A344650 -id:A344650 - cp's OEIS frontend

A345961 Numbers whose prime indices have reverse-alternating sum 2.

3, 10, 12, 21, 27, 30, 40, 48, 55, 70, 75, 84, 90, 91, 108, 120, 147, 154, 160, 187, 189, 192, 210, 220, 243, 247, 250, 270, 280, 286, 300, 336, 360, 363, 364, 391, 432, 442, 462, 480, 490, 495, 507, 525, 551, 588, 616, 630, 640, 646, 675, 713, 748, 750, 756

Offset: 1

Gus Wiseman, Jul 12 2021

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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly two odd conjugate prime indices. The restriction to odd omega is A345960, and the restriction to even omega is A345962.

			The initial terms and their prime indices:
    3: {2}
   10: {1,3}
   12: {1,1,2}
   21: {2,4}
   27: {2,2,2}
   30: {1,2,3}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   55: {3,5}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   91: {4,6}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}

Below we use k to indicate reverse-alternating sum.
The k > 0 version is A000037.
These multisets are counted by A000097.
The k = 0 version is A000290, counted by A000041.
These partitions are counted by A120452 (negative: A344741).
These are the positions of 2's in A344616.
The k = -1 version is A345912.
The k = 1 version is A345958.
The unreversed version is A345960 (negative: A345962).
A000070 counts partitions with alternating sum 1.
A002054/A345924/A345923 count/rank compositions with alternating sum -2.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices,  row sums of A112798.
A088218/A345925/A345922 count/rank compositions with alternating sum 2.
A088218 also counts compositions with alternating sum 0, ranked by A344619.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices.
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000984, A001791, A025047, A027193, A239830, A341446, A344611, A344650, A344651, A344743, A345910, A345911, A345918.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==2&]

A343942 Number of even-length strict integer partitions of 2n+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 96, 128, 170, 224, 292, 380, 491, 630, 805, 1024, 1295, 1632, 2049, 2560, 3189, 3959, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29249, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937, 172928

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

By conjugation, also the number of integer partitions of 2n+1 covering an initial interval of positive integers with greatest part even.

			The a(1) = 1 through a(7) = 13 strict partitions:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)      (7,6)      (8,7)
         (4,1)  (5,2)  (6,3)  (7,4)      (8,5)      (9,6)
                (6,1)  (7,2)  (8,3)      (9,4)      (10,5)
                       (8,1)  (9,2)      (10,3)     (11,4)
                              (10,1)     (11,2)     (12,3)
                              (5,3,2,1)  (12,1)     (13,2)
                                         (5,4,3,1)  (14,1)
                                         (6,4,2,1)  (6,4,3,2)
                                         (7,3,2,1)  (6,5,3,1)
                                                    (7,4,3,1)
                                                    (7,5,2,1)
                                                    (8,4,2,1)
                                                    (9,3,2,1)

Ranked by A005117 (strict), A028260 (even length), and A300063 (odd sum).
Odd bisection of A067661 (non-strict: A027187).
The non-strict version is A236914.
The opposite type of strict partition (odd length and even sum) is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A030229, A035294, A067659, A236559, A338907, A343941, A344649, A344654, A344739.

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,15}]

The Heinz numbers are A005117 /\ A028260 /\ A300063.

More terms from Bert Dobbelaere, Jun 12 2021

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

More terms from Alois P. Heinz, Aug 05 2021

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A345961 Numbers whose prime indices have reverse-alternating sum 2.

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A343942 Number of even-length strict integer partitions of 2n+1.

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A346634 Number of strict odd-length integer partitions of 2n + 1.

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