A096765 -id:A096765 - cp's OEIS frontend

A352832 Number of reversed integer partitions y of n with exactly one fixed point y(i) = i.

0, 1, 1, 1, 4, 3, 7, 7, 14, 19, 24, 32, 46, 60, 85, 109, 140, 179, 239, 300, 397, 495, 636, 790, 995, 1239, 1547, 1926, 2396, 2942, 3643, 4432, 5435, 6602, 8038, 9752, 11842, 14292, 17261, 20714, 24884, 29733, 35576, 42375, 50522, 60061, 71363, 84551, 100101

Offset: 0

Gus Wiseman, Apr 08 2022

nonn

A reversed integer partition of n is a finite weakly increasing sequence of positive integers summing to n.

			The a(0) = 0 through a(8) = 14 partitions (empty column indicated by dot):
  .  (1)  (11)  (111)  (13)    (14)     (15)      (16)       (17)
                       (22)    (1112)   (114)     (115)      (116)
                       (112)   (11111)  (222)     (1123)     (134)
                       (1111)           (1113)    (11113)    (224)
                                        (1122)    (11122)    (233)
                                        (11112)   (111112)   (1115)
                                        (111111)  (1111111)  (2222)
                                                             (11114)
                                                             (11123)
                                                             (11222)
                                                             (111113)
                                                             (111122)
                                                             (1111112)
                                                             (11111111)
For example, the reversed partition (2,2,4) has a unique fixed point at the second position.

* = unproved
*The non-reverse version is A001522, ranked by A352827, strict A352829.
*The non-reverse complement is A064428, ranked by A352826, strict A352828.
This is column k = 1 of A238352.
For no fixed point: counted by A238394, ranked by A352830, strict A025147.
For > 0 fixed points: counted by A238395, ranked by A352872, strict A096765.
The version for compositions is A240736, complement A352520.
These partitions are ranked by A352831.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, nonfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A352822 counts fixed points of prime indices.
A352833 counts partitions by fixed points.
Cf. A064410, A177510, A257990, A325187, A351983, A352824, A352825.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Reverse/@IntegerPartitions[n],pq[#]==1&]],{n,0,30}]

A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 2, 2, 1, 3, 3, 6, 3, 6, 9, 20, 13, 22, 22, 45, 47, 70, 75, 100, 107, 132, 157, 202, 229, 302, 396, 495, 536, 699, 820, 962, 1193, 1507, 1699, 2064, 2455, 2945, 3408, 4026, 4691, 5749, 6670, 7614, 9127, 10930, 12329, 14370, 16955, 19961, 22950, 26574, 30941

Offset: 0

Gus Wiseman, May 16 2024

nonn

Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
The a(2) = 2 through a(10) = 9 partitions:
(2)   (21)   (31)  (221)    (51)    (421)      (431)   (441)     (91)
(11)  (111)        (2111)   (321)   (2221)     (521)   (3321)    (631)
                   (11111)  (3111)  (4111)     (5111)  (4221)    (721)
                                    (22111)            (33111)   (3331)
                                    (211111)           (42111)   (7111)
                                    (1111111)          (411111)  (32221)
                                                                 (322111)
                                                                 (3211111)
                                                                 (31111111)

For all positive integers (not just prime) we get A000041.
For even instead of prime we have A087787, strict A025147, odd A096765.
These partitions have Heinz numbers A277319.
The strict case is A372687, ranks A372851.
The version counting only distinct parts is A372887, ranks A372850.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15

Offset: 0

Peter Bala, Mar 30 2017

Compare with A284592.

			Square array begins
  n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009
  1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765
  2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744
  3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18
  4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19
  5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28
  6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33
  7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41
  ...
T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).

Alois P. Heinz, Antidiagonals n = 0..200, flattened
H. S. Wilf, Lectures on Integer Partitions

Rows n=0..2 give A000009, A096765, A015744.
Main diagonal gives A365662.
Antidiagonal sums give A032302.
Cf. A284592, A322210.

# A284593 as a square array
ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Aug 24 2019

nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
T[n_, k_] := M[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A032302.

A339162 Number of compositions (ordered partitions) of n into distinct parts, the least being 1.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 8, 8, 14, 14, 44, 44, 74, 98, 128, 272, 326, 470, 644, 932, 1106, 2234, 2552, 3800, 4958, 7070, 9068, 12140, 20042, 24674, 34256, 45632, 61814, 80630, 109316, 135572, 217778, 262298, 362744, 466664, 636494, 805454, 1085804, 1375388, 1776938, 2591762

Offset: 0

Ilya Gutkovskiy, Nov 25 2020

nonn

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Index entries for sequences related to compositions

Cf. A032020, A096765, A339163, A339164, A339165, A339166.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-1)*(i+2)/2 `if`(n<1, 0, b(n-1$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020

nmax = 45; CoefficientList[Series[Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k! * x^(k*(k + 1)/2) / Product_{j=1..k-1} (1 - x^j).

A096749 Number of partitions of n into distinct parts, the least being 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 59, 68, 78, 91, 103, 118, 136, 155, 176, 201, 228, 259, 294, 332, 375, 425, 478, 538, 607, 681, 764, 858, 961, 1075, 1203, 1343, 1499, 1673, 1863, 2073, 2308, 2564, 2847, 3161, 3504

Offset: 0

N. J. A. Sloane, Sep 28 2008

nonn

The old entry with this sequence number was a duplicate of A071569.

a(n), n>2 is the Euler transform of [0,0,1,1,1] joined with period [0,1]. - Georg Fischer, Aug 15 2020

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A025147, A071569.

Cf. A096765 (least=1), A022824 (3), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-2)*(i+3)/2 `if`(n<2, 0, b(n-2$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 07 2014
# Using the function EULER from Transforms (see link at the bottom of the page).
[0,0,1,op(EULER([0,0,1,1,seq(irem(n,2),n=1..57)]))]; # Peter Luschny, Aug 19 2020

b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-2)*(i+3)/2Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 2], {n, 66}]] (* Robert Price, Jun 13 2020 *)

G.f.: x^2*Product_{j>=3} (1+x^j). - R. J. Mathar, Jul 31 2008
a(n) = A025148(n-2), n>1. - R. J. Mathar, Sep 30 2008
G.f.: Sum_{k>=1} x^(k*(k + 3)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157

Offset: 0

Gus Wiseman, May 15 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)

For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
For odd instead of prime we have A096765, even A025147, non-strict A087787
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 lists binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
- reverse A272020
A058698 counts partitions of prime numbers, strict A064688.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A372851 Squarefree numbers whose prime indices are the binary indices of some prime number.

Original entry on oeis.org

3, 6, 10, 22, 30, 42, 46, 66, 70, 102, 114, 118, 130, 182, 238, 246, 266, 318, 330, 354, 370, 402, 406, 434, 442, 510, 546, 646, 654, 690, 762, 770, 798, 930, 938, 946, 962, 986, 1066, 1102, 1122, 1178, 1218, 1222, 1246, 1258, 1334, 1378, 1430, 1482, 1578

Offset: 1

Gus Wiseman, May 16 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The prime indices of 70 are {1,3,4}, which are the binary indices of 13, which is prime, so 70 is in the sequence.
The prime indices of 15 are {2,3}, which are the binary indices of 6, which is not prime, so 15 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
   10: {1,3}
   22: {1,5}
   30: {1,2,3}
   42: {1,2,4}
   46: {1,9}
   66: {1,2,5}
   70: {1,3,4}
  102: {1,2,7}
  114: {1,2,8}
  118: {1,17}
  130: {1,3,6}
  182: {1,4,6}
  238: {1,4,7}
  246: {1,2,13}
  266: {1,4,8}
  318: {1,2,16}
  330: {1,2,3,5}
  354: {1,2,17}
  370: {1,3,12}
  402: {1,2,19}

[Warning: do not confuse A372887 with the strict case A372687.]
For odd instead of prime we have A039956.
For even instead of prime we have A056911.
Strict partitions of this type are counted by A372687.
Non-strict partitions of this type are counted by A372688, ranks A277319.
The nonsquarefree version is A372850, counted by A372887.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A025147, A035100, A071814, A096111, A096765, A231204, A372429, A372471, A372689.

Select[Range[100],SquareFreeQ[#] && PrimeQ[Total[2^(PrimePi/@First/@FactorInteger[#]-1)]]&]

Squarefree numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the (distinct) prime indices of k.

A026832 Number of partitions of n into distinct parts, the least being odd.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset: 0

Clark Kimberling

Fine's numbers L(n).

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 29 2006

			a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

Cf. A000009, A096765.

a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* Harvey P. Dale, May 14 2022 *)

G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - Emeric Deutsch, Mar 29 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

A026824 Number of partitions of n into distinct parts, the least being 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 23, 27, 31, 36, 41, 47, 55, 62, 71, 81, 93, 105, 120, 135, 154, 174, 197, 221, 251, 281, 317, 356, 400, 447, 502, 561, 628, 701, 782, 871, 972, 1081, 1202, 1336, 1483, 1645, 1825, 2021, 2237, 2476

Offset: 0

Clark Kimberling

nonn

Also, number of partitions of n such that if k is the largest part, then k occurs exactly 3 times and each of the numbers 1,2,...,k-1 occur at least once (these are the conjugates of the partitions described in the definition). Example: a(14)=3 because we have [3,3,3,2,2,1],[3,3,3,2,1,1,1] and [2,2,2,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

For n > 3, a(n) is the Euler transform of [0,0,0,1,1,1,1] joined with the period 2 sequence [0,1, ...]. - Georg Fischer, Aug 18 2020

			a(14) = 3 because we have [11,3], [7,4,3] and [6,5,3].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A025147, A025149.

Cf. A096765 (least=1), A096749 (2), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

g:=x^3*product(1+x^j,j=4..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..59); # Emeric Deutsch, Apr 17 2006
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-3)*(i+4)/2 `if`(n<3, 0, b(n-3$2)):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 07 2014

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[(i-3)(i+4)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<3, 0, b[n-3, n-3]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[x^3/((1+x)*(1+x^2)*(1+x^3)) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 3], {n, 1, 66}]] (* Robert Price, Jun 13 2020 *)

From Emeric Deutsch, Apr 17 2006: (Start)
G.f.: (x^3)*Product_{j=4..infinity} (1+x^j).
G.f.: Sum_{k=1..infinity} x^(k*(k+5)/2)/(Product_{j=1..k-1} (1-x^j)). (End)
a(n) = A025149(n-3), n>3. - R. J. Mathar, Jul 31 2008
a(n) ~ exp(Pi*sqrt(n/3)) / (32*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

More terms from Emeric Deutsch, Apr 17 2006

A329436 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 5, 6, 8, 7, 13, 10, 16, 18, 22, 21, 34, 29, 44, 45, 56, 56, 82, 78, 100, 109, 136, 137, 185, 181, 231, 247, 295, 317, 399, 404, 490, 533, 638, 669, 817, 853, 1020, 1108, 1276, 1371, 1638, 1728, 2017, 2186, 2519, 2702, 3153, 3371, 3885

Offset: 1

Ilya Gutkovskiy, Nov 13 2019

nonn

Inverse Moebius transform of A025147.

Number of uniform (constant multiplicity) partitions of n not containing 1, ranked by the odd terms of A072774. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:
  (2)  (3)  (4)    (5)    (6)      (7)    (8)        (9)      (10)
            (2,2)  (3,2)  (3,3)    (4,3)  (4,4)      (5,4)    (5,5)
                          (4,2)    (5,2)  (5,3)      (6,3)    (6,4)
                          (2,2,2)         (6,2)      (7,2)    (7,3)
                                          (2,2,2,2)  (3,3,3)  (8,2)
                                                     (4,3,2)  (5,3,2)
                                                              (3,3,2,2)
                                                              (2,2,2,2,2)
(End)

The strict case is A025147.
The version allowing 1 is A047966.
The version requiring 1 is A097986.
Cf. A023645, A047968, A072774, A096765, A329435, A329436, A367586.

nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&SameQ@@Length/@Split[#]&]], {n,0,30}] (* Gus Wiseman, Dec 01 2023 *)

G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A025147(d).

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