A167171 -id:A167171 - cp's OEIS frontend

A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A000701 at a(11) = 28, A000701(11) = 27
A vertical section is a partial Young diagram with at most one square in each row.
Conjecture: a(n) is the number of non-half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(2) = 1 through a(9) = 14 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the non-half-loop-graphical partitions up to n = 9:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (331)  (71)    (81)
                               (421)  (422)   (432)
                               (511)  (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (4311)
                                              (5211)
                                              (6111)
For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
  {{1,1},{1,2},{1,3},{2,2}}
  {{1},{2},{1,1},{1,2},{2,3}}
  {{1},{2},{1,1},{1,3},{2,2}}
  {{1},{3},{1,1},{1,2},{2,2}}
None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

The complement is counted by A321729.
Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339655.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 counts graphical partitions of 2n into k parts.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is zero, where m is monomial and e is elementary symmetric functions.

a(n) = A000041(n) - A321729(n).

A349801 Number of integer partitions of n into three or more parts or into two equal parts.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 18, 25, 37, 50, 71, 94, 128, 168, 223, 288, 376, 480, 617, 781, 991, 1243, 1563, 1945, 2423, 2996, 3704, 4550, 5589, 6826, 8333, 10126, 12293, 14865, 17959, 21618, 25996, 31165, 37318, 44562, 53153, 63239, 75153, 89111, 105535, 124730

Offset: 0

Gus Wiseman, Dec 23 2021

nonn

This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:
- The non-strict case is A047967, weak A349796, weak complement A349795.
- The complement is counted by A065033(n) = ceiling(n/2) for n > 0.
- These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.
- The version for compositions is A345192, ranked by A345168.
- The weak version for compositions is A349053, ranked by A349057.
- The weak version is A349061, complement A349060, ranked by A349794.

			The a(2) = 1 through a(7) = 11 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)
               (211)   (311)    (222)     (331)
               (1111)  (2111)   (321)     (421)
                       (11111)  (411)     (511)
                                (2211)    (2221)
                                (3111)    (3211)
                                (21111)   (4111)
                                (111111)  (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

A000041 counts partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A004250 counts partitions into three or more parts, strict A347548.
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A096441 counts weakly alternating 0-appended partitions.
A345165 counts partitions w/ no alternating permutation, complement A345170.
Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,x_}|{,,__}]&]],{n,0,10}]

a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).

A325399 Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A167171 in having 70. First differs from A325398 in lacking 42.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325393.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A056239, A112798, A320466, A320470, A325358, A325391, A325393, A325396, A325397, A325398, A325400, A325405, A325457, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A046682 at a(11) = 28, A046682(11) = 29.
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3
Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(1) = 1 through a(8) = 12 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the half-loop-graphical partitions up to n = 8:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (2221)     (2222)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(7).

The complement is counted by A321728.
Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339656.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 is a triangle counting graphical partitions by length.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]]>0&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is nonzero, where m is monomial symmetric functions and e is elementary symmetric functions.

a(n) = A000041(n) - A321728(n).

A289553 Numbers that are not the product of two distinct noncomposite numbers (A008578).

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 125, 126, 128, 130, 132

Offset: 1

Jaroslav Krizek, Jul 07 2017

nonn

Numbers k such that tau(k) = A000005(k) is not equal to 2*omega(k) = 2*A001221(k).

			Number 6 is not in the sequence because 6 = 2 * 3.

Complement of A167171.

Cf. A246716 (with two distinct primes), A008578 (with two distinct nonprimes), A289555 (with two distinct composites).

[n: n in [1..1000] | NumberOfDivisors(n) ne 2*#PrimeDivisors(n)];

A339843 Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.

Original entry on oeis.org

1, 1, 3, 9, 29, 97, 336, 1188, 4275, 15579, 57358, 212908, 795657, 2990221, 11291665, 42814783, 162920417, 621885767, 2380348729

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as half-loop-graphical partitions (A321729). An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops or edges;
(2) n can be factored into distinct primes or squarefree semiprimes;
(3) the prime signature of n is half-loop-graphical.

			The a(0) = 1 through a(3) = 9 sorted degree sequences:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (2,1,1)
           (2,2)  (2,2,1)
                  (2,2,2)
                  (3,1,1)
                  (3,2,1)
                  (3,2,2)
                  (3,3,2)
                  (3,3,3)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(3).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
The version for simple graphs is A004251, covering: A095268.
The non-covering version (it allows isolated vertices) is A029889.
The same partitions counted by sum are conjectured to be A321729.
These graphs are counted by A006125 shifted left, covering: A322661.
The version for full loops is A339844, covering: A339845.
These graphs are ranked by A340018 and A340019.
A006125 counts labeled simple graphs, covering: A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A339659 counts graphical partitions of 2n into k parts.
Cf. A062740, A096373, A167171, A320461, A320893, A320921, A320923, A338915, A339560, A339841, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n]&]]],{n,0,5}]

a(n) = A029889(n) - A029889(n-1) for n > 0. - Andrew Howroyd, Jan 10 2024

a(7)-a(18) added (using A029889) by Andrew Howroyd, Jan 10 2024

A339002 Numbers of the form prime(x) * prime(y) where x and y are distinct and have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 65, 87, 91, 111, 115, 129, 133, 159, 183, 185, 203, 213, 235, 237, 247, 259, 267, 299, 301, 303, 305, 319, 321, 339, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 543, 551, 553, 559, 565, 579, 597, 611, 623, 669, 685, 687

Offset: 1

Gus Wiseman, Nov 22 2020

nonn

			The sequence of terms together with their prime indices begins:
     21: {2,4}     235: {3,15}    393: {2,32}
     39: {2,6}     237: {2,22}    417: {2,34}
     57: {2,8}     247: {6,8}     427: {4,18}
     65: {3,6}     259: {4,12}    445: {3,24}
     87: {2,10}    267: {2,24}    453: {2,36}
     91: {4,6}     299: {6,9}     481: {6,12}
    111: {2,12}    301: {4,14}    489: {2,38}
    115: {3,9}     303: {2,26}    497: {4,20}
    129: {2,14}    305: {3,18}    515: {3,27}
    133: {4,8}     319: {5,10}    517: {5,15}
    159: {2,16}    321: {2,28}    519: {2,40}
    183: {2,18}    339: {2,30}    543: {2,42}
    185: {3,12}    365: {3,21}    551: {8,10}
    203: {4,10}    371: {4,16}    553: {4,22}
    213: {2,20}    377: {6,10}    559: {6,14}

A300912 is the complement in A001358.
A338909 is the not necessarily squarefree version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A339005 lists products of pairs of distinct primes of divisible index.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338910/A338911 list products of pairs of primes both of odd/even index.
A339003/A339004 list squarefree semiprimes of odd/even index.
Cf. A001221, A001222, A055684, A056239, A112798, A115392, A166237, A167171, A318990, A320892, A320911, A338901.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

A350352 Products of three or more distinct prime numbers.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Gus Wiseman, Jan 11 2022

nonn

First differs from A336568 in lacking 420.

			The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]>=3&]

is(n,f=factor(n))=my(e=f[,2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022

list(lim)=my(v=List()); forsquarefree(n=30,lim\1, if(#n[2][,2]>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Oct 27 2013

nonn

Dirichlet convolution of A080339(n) with itself, where A080339 = characteristic function of noncomposite numbers (A008578).
Dirichlet convolution of functions b(n) and c(n) is function a(n) = Sum_{d|n} b(d) * c(n/d).
a(n) = 0, 1 or 2. a(n) = 0 for numbers n from A033942 (numbers with least 3 prime factors (counted with multiplicity)); a(n) = 1 for n = p^2, p = prime; a(n) = 2 for numbers n from A167171 (A006881 union A000040).

			For n = 6: a(6) = Sum_(d|6) A080339(d) * A080339(6/d) = 1*0 + 1*1 + 1*1 + 0*1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A080339, A033942, A167171, A006881, A000040, A230595.

a[n_] := Switch[FactorInteger[n][[;;, 2]], {2}, 1, {1}, 2, {1, 1}, 2, , 0]; a[1] = 1; Array[a, 100] (* _Amiram Eldar, Aug 29 2023 *)

A080339(n) = ((1==n)||isprime(n));
A230594(n) = sumdiv(n,d,(A080339(d)*A080339(n/d))); \\ Antti Karttunen, Jul 27 2017

from sympy import factorint
def A230594(n): return 0 if sum(f:=factorint(n).values())>2 else (1 if n==1 or max(f)==2 else 2) # Chai Wah Wu, Jul 23 2024

a(n) = Sum_{d|n} A080339(d) * A080339(n/d).

A230851 Numbers with divisors which are half odious (A000069) and half evil (A001969).

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 23, 24, 29, 33, 34, 39, 40, 43, 46, 48, 53, 57, 58, 63, 65, 66, 68, 69, 71, 78, 80, 83, 86, 87, 89, 92, 95, 96, 101, 105, 106, 111, 113, 114, 115, 116, 117, 119, 123, 125, 126, 130, 132, 136, 138, 139, 141, 142, 145, 149, 156, 160, 163, 166, 171, 172, 174, 177, 178, 183

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2013

nonn

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

Cf. A000069, A001969, A167171, A227872.

aQ[n_] := DivisorSum[n, (-1)^DigitCount[#, 2][[1]] &] == 0; Select[Range[200], aQ] (* Amiram Eldar, Sep 23 2019 *)

is(n)=!sumdiv(n,d,(-1)^hammingweight(d)) \\ Charles R Greathouse IV, Oct 31 2013

Numbers n such that d(n) = 2*A227872(n) where A227872(n) is number of odious divisors of n.

Corrected by Charles R Greathouse IV, Oct 31 2013

cp's OEIS Frontend

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