3) = A000292(4*n+3) and a(4*n+4) = A000292(4*n+4) + a(n+1) for n >= 0.

A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

Original entry on oeis.org

1, 9, 27, 73, 125, 243, 343, 585, 729, 1125, 1331, 1971, 2197, 3087, 3375, 4681, 4913, 6561, 6859, 9125, 9261, 11979, 12167, 15795, 15625, 19773, 19683, 25039, 24389, 30375, 29791, 37449, 35937, 44217, 42875, 53217, 50653, 61731, 59319, 73125, 68921, 83349, 79507, 97163, 91125

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

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

Cf. A000578, A001511, A016755, A023872, A059376, A129527, A209229, A328407.

[n eq 1 select 1 else IsOdd(n) select n^3 else Self(n div 2)+n^3 :n in [1..45]]; // Marius A. Burtea, Oct 15 2019

nmax = 45; CoefficientList[Series[Sum[x^(2^k) (1 + 4 x^(2^k) + x^(2^(k + 1)))/(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n^3, n^3]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^3 &], {n, 1, 45}]
f[p_, e_] :=p^(3*e); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

G.f.: Sum_{k>=0} x^(2^k) * (1 + 4*x^(2^k) + x^(2^(k+1))) / (1 - x^(2^k))^4.
G.f.: (1/7) * Sum_{k>=1} J_3(2*k) * x^k / (1 - x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) / (1 - 2^(-s)).
a(2*n) = a(n) + 8*n^3, a(2*n+1) = (2*n + 1)^3.
a(n) = Sum_{d|n} A209229(n/d) * d^3.
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023872.
Sum_{k=1..n} a(k) ~ 4*n^4/15. - Vaclav Kotesovec, Oct 15 2019
Multiplicative with a(2^e) = (8^(e+1)-1)/7, and a(p^e) = p^(3*e) for an odd prime p. - Amiram Eldar, Oct 23 2023

A338045 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^3.

Original entry on oeis.org

1, 4, 6, 14, 15, 27, 28, 50, 45, 70, 66, 105, 91, 133, 120, 186, 153, 216, 190, 280, 231, 319, 276, 405, 325, 442, 378, 539, 435, 585, 496, 714, 561, 748, 630, 882, 703, 931, 780, 1100, 861, 1134, 946, 1309, 1035, 1357, 1128, 1581, 1225, 1600, 1326, 1820, 1431, 1863, 1540

Offset: 1

Ilya Gutkovskiy, Oct 08 2020

nonn

Cf. A000217, A000294, A000384, A001511, A014105, A129527, A195094, A209229, A328407, A338046.

nmax = 55; CoefficientList[Series[Sum[x^(2^k) /(1 - x^(2^k))^3, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n (n + 1)/2, n (n + 1)/2]; Table[a[n], {n, 1, 55}]
Table[(1/2) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) &], {n, 1, 55}]

G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^3.
a(2*n) = a(n) + A014105(n), a(2*n+1) = A000384(n+1).
a(n) = (1/2) * Sum_{d|n} A209229(n/d) * d * (d + 1).
a(n) = Sum_{d|n} A195094(d).
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000294.

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A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

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A328407 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + x) / (1 - x)^3.

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A373188 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^2.

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A331739 a(n) is n minus its largest odd divisor.

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A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.

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A129265 Triangle read by rows: T(n,k) is the number of power of two divisors of n that are less than or equal to n/k.

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A129559 A054523 * A115361.

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A373187 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^4.

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