A340692 -id:A340692 - cp's OEIS frontend

A064173 Number of partitions of n with positive rank.

0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 80, 106, 136, 178, 225, 291, 366, 466, 583, 735, 912, 1140, 1407, 1743, 2140, 2634, 3214, 3932, 4776, 5807, 7022, 8495, 10225, 12313, 14762, 17696, 21136, 25236, 30030, 35722, 42367, 50216, 59368, 70138, 82665

Offset: 1

Vladeta Jovovic, Sep 19 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
Also number of partitions of n with negative rank. - Omar E. Pol, Mar 05 2012
Column 1 of A208478. - Omar E. Pol, Mar 11 2012
Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p. - Clark Kimberling, Feb 28 2014
The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rank--they are a subsemigroup of the monoid of nonpositive rank partitions. - Richard Locke Peterson, Jul 15 2018

			a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
From _Gus Wiseman_, Feb 09 2021: (Start)
The a(2) = 1 through a(9) = 13 partitions of positive rank:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (421)  (71)    (81)
                               (511)  (422)   (432)
                                      (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (5211)
                                              (6111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
FindStat, St000145: The Dyson rank of a partition
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Note: A-numbers of ranking sequences are in parentheses below.
The negative-rank version is also A064173 (A340788).
The case of odd positive rank is A101707 (A340604).
The case of even positive rank is A101708 (A340605).
These partitions are ranked by (A340787).
A063995/A105806 count partitions by rank.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is a multiple of the greatest part.
A200750 counts partitions whose length and greatest part are coprime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
- Balance -
A047993 counts balanced partitions (A106529).
A340599 counts alt-balanced factorizations.
A340653 counts balanced factorizations.
Cf. A003114, A006141, A039900, A096401, A117193, A117409, A143773, A324516, A324518, A324520.

A064173 := proc(n)
    a := 0 ;
    for p in combinat[partition](n) do
        r := max(op(p))-nops(p) ;
        if r > 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A064173(n),n=0..40) ;# Emeric Deutsch, Dec 11 2004

Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* Clark Kimberling, Feb 12 2014 *)
Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* Clark Kimberling, Feb 28 2014 *)
P = PartitionsP;
a[n_] := (P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}])/2;
a /@ Range[48] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(k+j-2))/(1-x^j))))) \\ Seiichi Manyama, Jan 25 2022

a(n) = (A000041(n) - A047993(n))/2.
a(n) = p(n-2) - p(n-7) + p(n-15) - ... - (-1)^k*p(n-(3*k^2+k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Product_{k>=1} (1/(1-q^k)) * Sum_{k>=1} ( (-1)^k * (-q^(3*k^2/2+k/2))) (conjectured). - Thomas Baruchel, May 12 2018
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022
a(n)+A064174(n) = A000041(n). - R. J. Mathar, Feb 22 2023

A340604 Heinz numbers of integer partitions of odd positive rank.

Original entry on oeis.org

3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 33, 34, 37, 42, 43, 46, 51, 52, 53, 55, 61, 62, 63, 69, 70, 71, 76, 77, 78, 79, 82, 85, 88, 89, 93, 94, 98, 101, 105, 107, 113, 114, 115, 116, 117, 118, 119, 121, 123, 130, 131, 132, 134, 136, 139, 141, 146, 147, 148, 151

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      3: (2)         46: (9,1)       82: (13,1)
      7: (4)         51: (7,2)       85: (7,3)
     10: (3,1)       52: (6,1,1)     88: (5,1,1,1)
     13: (6)         53: (16)        89: (24)
     15: (3,2)       55: (5,3)       93: (11,2)
     19: (8)         61: (18)        94: (15,1)
     22: (5,1)       62: (11,1)      98: (4,4,1)
     25: (3,3)       63: (4,2,2)    101: (26)
     28: (4,1,1)     69: (9,2)      105: (4,3,2)
     29: (10)        70: (4,3,1)    107: (28)
     33: (5,2)       71: (20)       113: (30)
     34: (7,1)       76: (8,1,1)    114: (8,2,1)
     37: (12)        77: (5,4)      115: (9,3)
     42: (4,2,1)     78: (6,2,1)    116: (10,1,1)
     43: (14)        79: (22)       117: (6,2,2)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101707.
Allowing negative ranks gives A340692, counted by A340603.
The even version is A340605, counted by A101708.
The not necessarily odd case is A340787, counted by A064173.
A001222 gives number of prime indices.
A061395 gives maximum prime index.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],OddQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is odd and positive.

A340604 \/ A340605 = A340787.

A340601 Number of integer partitions of n of even rank.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
The positive case is A101708 (A340605).
The Heinz numbers of these partitions are A340602.
The odd version is A340692 (A340603).
- Rank -
A047993 counts partitions of rank 0 (A106529).
A072233 counts partitions by sum and length.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024

A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750

Offset: 0

Emeric Deutsch, Dec 12 2004

nonn

a(n) + A101708(n) = A064173(n).

			a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
From _Gus Wiseman_, Feb 07 2021: (Start)
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
  11   .  31     32   33       52     53         54       55
          1111        51       3211   71         72       73
                      3111            3221       3222     91
                      111111          3311       3321     3322
                                      5111       5211     3331
                                      311111     321111   5221
                                      11111111            5311
                                                          7111
                                                          322111
                                                          331111
                                                          511111
                                                          31111111
                                                          1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
  2   .  4     221   6       421     8         432       A
         211         222     22111   422       441       433
                     411             431       621       442
                     21111           611       22221     622
                                     22211     42111     631
                                     41111     2211111   811
                                     2111111             22222
                                                         42211
                                                         43111
                                                         61111
                                                         2221111
                                                         4111111
                                                         211111111
(End)

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of ranking sequences are in parentheses below.
The even-rank version is A101708 (A340605).
The even- but not necessarily positive-rank version is A340601 (A340602).
The Heinz numbers of these partitions are (A340604).
Allowing negative odd ranks gives A340692 (A340603).
- Rank -
A047993 counts balanced (rank zero) partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A000041, A027187, A101709, A101199, A101200, A117409, A200750.

b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1)/2:
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 29 2021

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
     If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
     If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
a /@ Range[0, 55] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

a(n) = (A000041(n) - A000025(n))/4. - Vladeta Jovovic, Dec 14 2004
G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
a(n) = A340692(n)/2. - Gus Wiseman, Feb 07 2021

More terms from Joerg Arndt, Oct 07 2012

a(0)=0 prepended by Alois P. Heinz, Jan 29 2021

A000025 Coefficients of the 3rd-order mock theta function f(q).

Original entry on oeis.org

1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.

			G.f. = 1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe and then corrected by Sean A. Irvine, Apr 25 2019)
George E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95.
Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
Leila A. Dragonette, Some asymptotic formulas for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015. [See f(q)]
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).
Ken Ono, The last words of a genius, Notices Amer. math. Soc., 57 (2010), 1410-1419.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
Eric Weisstein's World of Mathematics, Mock Theta Function.

Other '3rd-order' mock theta functions are at A013953, A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.

Cf. A340601, A340692.

a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
        (1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
seq(a(n), n=0..120);

CoefficientList[Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}], q] (* N. J. A. Sloane *)
sgn[P_ (* a partition *)] :=
Signature[
  PermutationList[
   Cycles[Flatten[
     SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@
       Accumulate[P]], Length[P] - 1]]]]
conjugate[P_List(* a partition *)] :=
Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];
  Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]
Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@
    IntegerPartitions[#]] & /@ Range[20]
(* George Beck, Oct 25 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / Product[ 1 + x^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]]; (* Michael Somos, Jun 30 2015 *)
rnk[prts_]:=Max[prts]-Length[prts]; mtf[n_]:=Module[{pn=IntegerPartitions[n]},Total[If[ EvenQ[ rnk[#]],1,-1]&/@pn]]; Join[{1},Array[mtf,60]] (* Harvey P. Dale, Sep 13 2024 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))}; /* Michael Somos, Sep 02 2007 */

my(N=60, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)^2/(1+x^k))) \\ Seiichi Manyama, May 23 2023

G.f.: 1 + Sum_{n>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).
G.f.: (1 + 4 * Sum_{n>=1} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>=1} (1 - q^i).
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - Vaclav Kotesovec, Jun 10 2019
G.f.: 1 - Sum_{n >= 1} (-1)^n*x^n/Product_{k = 1..n} 1 + x^k. See Fine, equation 26.22, p. 55. - Peter Bala, Feb 04 2021
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A340601(n) - A340692(n).
G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k)^2 / (1+x^k). (End)

Entry improved by comments from Dean Hickerson

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
     3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
     5: {3}          29: {10}         49: {4,4}
     7: {4}          30: {1,2,3}      50: {1,3,3}
     9: {2,2}        31: {11}         51: {2,7}
    11: {5}          33: {2,5}        53: {16}
    12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
    13: {6}          36: {1,1,2,2}    55: {3,5}
    15: {2,3}        37: {12}         56: {1,1,1,4}
    17: {7}          39: {2,6}        57: {2,8}
    18: {1,2,2}      40: {1,1,1,3}    59: {17}
    19: {8}          41: {13}         60: {1,1,2,3}
    21: {2,4}        42: {1,2,4}      61: {18}
    23: {9}          43: {14}         63: {2,2,4}
    24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
    25: {3,3}        47: {15}         66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

The version looking at greatest factor is A057716.
The version for twice-balanced is A340657, with complement A340656.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors, listed by A161906.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]

A340602 Heinz numbers of integer partitions of even rank.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)

FindStat, St000145: The Dyson rank of a partition

Taking only length gives A001222.
Taking only maximum part gives A061395.
These partitions are counted by A340601.
The complement is A340603.
The case of positive rank is A340605.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank = maximum minus minimum part (A324515).
A340653 counts factorizations of rank 0.
A340692 counts partitions of odd rank (A340603).
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

Either n = 1 or A061395(n) - A001222(n) is even.

A340832 Number of factorizations of n into factors > 1 with odd least factor.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 1, 0, 4, 0, 1, 2, 2, 1, 2, 0, 1, 2, 2, 1, 2, 0, 1, 3, 1, 0, 4, 0, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 4, 0, 2, 1, 1, 1, 5, 0, 1, 3, 2, 0, 2, 0, 1, 5, 2, 0, 2, 0, 2, 2, 1, 1, 4, 1, 1, 1, 1, 0, 5, 0, 1, 6

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)     (3*36)     (135)      (3*60)     (3*84)
  (5*9)    (9*12)     (3*45)     (5*36)     (7*36)
  (3*15)   (3*4*9)    (5*27)     (9*20)     (9*28)
  (3*3*5)  (3*6*6)    (9*15)     (5*6*6)    (3*3*28)
           (3*3*12)   (3*5*9)    (3*3*20)   (3*4*21)
           (3*3*3*4)  (3*3*15)   (3*4*15)   (3*6*14)
                      (3*3*3*5)  (3*5*12)   (3*7*12)
                                 (3*6*10)   (3*3*4*7)
                                 (3*3*4*5)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A340854.
Positions of nonzero terms are A340855.
The version for partitions is A026804.
Odd-length factorizations are counted by A339890.
The version looking at greatest factor is A340831.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340607 counts factorizations with odd length and greatest factor.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A026424 lists numbers with odd Omega.
A027193 counts partitions of odd length.
A058695 counts partitions of odd numbers (A300063).
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A244991 lists numbers whose greatest prime index is odd.
A340692 counts partitions of odd rank.
Cf. A050320, A160786, A340385, A340596, A340599, A340654, A340655, A340931.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@*Min]],{n,100}]

A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

For all terms A061395(a(n)) < A001222(a(n)).

A340605 Heinz numbers of integer partitions of even positive rank.

Original entry on oeis.org

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
Allowing any positive rank gives A064173 (A340787).
The odd version is counted by A101707 (A340604).
These partitions are counted by A101708.
The not necessarily positive case is counted by A340601 (A340602).
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340692 counts partitions of odd rank (A340603).
- Even -
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is even and positive.

A340604 \/ A340605 = A340787.

cp's OEIS Frontend

A064173 Number of partitions of n with positive rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A340604 Heinz numbers of integer partitions of odd positive rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A340601 Number of integer partitions of n of even rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A000025 Coefficients of the 3rd-order mock theta function f(q).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A340602 Heinz numbers of integer partitions of even rank.

Views

Author