A026424 -id:A026424 - cp's OEIS frontend

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Gus Wiseman, Jan 22 2023

nonn

First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.

These partitions are counted by A240219, strict A359897.
The LHS (mean of prime indices) is A326567/A326568.
The complement is A359890, counted by A359894.
The odd-length case is A359891, complement A359892, counted by A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359893 and A359901 count partitions by median, odd-length A359902.
A359908 lists numbers whose prime indices have integer median.
Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||Mean[prix[#]]==Median[prix[#]]&]

Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Characteristic function of A028260 (positions of 1's). Cf. also A026424 (positions of 0's) and A320655.
One less than A007421.
Cf. A003961, A008836, A010052, A038548 (inverse Möbius transform), A046523, A055037 (partial sums), A343784, A347102, A353337, A353338, A353555, A353557, A353629, A353669, A358750, A358752, A353374, A358775, A356163, A356170.
Cf. also A066829, A353374.

A065043 := proc(n)
    if type(numtheory[bigomega](n),'even') then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 26 2013

Table[(LiouvilleLambda[n]+1)/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)

{ for (n=1, 1000, a=1 - bigomega(n)%2; write("b065043.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

A065043(n) = (1 - (bigomega(n)%2)); \\ Antti Karttunen, Apr 19 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A065043(n): return (reduce(ixor, factorint(n).values(),0)&1)^1 # Chai Wah Wu, Jan 01 2023

a(n) = 1 - A001222(n) mod 2.
a(n) = A007421(n) - 1.
a(n) = 1 - A066829(n).
a(A028260(k)) = 1 and a(A026424(k)) = 0 for all k.
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012
a(n) = (A008836(n) + 1)/2. - Enrique Pérez Herrero, Jul 07 2012
a(n) = A001222(2n) mod 2. - Wesley Ivan Hurt, Jun 22 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^2)/(1 - x^n). - Ilya Gutkovskiy, Apr 25 2017
From Antti Karttunen, Dec 01 2022: (Start)
For x, y >= 1, a(x*y) = 1 - abs(a(x)-a(y)).
a(n) = a(A046523(n)) = A356163(A003961(n)).
a(n) = A000035(A356163(n)+A347102(n)).
a(n) = A010052(n) + A353669(n).
a(n) = A353555(n) + A353557(n).
a(n) = A358750(n) + A358752(n).
a(n) = A353374(n) + A358775(n).
a(n) >= A356170(n).
(End)

Corrected by Charles R Greathouse IV, Sep 02 2009

A063655 Smallest semiperimeter of integral rectangle with area n.

Original entry on oeis.org

2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 14, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 16, 18, 17, 68, 21, 26

Offset: 1

Floor van Lamoen, Jul 24 2001

nonn

Similar to A027709, which is minimal perimeter of polyomino of n cells, or equivalently, minimal perimeter of rectangle of area at least n and with integer sides. Present sequence is minimal semiperimeter of rectangle with area exactly n and with integer sides. - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 03 2002
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n. That is, b = largest divisor of n <= sqrt(n), d = smallest divisor of n >= sqrt(n). a(n) = n+1 iff n is noncomposite (1 or prime). - Daniel Forgues, Nov 22 2009
From Juhani Heino, Feb 05 2019: (Start)
Basis for any thickness "frames" around the minimal area. Perimeter can be thought as the 0-thick frame, it is obviously 2a(n). Thickness 1 is achieved by laying unit tiles around the area, there are 2(a(n)+2) of them. Thickness 2 comes from the second such layer, now there are 4(a(n)+4) and so on. They all depend only on a(n), so they share this structure:
Every n > 1 is included. (For different thicknesses, every integer that can be derived from these with the respective formula. So, the perimeter has every even n > 2.)
For each square n > 1, a(n) = a(n-1).
a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
(End)
Gives a discrete Uncertainty Principle. A complex function on an abelian group of order n and its Discrete Fourier Transform must have at least a(n) nonzero entries between them. This bound is achieved by the indicator function on a subgroup of size closest to sqrt(n). - Oscar Cunningham, Oct 10 2021
Also two times the median divisor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The version for mean instead of median is A057020/A057021. Other doubled medians of multisets are: A360005 (prime indices), A360457 (distinct prime indices), A360458 (distinct prime factors), A360459 (prime factors), A360460 (prime multiplicities), A360555 (0-prepended differences). - Gus Wiseman, Mar 18 2023

			Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Roy Meshulam, An uncertainty inequality for finite abelian groups, arXiv:math/0312407 [math.CO], 2003.
Roy Meshulam, An uncertainty inequality for finite abelian groups, European Journal of Combinatorics, 27 (2006) 63-67.

Cf. A027709, A033676, A033677, A092510, A162348.
Positions of odd terms are A139710.
Positions of even terms are A139711.
A000005 counts divisors, listed by A027750.
A000975 counts subsets with integer median.
Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.

A063655 := proc(n)
    local i,j;
    for i from floor(sqrt(n)) to 1 by -1 do
        j := floor(n/i) ;
        if i*j = n then
            return i+j;
        end if;
    end do:
end proc:
seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002

Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 2*Sqrt[n], d[[len/2]] + d[[1 + len/2]]], {n, 100}] (* T. D. Noe, Mar 06 2012 *)
Table[2*Median[Divisors[n]],{n,100}] (* Gus Wiseman, Mar 18 2023 *)

A063655(n) = { my(c=1); fordiv(n,d,if((d*d)>=n,if((d*d)==n,return(2*d),return(c+d))); c=d); (0); }; \\ Antti Karttunen, Oct 20 2017

from sympy import divisors
def A063655(n):
    d = divisors(n)
    l = len(d)
    return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n-1) + A162348(2n). - Daniel Forgues, Sep 29 2014
a(n) = Min_{d|n} (n/d + d). - Ridouane Oudra, Mar 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001

A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 23, 25, 30, 31, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Equally, numbers n for which A061395(n) is odd.
A122111 maps each one of these numbers to a unique term of A026424 and vice versa.
If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - Gus Wiseman, Feb 08 2021

			From _Gus Wiseman_, Feb 08 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}           32: {1,1,1,1,1}     64: {1,1,1,1,1,1}
      4: {1,1}         33: {2,5}           66: {1,2,5}
      5: {3}           34: {1,7}           67: {19}
      8: {1,1,1}       40: {1,1,1,3}       68: {1,1,7}
     10: {1,3}         41: {13}            69: {2,9}
     11: {5}           44: {1,1,5}         73: {21}
     15: {2,3}         45: {2,2,3}         75: {2,3,3}
     16: {1,1,1,1}     46: {1,9}           77: {4,5}
     17: {7}           47: {15}            80: {1,1,1,1,3}
     20: {1,1,3}       50: {1,3,3}         82: {1,13}
     22: {1,5}         51: {2,7}           83: {23}
     23: {9}           55: {3,5}           85: {3,7}
     25: {3,3}         59: {17}            88: {1,1,1,5}
     30: {1,2,3}       60: {1,1,2,3}       90: {1,2,2,3}
     31: {11}          62: {1,11}          92: {1,1,9}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A244990.
Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.
Looking at least instead of greatest prime index gives A026804.
The partitions with these Heinz numbers are counted by A027193.
The case where Omega is odd also is A340386.
A001222 counts prime factors.
A056239 adds up prime indices.
A300063 ranks partitions of odd numbers.
A061395 selects maximum prime index.
A066208 ranks partitions into odd parts.
A112798 lists the prime indices of each positive integer.
A340931 ranks odd-length partitions of odd numbers.
Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]]&] (* Gus Wiseman, Feb 08 2021 *)

For all n, A244989(a(n)) = n.

A360457 Two times the median of the set of distinct prime indices of n; a(1) = 1.

Original entry on oeis.org

1, 2, 4, 2, 6, 3, 8, 2, 4, 4, 10, 3, 12, 5, 5, 2, 14, 3, 16, 4, 6, 6, 18, 3, 6, 7, 4, 5, 20, 4, 22, 2, 7, 8, 7, 3, 24, 9, 8, 4, 26, 4, 28, 6, 5, 10, 30, 3, 8, 4, 9, 7, 32, 3, 8, 5, 10, 11, 34, 4, 36, 12, 6, 2, 9, 4, 38, 8, 11, 6, 40, 3, 42, 13, 5, 9, 9, 4, 44, 4

Offset: 1

Gus Wiseman, Feb 14 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Distinct prime indices are listed by A304038.

			The prime indices of 65 are {3,6}, with distinct parts {3,6}, with median 9/2, so a(65) = 9.
The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so a(900) = 4.

The version for divisors is A063655.
For mean instead of two times median we have A326619/A326620.
The version for all prime indices is A360005.
Positions of first appearances are A360006, sorted A360007.
The version for distinct prime factors is A360458.
The version for all prime factors is A360459.
The version for prime multiplicities is A360460.
Positions of even terms are A360550.
Positions of odd terms are A360551.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A359907, A359908, A359912, A360248, A360453.

Table[If[n==1,1,2*Median[PrimePi/@First/@FactorInteger[n]]],{n,100}]

A069288 Number of odd divisors of n <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

a(n) = #{d : d = A182469(n,k), d <= A000196(n), k=1..A001227(n)}. - Reinhard Zumkeller, Apr 05 2015

			From _Gus Wiseman_, Feb 11 2021: (Start)
The inferior odd divisors for selected n are the columns below:
n: 1    9   30   90  225  315  630  945 1575 2835 4410 3465 8190 6930
  --------------------------------------------------------------------
   1    3    5    9   15   15   21   27   35   45   63   55   65   77
        1    3    5    9    9   15   21   25   35   49   45   63   63
             1    3    5    7    9   15   21   27   45   35   45   55
                  1    3    5    7    9   15   21   35   33   39   45
                       1    3    5    7    9   15   21   21   35   35
                            1    3    5    7    9   15   15   21   33
                                 1    3    5    7    9   11   15   21
                                      1    3    5    7    9   13   15
                                           1    3    5    7    9   11
                                                1    3    5    7    9
                                                     1    3    5    7
                                                          1    3    5
                                                               1    3
                                                                    1
(End)

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

Cf. A000005, A000196, A001227, A069289, A182469.
Positions of first appearances are A334853.
A055396 selects the least prime index.
A061395 selects the greatest prime index.
- Odd -
A000009 counts partitions into odd parts (A066208).
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A067659 counts strict partitions of odd length (A030059).
- Inferior divisors -
A033676 selects the greatest inferior divisor.
A033677 selects the least superior divisor.
A038548 counts inferior divisors.
A060775 selects the greatest strictly inferior divisor.
A063538 lists numbers with a superior prime divisor.
A063539 lists numbers without a superior prime divisor.
A063962 counts inferior prime divisors.
A064052 lists numbers with a properly superior prime divisor.
A140271 selects the least properly superior divisor.
A217581 selects the greatest inferior divisor.
A333806 counts strictly inferior prime divisors.
Cf. A001055, A244991, A300272, A340101, A340607, A340832, A340854/A340855.

a069288 n = length $ takeWhile (<= a000196 n) $ a182469_row n
-- Reinhard Zumkeller, Apr 05 2015

odn[n_]:=Count[Divisors[n],?(OddQ[#]&&#<=Sqrt[n ]&)]; Array[odn,100] (* _Harvey P. Dale, Nov 04 2017 *)

a(n) = my(ir = sqrtint(n)); sumdiv(n, d, (d % 2) * (d <= ir)); \\ Michel Marcus, Jan 14 2014

G.f.: Sum_{n>=1} 1/(1-q^(2*n-1)) * q^((2*n-1)^2). [Joerg Arndt, Mar 04 2010]

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)

The complement is counted by A240219.
These partitions are ranked by A359890, complement A359889.
The odd-length case is ranked by A359892, complement A359891.
The odd-length case is A359896, complement A359895.
The strict case is A359898, complement A359897.
The odd-length strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284 and A058398 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
A359909 counts factorizations with the same mean as median, odd-len A359910.
Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Median[#]&]],{n,0,30}]

A066829 Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 1

G. L. Honaker, Jr., Jan 17 2002

From Reinhard Zumkeller, Jul 01 2009: (Start)
The first N Terms are constructed by the following sieving process:
for j:=1 until N do a(j):=0,
for i:=1 until N/2 do
for j:=2*i step i until N do a(j):=1-a(i). (End)
Omega is also written in the OEIS as bigomega. See also comments, references and formulas in A008836 (Liouville's lambda), A007421 and A065043, that all contain the same information as this sequence. - Antti Karttunen, Apr 30 2022

			From _Reinhard Zumkeller_, Jul 01 2009: (Start)
Sieve for N = 30, also demonstrating the affinity to the Sieve of Eratosthenes:
[initial] a(j):=0, 1<=j<=30:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[i=1] a(1)=0 --> a(j):=1, 2<=j<=30:
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[i=2] a(2)=1 --> a(2*j):=0, 2<=j<=[30/2]:
0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
[i=3] a(3)=1 --> a(3*j):=0, 2<=j<=[30/3]:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0
[i=4] a(4)=0 --> a(4*j):=1, 2<=j<=[30/4]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0
[i=5] a(5)=1 --> a(5*j):=0, 2<=j<=[30/5]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0
[i=6] a(6)=0 --> a(6*j):=1, 2<=j<=[30/6]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1
[i=7] a(7)=1 --> a(7*j):=0, 2<=j<=[30/7]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1
[i=8] a(8)=1 --> a(8*j):=0, 2<=j<=[30/8]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1
[i=9] a(9)=0 --> a(9*j):=1, 2<=j<=[30/9]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1
[i=10] a(10)=0 --> a(10*j):=1, 2<=j<=[30/10]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1
and so on: a(22):=0 in [i=11], a(24):=0 in [i=12], a(26):=0 in [i=13], a(28):=1 in [i=14], and a(30):=1 in [i=15]. (End)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21 (provides Dirichlet g.f.)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences generated by sieves

Characteristic function of A026424 (positions of 1's). Cf. also A028260 (its complement, positions of 0's).
Cf. A001222 (bigomega), A007421, A008836, A055038 (partial sums), A065043, A069545 (run lengths), A072203, A349905, A353556, A353558, A358751, A358753.
Cf. A000035.

a066829 = (`mod` 2) . a001222 -- Reinhard Zumkeller, Nov 19 2011

A066829 := proc(n)
    modp(numtheory[bigomega](n) ,2) ;
end proc:
seq(A066829(n),n=1..80) ; # R. J. Mathar, Jul 15 2017

Table[(1-LiouvilleLambda[n])/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)
Table[If[OddQ[PrimeOmega[n]],1,0],{n,120}] (* Harvey P. Dale, Mar 12 2016 *)

A066829(n) = (bigomega(n)%2); \\ Simplified by Antti Karttunen, Apr 30 2022

from sympy import primeomega as Omega
def a(n): return Omega(n)%2
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Apr 30 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A066829(n): return reduce(ixor, factorint(n).values(),0)&1 # Chai Wah Wu, Jan 01 2023

a(A026424(n)) = 1; a(A028260(n)) = 0.
Dirichlet g.f.: (zeta(s)^2 - zeta(2*s)) / (2*zeta(s)). [Typo corrected by Vaclav Kotesovec, Jan 30 2024]
a(n) = (1-A008836(n)) / 2. - Corrected by Antti Karttunen, Apr 30 2022
a(m*n) = a(m) XOR a(n). - Reinhard Zumkeller, Aug 28 2008
a(n) = A001222(n) mod 2. - Reinhard Zumkeller, Nov 19 2011
From Antti Karttunen, May 01 & Nov 30 2022: (Start)
a(n) = 1 - A065043(n) = A349905(n) mod 2.
a(n) = A353556(n) + A353558(n).
a(n) = A358751(n) + A358753(n). (End)
a(n) = A000035(A001222(n)). - Omar E. Pol, Apr 09 2025

Corrected and comment added by Reinhard Zumkeller, Jun 26 2009

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
The strict case (for n > 2) is A067659 (odd bisection: A344650).
The Heinz numbers of these partitions are A119899 (complement: A344609).
The bisections are A236914 (odd) and A344743 (even).
The ordered version appears to be A294175 (even bisection: A008549).
The complement is counted by A344607 (even bisection: A344611).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with alternating sum <= 0, ranked by A028260.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30}]

A359912 Numbers whose prime indices do not have integer median.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 60, 65, 69, 74, 77, 84, 86, 93, 95, 106, 119, 122, 123, 132, 141, 142, 143, 145, 150, 156, 158, 161, 177, 178, 185, 196, 201, 202, 204, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 276, 278

Offset: 1

Gus Wiseman, Jan 24 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
   1: {}
   6: {1,2}
  14: {1,4}
  15: {2,3}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  51: {2,7}
  58: {1,10}
  60: {1,1,2,3}

For prime factors instead of indices we have A072978, complement A359913.
These partitions are counted by A307683.
For mean instead of median: A348551, complement A316413, counted by A349156.
The complement is A359908, counted by A325347.
Positions of odd terms in A360005.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A051293, A067538, A175352, A175761, A289509, A359890, A359905, A360006, A359907, A360009.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Median[prix[#]]]&]

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A359889 Numbers that are 1 or whose prime indices have the same mean as median.

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A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

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A063655 Smallest semiperimeter of integral rectangle with area n.

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A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

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A360457 Two times the median of the set of distinct prime indices of n; a(1) = 1.

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A069288 Number of odd divisors of n <= sqrt(n).

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A359894 Number of integer partitions of n whose parts do not have the same mean as median.

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