A034729 -id:A034729 - cp's OEIS frontend

A323764 Dirichlet self-convolution of the integer partition numbers A000041.

1, 1, 4, 6, 14, 14, 34, 30, 64, 69, 112, 112, 228, 202, 330, 394, 575, 594, 956, 980, 1492, 1674, 2228, 2510, 3700, 3965, 5276, 6200, 8126, 9130, 12318, 13684, 17842, 20622, 25808, 29976, 38377, 43274, 53990, 62976, 77912, 89166, 110656, 126522, 154918, 179744

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of multiset partitions of constant multiset partitions of integer partitions of n.

			The a(4) = 14 multiset partitions of constant multiset partitions:
  ((1111))              ((22))      ((4))  ((31))  ((211))
  ((11)(11))            ((2)(2))
  ((11))((11))          ((2))((2))
  ((1)(1)(1)(1))
  ((1))((1)(1)(1))
  ((1)(1))((1)(1))
  ((1))((1))((1)(1))
  ((1))((1))((1))((1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A000837, A001970, A002033, A003238, A034729, A047968, A279787, A305551, A306017, A319056, A323774.

Join[{1},Table[Sum[PartitionsP[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 1, 4, 11, 24, 57, 112, 244, 480, 1013, 1972, 4083, 8064, 16331, 32512, 65519, 130488, 262125, 523244, 1048377, 2095104, 4194281, 8384176, 16777136, 33546240, 67108096, 134201316, 268435427, 536836584, 1073741793, 2147418112, 4294964213, 8589803520, 17179868787, 34359470272, 68719476699, 137438429184, 274877894643

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Dirichlet convolution of this sequence with phi (A000010) is A000740, with sigma (A000203) it is A034729, and with A018804 it is A034738.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A011782, A055615, A349569 (Dirichlet inverse).

Cf. also A000010, A000740, A000203, A018804, A034729, A034738, A349564, A349566, A349568.

a[n_] := DivisorSum[n, # * MoebiusMu[#] * 2^(n/# - 1) &]; Array[a, 40] (* Amiram Eldar, Nov 22 2021 *)

A055615(n) = (n*moebius(n));
A349570(n) = sumdiv(n,d,(2^(d-1)) * A055615(n/d));

a(n) = Sum_{d|n} 2^(d-1) * A055615(n/d).

A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 11, 5, 19, 1, 47, 1, 67, 21, 139, 1, 295, 1, 539, 69, 1027, 1, 2223, 17, 4099, 261, 8267, 1, 16951, 1, 32907, 1029, 65539, 81, 133423, 1, 262147, 4101, 524955, 1, 1056871, 1, 2098187, 16661, 4194307, 1, 8423599, 65, 16777747, 65541

Offset: 1

Morgan L. Owens, Nov 21 2014

nonn

a(n)==1 iff n is prime.
Apparently Moebius transform of A178472.
For n>1, the binary representation of a(n) is given by row (n-1) of A077049 (when read as a triangular array). - Tom Edgar, Nov 28 2014

Cf. A034729, A077049, A178472.

With[{n=Range[100]},(1/2) ((Total/@(2^Divisors[n])) - 2^n)]

a(n) = sumdiv(n, k, 2^(k-1)) - 2^(n-1); \\ Michel Marcus, Nov 25 2014

from sympy import divisors
def A247146(n): return sum(1<Chai Wah Wu, Jul 15 2022

a(n) = A034729(n) - 2^(n-1). - Michel Marcus, Nov 22 2014

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 16, 37, 66, 9, 128, 7, 256, 17, 10, 549, 1024, 67, 2048, 13, 18, 129, 4096, 39, 8200, 257, 16450, 21, 32768, 11, 65536, 131621, 130, 1025, 24, 71, 262144, 2049, 258, 45, 524288, 19, 1048576, 133, 74, 4097, 2097152, 551, 4194320, 8201

Offset: 1

Franklin T. Adams-Watters, Mar 06 2015

nonn

			The prime power divisors of 12 are 2, 3, and 4. These are indices 1, 2, and 3 in the list of prime powers, so a(12) = 2^(1-1) + 2^(2-1) + 2^(3-1) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row
-- Reinhard Zumkeller, Mar 07 2015

al(n) = my(r=vector(n),pps=[p| p <- [1..n], isprimepower(p)],p2); for(k=1,#pps,p2=2^(k-1);forstep(j=pps[k],n,pps[k],r[j]+=p2));r

Additive with a(p^k) = Sum_{j=1..k} 2^(A065515(p^j)-1).
a(A051451(k)) = 2^k - 1.
a(n) = Sum_{k=1..A073093(n)} 2^(A095874(A210208(n,k))-2). - Reinhard Zumkeller, Mar 07 2015

A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 3, 5, 9, 10, 22, 20, 37, 44, 65, 68, 127, 119, 182, 226, 307, 335, 511, 544, 782, 913, 1171, 1359, 1908, 2121, 2738, 3286, 4174, 4821, 6305, 7182, 9108, 10739, 13195, 15548, 19465, 22397, 27477, 32423, 39448, 45843, 55995, 64871, 78343, 91761, 109325

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of strict multiset partitions of constant multiset partitions of integer partitions of n.

			The a(1) = 1 through a(5) = 10 strict multiset partitions of constant multiset partitions of integer partitions:
  ((1))  ((2))     ((3))          ((4))             ((5))
         ((11))    ((21))         ((31))            ((41))
         ((1)(1))  ((111))        ((22))            ((32))
                   ((1)(1)(1))    ((211))           ((311))
                   ((1))((1)(1))  ((1111))          ((221))
                                  ((2)(2))          ((2111))
                                  ((11)(11))        ((11111))
                                  ((1)(1)(1)(1))    ((1)(1)(1)(1)(1))
                                  ((1))((1)(1)(1))  ((1))((1)(1)(1)(1))
                                                    ((1)(1))((1)(1)(1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A001970, A034729, A047968, A050343, A316980, A319066, A323764, A323766, A323774.

Join[{1}, Table[Sum[PartitionsQ[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A143425 Triangle read by rows A051731 * A130123, 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 0, 4, 1, 2, 0, 8, 1, 0, 0, 0, 16, 1, 2, 4, 0, 0, 32, 1, 0, 0, 0, 0, 0, 64, 1, 2, 0, 8, 0, 0, 0, 128, 1, 0, 4, 0, 0, 0, 0, 0, 256, 1, 2, 0, 0, 16, 0, 0, 0, 0, 512, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 1, 2, 4, 8, 0, 32, 0, 0, 0, 0, 0, 2048

Offset: 1

Gary W. Adamson, Aug 14 2008

Row sums = A034729: (1, 3, 5, 11, 17, ...).

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 4;
  1, 2, 0, 8;
  1, 0, 0, 0, 16;
  1, 2, 4, 0,  0, 32;
  ...

Cf. A034729, A051731, A130123.

tabl(nn) = {ma = matrix(nn, nn, n, k, !(n%k)); mb = matrix(nn, nn, n, k, n--; k--; if (n==k, 2^n, 0)); m = ma*mb; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Jun 30 2017

Triangle read by rows A051731 * A130123, 1<=k<=n, where A130123 = an infinite lower triangular matrix with (1, 2, 4, 8, ...) in the main diagonal and the rest zeros. A051731 = the inverse Mobius transform.

Typo in data corrected by Michel Marcus, Jun 30 2017

A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 10, 4, 18, 0, 46, 0, 66, 20, 138, 0, 294, 0, 538, 68, 1026, 0, 2222, 16, 4098, 260, 8266, 0, 16950, 0, 32906, 1028, 65538, 80, 133422, 0, 262146, 4100, 524954, 0, 1056870, 0, 2098186, 16660, 4194306, 0, 8423598, 64, 16777746, 65540

Offset: 1

Gus Wiseman, Oct 15 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The nontrivial divisors of 18 are {2, 3, 6, 9}, so a(18) = 2^1 + 2^2 + 2^5 + 2^8 = 294.

Removing zeros gives binary indices of rows of A163870.
The version for all divisors is A034729.
The version for proper divisors is A247146.
Cf. A000005, A000120, A027750, A029931, A033676, A048793, A060775, A070824, A070939, A275700, A326031.

Table[Total[(2^DeleteCases[Divisors[n],1|n])/2],{n,100}]

from sympy import divisors
def A328337(n): return sum(1<<(d-1) for d in divisors(n,generator=True) if 1Chai Wah Wu, Jul 15 2022

A000120(a(n)) = A070824(n).
A070939(a(n)) = A032742(n).
A001511(a(n)) = A107286(n).

A336997 a(n) = n! * Sum_{d|n} 2^(d - 1) / d!.

Original entry on oeis.org

1, 4, 10, 56, 136, 1952, 5104, 94208, 605056, 7741952, 39917824, 1458295808, 6227024896, 175463616512, 2353813878784, 48886264659968, 355687428161536, 17362063156969472, 121645100409094144, 6001501553433509888, 85800344155030552576, 2248030289949388439552

Offset: 1

Ilya Gutkovskiy, Aug 10 2020

nonn

Cf. A010842, A034729, A057625, A336998.

Table[n! Sum[2^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[(Exp[2 x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sumdiv(n, d, 2^(d-1)/d!); \\ Michel Marcus, Aug 12 2020

E.g.f.: Sum_{k>=1} (exp(2*x^k) - 1) / 2.

a(p) = p! + 2^(p - 1), where p is prime.

A360303 a(n) = Sum_{k=1..floor(sqrt(n))} 2^floor(n/k-k).

Original entry on oeis.org

0, 1, 2, 4, 9, 17, 34, 66, 132, 261, 521, 1033, 2066, 4114, 8226, 16420, 32837, 65605, 131209, 262281, 524554, 1048850, 2097682, 4194834, 8389668, 16778277, 33556517, 67110981, 134221897, 268439625, 536879242, 1073750154, 2147500178, 4294983954, 8589967634, 17179902228, 34359804453

Offset: 0

Luc Rousseau, Feb 02 2023

This sequence corresponds to the left half of a drawing, the whole drawing being reconstituted by symmetry (see the Illustration link). The divisors of n are closely related to the occurrences of the bit pattern "01 over 10" in the 2 X 2 squares along the (n-1)th and n-th lines (see the pattern link). In particular, n is a prime number if and only if a(n) - a(n-1) = 2^(n-2).

			For n = 5, floor(sqrt(n)) = 2. So, two bits are set in a(n); they are the bits number floor(5/1-1)=4 and floor(5/2-2)=0, so a(n) = 10001_2 = 17.

Luc Rousseau, Illustration in black and white, n = 0..100.
Luc Rousseau, Illustrated bit pattern for the detection of divisors, n = 1..9.

Cf. A034729.

a(n)=sum(k=1,floor(sqrt(n)),2^floor(n/k-k))

a(n) = Sum_{k=1..floor(sqrt(n))} 2^floor(n/k - k).

cp's OEIS Frontend

A323764 Dirichlet self-convolution of the integer partition numbers A000041.

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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

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A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

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A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

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A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

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A143425 Triangle read by rows A051731 * A130123, 1<=k<=n.

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A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

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