A048166 -id:A048166 - cp's OEIS frontend

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

A023534 Numbers k such that the largest power of 2 dividing k equals 2^omega(k).

Original entry on oeis.org

1, 2, 12, 20, 28, 36, 44, 52, 68, 76, 92, 100, 108, 116, 120, 124, 148, 164, 168, 172, 188, 196, 212, 236, 244, 264, 268, 280, 284, 292, 312, 316, 324, 332, 356, 360, 388, 404, 408, 412, 428, 436, 440, 452, 456, 484, 500, 504, 508, 520, 524, 548, 552, 556

Offset: 1

Benoit Cloitre, Sep 04 2002

nonn

That is, numbers k such that A001221(k) is equal to A007814(k), where A001221(k) = omega(k) is the number of distinct primes dividing k, and A007814(k) is the exponent of the highest power of 2 dividing k.

Numbers k that are unitarily divisible by the number of unitary divisors of k (A034444), i.e., numbers k such that A034444(k) is a unitary divisor of k. Subsequence of A048166. - Amiram Eldar, Aug 15 2025

			omega(12) = 2 and 4 = 2^2 is the largest power of 2 dividing 12, hence 12 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Cf. A039700, A001221 (omega), A007814 (2-adic valuation), A034444, A048166.

Select[Range[600],IntegerExponent[#,2]==PrimeNu[#]&] (* Harvey P. Dale, Jun 26 2011 *)

isok(n) = omega(n) == valuation(n, 2); \\ Michel Marcus, Apr 16 2015

from itertools import count, islice
from sympy.ntheory.factor_ import primenu
def A023534_gen(startvalue=1): # generator of terms
    return filter(lambda n:(~n & n-1).bit_length() == primenu(n),count(max(startvalue,1)))
A023534_list = list(islice(A023534_gen(),30)) # Chai Wah Wu, Jul 05 2022

A048169 n is divisible by the cube of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 8, 16, 32, 64, 128, 192, 256, 320, 384, 448, 512, 576, 640, 704, 768, 832, 896, 1024, 1088, 1152, 1216, 1280, 1408, 1472, 1536, 1600, 1664, 1728, 1792, 1856, 1984, 2048, 2176, 2304, 2368, 2432, 2560, 2624, 2752, 2816, 2944, 3008, 3072, 3136, 3200, 3328

Offset: 1

Labos Elemer

nonn

			a[ 54 ]=x=3968=128*31 has 4 unitary divisors: {1,3968,128,31} and the cube, 64, divides 3968: q=3968/64=62.

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

A034444, A046755, A048166-A048168.

Select[Range[3400], Divisible[#, 8^PrimeNu[#]] &] (* Amiram Eldar, Aug 05 2019 *)

A048481 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048472.

Original entry on oeis.org

1, 3, 9, 27, 77, 207, 529, 1299, 3093, 7191, 16409, 36891, 81949, 180255, 393249, 852003, 1835045, 3932199, 8388649, 17825835, 37748781, 79691823, 167772209, 352321587, 738197557, 1543503927, 3221225529, 6710886459, 13958643773, 28991029311, 60129542209

Offset: 0

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Partial sums of A048495.

Cf. A134397, A048166.

LinearRecurrence[{6,-13,12,-4},{1,3,9,27},40] (* Harvey P. Dale, Aug 13 2015 *)

Vec((4*x^2-3*x+1)/((x-1)^2*(2*x-1)^2) + O(x^100)) \\ Colin Barker, Dec 04 2014

Row sums of triangle A134397. Also, binomial transform of A048166. - Gary W. Adamson, Oct 23 2007
a(n) = 6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4). - Colin Barker, Dec 04 2014
G.f.: (4*x^2-3*x+1) / ((x-1)^2*(2*x-1)^2). - Colin Barker, Dec 04 2014
a(n) = 2^(n+1)*(n-2) + 2*n + 5. - Christian Krause, Oct 31 2023

Corrected by T. D. Noe, Nov 08 2006

A351851 Numbers that are divisible by the number of their divisors over the Gaussian integers.

Original entry on oeis.org

1, 6, 9, 18, 20, 56, 126, 168, 180, 198, 280, 342, 352, 414, 432, 441, 486, 504, 558, 616, 625, 728, 774, 832, 846, 952, 1056, 1062, 1064, 1089, 1176, 1206, 1278, 1288, 1422, 1494, 1512, 1624, 1736, 1760, 1848, 1854, 1920, 1926, 2025, 2072, 2160, 2286, 2296, 2358

Offset: 1

Amiram Eldar, Feb 22 2022

nonn

Numbers k such that A062327(k) | k.

All the odd terms are squares.

			6 is a term since it is divisible by A062327(6) = 6.

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

Cf. A033950, A048166, A062327, A351853.

Select[Range[2400], Divisible[#, DivisorSigma[0, #, GaussianIntegers -> True]] &]

A351853 Numbers that are divisible by the number of their divisors over the Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 6, 8, 24, 40, 80, 81, 88, 120, 128, 136, 162, 180, 184, 225, 232, 240, 264, 324, 328, 360, 376, 384, 408, 424, 448, 450, 472, 552, 560, 568, 625, 640, 648, 664, 696, 712, 756, 808, 856, 880, 896, 900, 904, 984, 1040, 1048, 1096, 1128, 1192, 1250, 1272

Offset: 1

Amiram Eldar, Feb 22 2022

nonn

Numbers k such that A319442(k) | k.

All the odd terms are squares or numbers of the form 3 times a square.

			6 is a term since it is divisible by A319442(6) = 6.

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

Cf. A033950, A048166, A319442, A351851.

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, eisNumDiv[n]]; Select[Range[1000], q]

A048167 Integer quotients of numbers k divisible by the number of unitary divisors of k (A034444).

Original entry on oeis.org

1, 1, 2, 4, 3, 8, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 32, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 15, 31, 64, 34, 36, 37, 38, 40, 41, 21, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 30, 61, 62, 128, 33, 67, 68, 35, 71, 72, 73, 74, 76, 39, 79, 80, 81, 82, 83, 42

Offset: 1

Labos Elemer

nonn

			1008 = 2*2*2*3*3*7 has 8 unitary divisors {1,1008,7,144,9,112,16,63}, 8 divides 1008 and the quotient is 126, a term in the sequence.

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

Cf. A034444, A036762, A000005, A048166.

f[n_] := n / 2^PrimeNu[n]; f /@ Select[Range[340], IntegerQ[f[#]] &] (* Amiram Eldar, Jul 16 2019 *)

a(n) = A048166(n)/A034444(A048166(n)). - Amiram Eldar, Jul 16 2019

A048168 n is divisible by the square of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 4, 8, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 256, 272, 288, 304, 320, 352, 368, 384, 400, 416, 432, 448, 464, 496, 512, 544, 576, 592, 608, 640, 656, 688, 704, 736, 752, 768, 784, 800, 832, 848, 864, 896, 928, 944, 960, 976, 992

Offset: 1

Labos Elemer

nonn

			x=2032=16*127 has 4 unitary divisors: {1,16,127,2032} and ud[ x ]^2=16 divides 2032.

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

A034444, A046754, A048166.

Select[Range[10^3], Divisible[#, 4^PrimeNu[#]] &] (* Amiram Eldar, Aug 05 2019 *)

A387056 Numbers k that are divisible by the number of infinitary divisors of k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 100, 104, 112, 116, 124, 128, 136, 144, 148, 152, 160, 164, 172, 176, 184, 188, 192, 196, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 268, 272, 284, 288, 292, 296

Offset: 1

Amiram Eldar, Aug 15 2025

First differs from A048166 at n = 27: A048166(27) = 108 is not a term of this sequence.

This sequence is infinite. For example, if p is a prime, then 8*p is a term.

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

A387057 is a subsequence.

Cf. A033950, A037445, A048166.

f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ f @@@ FactorInteger[n]; q[k_] := Divisible[k, id[k]]; Select[Range[300], q]

isok(k) = !(k % vecprod(apply(x -> 2^hammingweight(x), factor(k)[, 2])));

A048170 n is divisible by the 4th power of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 16, 32, 64, 128, 256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 4096, 4352, 4608, 4864, 5120, 5632, 5888, 6144, 6400, 6656, 6912, 7168, 7424, 7936, 8192, 8704, 9216, 9472, 9728, 10240, 10496, 11008, 11264, 11776

Offset: 1

Labos Elemer

nonn

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

A034444, A046756, A048166-A048169.

Select[Range[12000], Divisible[#, 16^PrimeNu[#]] &] (* Amiram Eldar, Aug 05 2019 *)

cp's OEIS Frontend

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

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A023534 Numbers k such that the largest power of 2 dividing k equals 2^omega(k).

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A048169 n is divisible by the cube of the number of unitary divisors of n (A034444).

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A048481 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048472.

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A351851 Numbers that are divisible by the number of their divisors over the Gaussian integers.

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A351853 Numbers that are divisible by the number of their divisors over the Eisenstein integers.

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A048167 Integer quotients of numbers k divisible by the number of unitary divisors of k (A034444).

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A048168 n is divisible by the square of the number of unitary divisors of n (A034444).

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