A347452 -id:A347452 - cp's OEIS frontend

A028983 Numbers whose sum of divisors is even.

3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Patrick De Geest

The even terms of this sequence are the even terms appearing in A178910. [Edited by M. F. Hasler, Oct 02 2014]
A071324(a(n)) is even. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152678(n). - Jaroslav Krizek, Oct 06 2009
A083207 is a subsequence. - Reinhard Zumkeller, Jul 19 2010
Numbers k such that the number of odd divisors of k (A001227) is even. - Omar E. Pol, Apr 04 2016
Numbers k such that the sum of odd divisors of k (A000593) is even. - Omar E. Pol, Jul 05 2016
Numbers with a squarefree part greater than 2. - Peter Munn, Apr 26 2020
Equivalently, numbers whose odd part is nonsquare. Compare with the numbers whose square part is even (i.e., nonodd): these are the positive multiples of 4, A008586\{0}, and A225546 provides a self-inverse bijection between the two sets. - Peter Munn, Jul 19 2020
Also numbers whose reversed prime indices have alternating product > 1, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). Also Heinz numbers of the partitions counted by A347448. - Gus Wiseman, Oct 29 2021
Numbers whose number of middle divisors is not odd (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

The complement is A028982 = A000290 U A001105.
Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.
Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.
Cf. A334748 (a permutation).
Related to A008586 via A225546.
Ranks the partitions counted by A347448, complement A119620.
Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.

Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* Jayanta Basu, Jun 05 2013 *)

is(n)=!issquare(n)&&!issquare(n/2) \\ Charles R Greathouse IV, Jan 11 2013

from math import isqrt
def A028983(n):
    def f(x): return n-1+isqrt(x)+isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

a(n) ~ n. - Charles R Greathouse IV, Jan 11 2013
a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - Charles R Greathouse IV, Sep 01 2015
A007913(a(n)) > 2. - Peter Munn, May 05 2020

A119620 Number of partitions of floor(3n/2) into n parts each from {1,2,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 15, 15, 22, 22, 30, 30, 42, 42, 56, 56, 77, 77, 101, 101, 135, 135, 176, 176, 231, 231, 297, 297, 385, 385, 490, 490, 627, 627, 792, 792, 1002, 1002, 1255, 1255, 1575, 1575, 1958, 1958, 2436, 2436, 3010, 3010, 3718, 3718

Offset: 0

John W. Layman, Jun 07 2006

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.
a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006
Number of partitions of n where all parts (except for possibly the first part) are even; see example. - Joerg Arndt, Apr 22 2013
For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p.  For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014
From Gus Wiseman, Oct 28 2021: (Start)
If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.
Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:
  11  111  22    221    33      331      44        441        55
           1111  11111  2211    22111    2222      22221      3322
                        111111  1111111  3311      33111      4411
                                         221111    2211111    222211
                                         11111111  111111111  331111
                                                              22111111
                                                              1111111111
These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.
(End)

			For n=8, floor(3*n/2) is 12 and there are five partitions of 12 into 8 parts each in the range 1-8 inclusive, namely: {5,1,1,1,1,1,1,1}, {4,2,1,1,1,1,1,1}, {3,3,1,1,1,1,1,1}, {3,2,2,1,1,1,1,1} and {2,2,2,2,1,1,1,1}. Thus a(8)=5.
From _Joerg Arndt_, Apr 22 2013: (Start)
a(8) = a(9) = 5, counting the following partitions where all parts (except for possibly the first part) are even:
01:  [ 2 2 2 2 ]
02:  [ 4 2 2 ]
03:  [ 4 4 ]
04:  [ 6 2 ]
05:  [ 8 ]
and
01:  [ 3 2 2 2 ]
02:  [ 5 2 2 ]
03:  [ 5 4 ]
04:  [ 7 2 ]
05:  [ 9 ]
(End)
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + ...

Cf. A008641, A008635.
Both bisections are A000041.
An adjoint version is A108711.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
Cf. A000070, A001700, A028983, A086543, A182616, A236559, A236914, A344654, A347444, A347452.

# Using the function EULER from Transforms (see link at the bottom of the page).
[1, op(EULER([1,0,seq(irem(n,2),n=2..55)]))]; # Peter Luschny, Aug 19 2020

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := f[n] = Length@ Select[ Partitions[ Floor[3n/2], n], Length@# == n &]; Table[ If[n > 1, f[2Floor[n/2]], f[n]], {n, 57}] (* Robert G. Wilson v, Jun 09 2006 *)
Table[ PartitionsP[ Floor[n/2]], {n, 57}] (* Robert G. Wilson v, Jun 09 2006 *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Ceiling[n/2]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 + x) / QPochhammer[x^2], {x, 0, n}]; (* Michael Somos, Mar 01 2014 *)

a(n)=numbpart(n\2); \\ Joerg Arndt, Apr 22 2013

a(n) = A000041(floor(n/2)). - Vladeta Jovovic, Jun 10 2006

G.f.: (Sum_{n>=0} x^(4*n) / Product_{k=1..n} (1-x^(2*k))) / (1 - x). - Michael Somos, Mar 01 2014 [corrected by Jason Yuen, Jan 24 2025]

More terms from Robert G. Wilson v, Jun 09 2006

Added a(0)=1. - Michael Somos, Mar 01 2014

A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

Original entry on oeis.org

1, 3, 6, 9, 10, 18, 20, 36, 40, 54, 56, 60, 108, 112, 120, 216, 224, 240, 324, 336, 352, 360, 400, 648, 672, 704, 720, 800, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2240, 2400, 3328, 3888, 4032, 4224, 4320, 4480, 4800, 6656, 7776, 8064, 8448

Offset: 1

Gus Wiseman, Nov 05 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
    1: {}
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   18: {1,2,2}
   20: {1,1,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}

The partitions with these as Heinz numbers are counted by A108711.
An adjoint version is A347452, counted by A119620.
The unrounded version is A348384, counted by A035377.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices, reverse A344616.
A344606 counts alternating permutations of prime factors.
Cf. A001105, A028982, A028260, A119899, A316413, A346703, A346704.

Select[Range[1000],Floor[2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/3]==PrimeOmega[#]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
isA348550(n) = (bigomega(n)==floor((2/3)*A056239(n))); \\ Antti Karttunen, Nov 08 2021

A001222(a(n)) = floor(2*A056239(a(n))/3).

A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

Original entry on oeis.org

1, 6, 36, 40, 216, 224, 240, 1296, 1344, 1408, 1440, 1600, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 34816, 39936, 46656, 48384, 50176, 50688, 51840, 53760, 56320, 57600, 64000, 155648, 208896, 239616, 266240, 279936, 290304, 301056, 304128, 311040, 315392

Offset: 1

Gus Wiseman, Nov 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.

			The terms and their prime indices begin:
     1: {}
     6: {1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   216: {1,1,1,2,2,2}
   224: {1,1,1,1,1,4}
   240: {1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  1344: {1,1,1,1,1,1,2,4}
  1408: {1,1,1,1,1,1,1,5}
  1440: {1,1,1,1,1,2,2,3}
  1600: {1,1,1,1,1,1,3,3}
  6656: {1,1,1,1,1,1,1,1,1,6}
  7776: {1,1,1,1,1,2,2,2,2,2}

David A. Corneth, Table of n, a(n) for n = 1..10000

These partitions are counted by A035377.
Rounding down gives A348550 or A347452, counted by A108711 or A119620.
A000041 counts integer partitions.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
Cf. A000070, A000097, A028260, A028982, A032766, A236914, A316413, A347457, A348551.

Select[Range[1000],2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
isA348384(n) = (A056239(n)==(3/2)*bigomega(n)); \\ Antti Karttunen, Nov 22 2021

The sequence contains n iff A056239(n) = 3*A001222(n)/2. Here, A056239 adds up prime indices, while A001222 counts them with multiplicity.

Intersection of A028260 and A347452.

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A028983 Numbers whose sum of divisors is even.

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A119620 Number of partitions of floor(3n/2) into n parts each from {1,2,...,n}.

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A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

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A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

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