A090858 -id:A090858 - cp's OEIS frontend

A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

0, 0, 0, 1, 1, 2, 3, 4, 7, 12, 16, 21, 33, 38, 50, 75, 87, 111, 150, 185, 244, 307, 373, 461, 585, 702, 856, 1043, 1255, 1498, 1822, 2143, 2565, 3064, 3607, 4251, 5064, 5920, 6953, 8174, 9503, 11064, 12927, 14921, 17320, 19986, 23067, 26485, 30499, 34894

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).

The Heinz numbers of these partitions are given by A325259.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)      (64)
              (41)  (51)    (52)    (62)     (63)      (73)
                    (2211)  (61)    (71)     (72)      (82)
                            (3211)  (3221)   (81)      (91)
                                    (3311)   (3321)    (3322)
                                    (4211)   (4221)    (4411)
                                    (32111)  (4311)    (5221)
                                             (5211)    (5311)
                                             (32211)   (6211)
                                             (42111)   (32221)
                                             (222111)  (33211)
                                             (321111)  (42211)
                                                       (43111)
                                                       (52111)
                                                       (421111)
                                                       (3211111)

Cf. A008284, A071625, A090858, A098859, A116608, A117571, A127002, A325242, A325259, A325270.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]],{n,0,30}]

A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 3, 3, 7, 6, 11, 12, 15, 21, 25, 31, 43, 49, 58, 79, 89, 108, 135, 165, 190, 232, 279, 328, 387, 461, 536, 650, 743, 870, 1029, 1202, 1381, 1613, 1864, 2163, 2505, 2875, 3292, 3829, 4367, 5001, 5746, 6538, 7462, 8533, 9714, 11008, 12527, 14196

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324519.

			The a(2) = 1 through a(11) = 11 integer partitions:
  (11)  (211)  (221)  (222)  (331)   (611)   (441)   (811)   (551)
               (311)  (411)  (511)   (3221)  (711)   (3322)  (911)
                             (3211)  (4211)  (3222)  (4222)  (3332)
                                             (3321)  (5221)  (4331)
                                             (4221)  (5311)  (4421)
                                             (4311)  (6211)  (5222)
                                             (5211)          (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)

Cf. A003114, A006141, A039900, A046660, A047993, A064174, A090858, A133121.

Cf. A324516, A324518, A324519, A324520.

Table[Length[Select[IntegerPartitions[n],Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Gus Wiseman, Jan 24 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)

Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Zeros of A328958.
The complement is A328957.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - omega(n) * Omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]==PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) = A001222(a(n)) * A001221(a(n)).

A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Clark Kimberling, Jan 01 2007

From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
  (21)  (31)   (32)   (42)   (43)    (53)    (54)
        (211)  (41)   (51)   (52)    (62)    (63)
               (221)  (411)  (61)    (71)    (72)
               (311)         (322)   (332)   (81)
                             (331)   (422)   (441)
                             (511)   (611)   (522)
                             (3211)  (3221)  (711)
                                     (4211)  (3321)
                                             (4221)
                                             (4311)
                                             (5211)
(End)

			a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
  (3211)  (3221)  (3321)  (5221)  (4322)  (4332)  (4432)
          (4211)  (4221)  (5311)  (4331)  (4431)  (5332)
                  (4311)  (6211)  (4421)  (5322)  (5422)
                  (5211)          (5411)  (5331)  (5521)
                                  (6221)  (6411)  (6322)
                                  (6311)  (7221)  (6331)
                                  (7211)  (7311)  (6511)
                                          (8211)  (7411)
                                                  (8221)
                                                  (8311)
                                                  (9211)
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1)

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019

g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k),i=1..j-1),j=2..k-1),k=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p,set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part,res,p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p,part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007

Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,70}], x]] (* G. C. Greubel, May 30 2019 *)

my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019

a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019

G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013

A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 63, 68, 76, 80, 84, 88, 92, 99, 100, 104, 112, 116, 117, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 164, 168, 171, 172, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 244, 248, 252, 260, 261

Offset: 1

Gus Wiseman, Jan 26 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The terms together with their prime indices begin:
   20: (3,1,1)
   28: (4,1,1)
   40: (3,1,1,1)
   44: (5,1,1)
   52: (6,1,1)
   56: (4,1,1,1)
   63: (4,2,2)
   68: (7,1,1)
   76: (8,1,1)
   80: (3,1,1,1,1)
   84: (4,2,1,1)
   88: (5,1,1,1)
   92: (9,1,1)
   99: (5,2,2)

Heinz number rankings are in parentheses below.
Taking just one condition gives (A073492) and (A065201), counted by A239955.
These partitions are counted by A350839.
A000041 = integer partitions, strict A000009.
A034296 = partitions with no gaps (A073491), strict A001227 (A073485).
A090858 = partitions with a single gap of size 1 (A325284).
A116931 = partitions with no successions (A319630), strict A003114.
A116932 = partitions with no successions or gaps of size 1, strict A025157.
A350842 = partitions with no gaps of size 1, strict A350844, sets A005314.
Cf. A000070, A024619, A026424, A055932, A183558, A277103, A305148, A321440, A350838.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]

A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 3, 1, 0, 1, 5, 4, 2, 2, 1, 0, 1, 6, 6, 3, 3, 2, 1, 0, 1, 8, 7, 5, 4, 2, 2, 1, 0, 1, 10, 8, 10, 3, 5, 2, 2, 1, 0, 1, 12, 13, 8, 9, 4, 4, 2, 2, 1, 0, 1, 15, 15, 14, 10, 8, 5, 4, 2, 2, 1, 0, 1, 18, 21, 15, 16, 8, 9, 4, 4, 2, 2, 1, 0, 1, 22, 25, 23, 17, 17, 7, 10, 4, 4, 2, 2, 1, 0, 1

Offset: 1

Vladeta Jovovic, Sep 18 2007

			1
1,1
2,0,1
2,2,0,1
3,2,1,0,1
4,2,3,1,0,1
5,4,2,2,1,0,1
6,6,3,3,2,1,0,1
8,7,5,4,2,2,1,0,1
10,8,10,3,5,2,2,1,0,1
12,13,8,9,4,4,2,2,1,0,1
15,15,14,10,8,5,4,2,2,1,0,1
18,21,15,16,8,9,4,4,2,2,1,0,1
From _Gus Wiseman_, Jan 23 2019: (Start)
It is possible to augment the triangle to cover the n = 0 and k = n cases, giving:
   1
   1  0
   1  1  0
   2  0  1  0
   2  2  0  1  0
   3  2  1  0  1  0
   4  2  3  1  0  1  0
   5  4  2  2  1  0  1  0
   6  6  3  3  2  1  0  1  0
   8  7  5  4  2  2  1  0  1  0
  10  8 10  3  5  2  2  1  0  1  0
  12 13  8  9  4  4  2  2  1  0  1  0
  15 15 14 10  8  5  4  2  2  1  0  1  0
  18 21 15 16  8  9  4  4  2  2  1  0  1  0
  22 25 23 17 17  7 10  4  4  2  2  1  0  1  0
  27 30 32 21 19 16  8  9  4  4  2  2  1  0  1  0
Row seven {5, 4, 2, 2, 1, 0, 1, 0} counts the following integer partitions (empty columns not shown).
  (7)    (322)   (2221)  (22111)  (211111)  (1111111)
  (43)   (331)   (4111)  (31111)
  (52)   (511)
  (61)   (3211)
  (421)
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Row sums are A000041. Row polynomials evaluated at -1 are A268498. Row polynomials evaluated at 2 are A006951.

Cf. A000009, A090858, A100471, A116608.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
       add(x^`if`(j=0, 0, j-1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2015

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^If[j == 0, 0, j-1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[ Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==k&]],{n,0,15},{k,0,n}] (* augmented version, Gus Wiseman, Jan 23 2019 *)

partitm(n,m,nmin)={ local(resul,partj) ; if( n < 0 || m <0, return([;]) ; ) ; resul=matrix(0,m); if(m==0, return(resul); ) ; for(j=max(1,nmin),n\m, partj=partitm(n-j,m-1,j) ; for(r1=1,matsize(partj)[1], resul=concat(resul,concat([j],partj[r1,])) ; ) ; ) ; if(m==1 && n >= nmin, resul=concat(resul,[[n]]) ; ) ; return(resul) ; }
partit(n)={ local(resul,partm,filr) ; if( n < 0, return([;]) ; ) ; resul=matrix(0,n) ; for(m=1,n, partm=partitm(n,m,1) ; filr=vector(n-m) ; for(r1=1,matsize(partm)[1], resul=concat( resul,concat(partm[r1,],filr) ) ; ) ; ) ; return(resul) ; }
A133121row(n)={ local(p=partit(n),resul=vector(n),nprts,ndprts) ; for(r=1,matsize(p)[1], nprts=0 ; ndprts=0 ; for(c=1,n, if( p[r,c]==0, break, nprts++ ; if(c==1, ndprts++, if(p[r,c]!=p[r,c-1], ndprts++ ) ; ) ; ) ; ) ; k=nprts-ndprts; resul[k+1]++ ; ) ; return(resul) ; }
A133121()={ for(n=1,20, arow=A133121row(n) ; for(k=1,n, print1(arow[k],",") ; ) ; ) ; }
A133121() ; \\ R. J. Mathar, Sep 28 2007

tabl(nn) = my(pl = prod(n=1, nn, 1+x^n/(1-y*x^n)) + O(x^nn)); for (k=1, nn-1, print(Vecrev(polcoeff(pl,k,x)))); \\ Michel Marcus, Aug 23 2015

G.f.: Product_{n>=1} 1 + x^n/(1-y*x^n).

More terms from R. J. Mathar, Sep 28 2007

A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with one fewer distinct multiplicities than distinct parts. The enumeration of these partitions by sum is given by A325244.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Cf. A056239, A060687, A090858, A112798, A116608, A118914, A130091, A323023, A325241, A325242, A325244, A325270, A325281.

Select[Range[100],PrimeNu[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A001221(a(n)) = A071625(a(n)) + 1.

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 133

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also Heinz numbers of integer partitions with 1 fewer distinct multiplicities than parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The enumeration of these partitions by sum is given by A117571.

Also numbers whose sorted prime signature is (1,1), (2), or (1,2). - Gus Wiseman, Jul 03 2019

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   44: {1,1,5}

Cf. A001221, A001222, A000961, A005117, A060687, A062770, A071625, A072774, A090858, A117571, A118914, A130091, A325244, A325259.

Select[Range[100],PrimeOmega[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 42, 43, 47, 49, 53, 59, 61, 64, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 89, 97, 100, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 120, 121, 125, 127, 128, 130, 131, 137

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Nonzeros of A328958.
The complement is A328956.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - Omega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]!=PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) != omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) != A001222(a(n)) * A001221(a(n)).

cp's OEIS Frontend

A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

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A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

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A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

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A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

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A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

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A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.

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A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

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