A325700 -id:A325700 - cp's OEIS frontend

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are Heinz numbers of the integer partitions counted by A045931.
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 integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   51: {2,7}
   58: {1,10}
   65: {3,6}
   69: {2,9}
   74: {1,12}
   77: {4,5}
   84: {1,1,2,4}
   86: {1,14}
   90: {1,2,2,3}
   93: {2,11}
   95: {3,8}

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

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]

is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ David A. Corneth, Oct 06 2020

from sympy import factorint, primepi
def ok(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return v[0] == v[1]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth

A045931 Number of partitions of n with equal number of even and odd parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 5, 7, 9, 11, 16, 18, 25, 28, 41, 44, 62, 70, 94, 107, 140, 163, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780

Offset: 0

David W. Wilson

nonn

The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006

			a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
From _Gus Wiseman_, Jan 23 2022: (Start)
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
  ()  .  .  21   .  32   2211   43   3221   54       3322   65       4332
                    41          52   4211   63       4321   74       4431
                                61          72       4411   83       5322
                                            81       5221   92       5421
                                            222111   6211   A1       6321
                                                            322211   6411
                                                            422111   7221
                                                                     8211
                                                                     22221111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3500 (first 1001 terms from David W. Wilson)

The version for subsets of {1..n} is A001405.
Dominated by A027187 (partitions of even length).
More odd/even parts: A108950/A108949.
More or same number of odd/even parts: A130780/A171966.
The strict case is A239241.
This is column k = 0 of the triangle A240009.
Counting only distinct parts gives A241638, ranked by A325700.
A half-conjugate version is A277579.
These partitions are ranked by A325698.
A000041 counts integer partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A257991/A257992 count odd/even parts by Heinz number.
Cf. A000070, A000700, A000712, A027193, A035363, A097613, A195017.

g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t,coeff(gser,x^n)) od: seq(coeff(t*P[n],t),n=0..53); # Emeric Deutsch, Mar 30 2006

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
(* Peter J. C. Moses, Mar 10 2014 *)
nmax = 100; CoefficientList[Series[Sum[x^(3*k) / Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2025 *)

G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 18 2007
a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = A000041(n) - A108950(n) - A108949(n) = A130780(n) + A171966(n) - A000041(n). - Gus Wiseman, Jan 23 2022
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (48*n^(3/2)). - Vaclav Kotesovec, Jun 15 2025

A195017 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} c_k*((-1)^(k-1)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 06 2012

Let p(n,x) be the completely additive polynomial-valued function such that p(1,x) = 0 and p(prime(n),x) = x^(n-1), like is defined in A206284 (although here we are not limited to just irreducible polynomials). Then a(n) is the value of the polynomial encoded in such a manner by n, when it is evaluated at x=-1. - The original definition rewritten and clarified by Antti Karttunen, Oct 03 2018
Positions of 0 give the values of n for which the polynomial p(n,x) is divisible by x+1. For related sequences, see the Mathematica section.
Also the number of odd prime indices of n minus the number of even prime indices of n (both counted with multiplicity), where 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. - Gus Wiseman, Oct 24 2023

			The sequence can be read from a list of the polynomials:
  p(n,x)      with x = -1, gives a(n)
------------------------------------------
  p(1,x) = 0           0
  p(2,x) = 1x^0        1
  p(3,x) = x          -1
  p(4,x) = 2x^0        2
  p(5,x) = x^2         1
  p(6,x) = 1+x         0
  p(7,x) = x^3        -1
  p(8,x) = 3x^0        3
  p(9,x) = 2x         -2
  p(10,x) = x^2 + 1    2.
(The list runs through all the polynomials whose coefficients are nonnegative integers.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206284, A277322, A284010.
For other evaluation functions of such encoded polynomials, see A001222, A048675, A056239, A090880, A248663.
Cf. A006093, A036689, A078437.
Zeros are A325698, distinct A325700.
For sum instead of count we have A366749 = A366531 - A366528.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A019507, A106529, A239241, A239261, A369180.

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 200;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x] /. x -> 0, {n, 1, z/2}]   (* A007814 *)
Table[p[2 n, x] /. x -> 0, {n, 1, z/2}] (* A001511 *)
Table[p[n, x] /. x -> 1, {n, 1, z}]     (* A001222 *)
Table[p[n, x] /. x -> 2, {n, 1, z}]     (* A048675 *)
Table[p[n, x] /. x -> 3, {n, 1, z}]     (* A090880 *)
Table[p[n, x] /. x -> -1, {n, 1, z}]    (* A195017 *)
z = 100; Sum[-(-1)^k IntegerExponent[Range[z], Prime[k]], {k, 1, PrimePi[z]}] (* Friedjof Tellkamp, Aug 05 2024 *)

A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); } \\ Antti Karttunen, Oct 03 2018

Totally additive with a(p^e) = e * (-1)^(1+PrimePi(p)), where PrimePi(n) = A000720(n). - Antti Karttunen, Oct 03 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} = (-1)^(primepi(p)+1)/(p-1) = Sum_{k>=1} (-1)^(k+1)/A006093(k) = A078437 +  Sum_{k>=1} (-1)^(k+1)/A036689(k) = 0.6339266524059... . - Amiram Eldar, Sep 29 2023
a(n) = A257991(n) - A257992(n). - Gus Wiseman, Oct 24 2023
a(n) = -Sum_{k=1..pi(n)} (-1)^k * valuation(n, prime(k)). - Friedjof Tellkamp, Aug 05 2024

More terms, name changed and example-section edited by Antti Karttunen, Oct 03 2018

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

Offset: 1

Gus Wiseman, Oct 22 2023

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, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366531(n).

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726

Offset: 1

Gus Wiseman, Jan 21 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)

A subset of A028260 (even bigomega), counted by A027187.
These partitions are counted by A277579.
This is the half-conjugate version of A325698, counted by A045931.
A000041 counts partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A100824 counts partitions with at most one odd part, ranked by A349150.
A108950/A108949 count partitions with more odd/even parts.
A122111 represents conjugation using Heinz numbers.
A130780/A171966 count partitions with more or equal odd/even parts.
A257991/A257992 count odd/even prime indices.
A316524 gives the alternating sum of prime indices (reverse: A344616).
Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

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],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(A122111(a(n))).

A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 1, 3, 1, 2, 1, 0, 2, 0, 1, 0, 1, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 0, 4, 1, 0, 0, 2, 1, 0, 2, 3, 1, 2

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.

Positions of first appearances are A001248.
The triangular version is A116482.
Positions of zeros are A346635.
Subtracting from the number of odd conjugate parts gives A350941.
Subtracting from the number of odd parts gives A350942.
Subtracting from the number of even parts gives A350950.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847 (this sequence).
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.

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]}]];
Table[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

a(n) = A344616(n) - A350941(n).

a(n) = A257992(A122111(n)).

A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 6, 18, 21, 24, 54, 65, 70, 72, 84, 96, 133, 147, 162, 182, 189, 210, 216, 260, 280, 288, 319, 336, 384, 418, 429, 481, 486, 490, 525, 532, 546, 585, 588, 630, 648, 728, 731, 741, 754, 756, 840, 845, 864, 1007, 1029, 1040, 1120, 1152, 1197, 1254, 1258, 1276

Offset: 1

Gus Wiseman, Jan 27 2022

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 together with their prime indices begin:
   1: ()
   6: (2,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  54: (2,2,2,1)
  65: (6,3)
  70: (4,3,1)
  72: (2,2,1,1,1)
  84: (4,2,1,1)
  96: (2,1,1,1,1,1)

These partitions are counted by A045931.
The conjugate strict version is counted by A239241.
The conjugate version is A325698.
These are the positions of 0's in A350941.
Adding the conjugate condition gives A350946, all four equal A350947.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A325698: # of even parts = # of odd parts.
A349157: # of even parts = # of odd conjugate parts, counted by A277579.
A350848: # even conjugate parts = # odd conjugate parts, counted by A045931.
A350943: # of even conjugate parts = # of odd parts, counted by A277579.
A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A316524 = alternating sum of prime indices, reverse A344616.
Cf. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.

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],Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A344616(a(n)) = A350847(a(n)).

A257991(A122111(a(n))) = A257992(A122111(a(n))).

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1

Offset: 1

Gus Wiseman, Jan 28 2022

sign

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.

			First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
  192: (2,1,1,1,1,1,1)
   32: (1,1,1,1,1)
   48: (2,1,1,1,1)
    8: (1,1,1)
   12: (2,1,1)
    2: (1)
    1: ()
   15: (3,2)
    9: (2,2)
   77: (5,4)
   49: (4,4)
  221: (7,6)
  169: (6,6)

The conjugate version is A350849.
This is a hybrid of A195017 and A350941.
Positions of 0's are A350943.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325698: # of even parts = # of odd parts.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350943: # of even conjugate parts = # of odd parts, counted by A277579.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.

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]}]];
Table[Count[primeMS[n],?OddQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

Original entry on oeis.org

1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920

Offset: 1

Gus Wiseman, Mar 14 2022

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 together with their prime indices begin:
      1: ()
      6: (2,1)
     84: (4,2,1,1)
    210: (4,3,2,1)
    490: (4,4,3,1)
    525: (4,3,3,2)
   2184: (6,4,2,1,1,1)
   2340: (6,3,2,2,1,1)
   5460: (6,4,3,2,1,1)
   9464: (6,6,4,1,1,1)
  12012: (6,5,4,2,1,1)
  12740: (6,4,4,3,1,1)
  12870: (6,5,3,2,2,1)
  13650: (6,4,3,3,2,1)
  14625: (6,3,3,3,2,2)
  19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.

These partitions are counted by A351978.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A239241, A241638, A325700, A350849, A350941, A350942, A350950, A350951.

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[1000],Count[primeMS[#],?EvenQ]==Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).

A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 294, 315, 350, 416, 441, 490, 525, 624, 660, 735, 924, 990, 1088, 1100, 1386, 1540, 1560, 1632, 1650, 1715, 2184, 2310, 2340, 2401, 2432, 2600, 3267, 3276, 3388, 3640, 3648, 3900, 4080, 4125, 5082, 5324, 5390

Offset: 1

Gus Wiseman, Mar 14 2022

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 together with their prime indices begin:
     1: ()
     2: (1)
     6: (2,1)
     9: (2,2)
    20: (3,1,1)
    30: (3,2,1)
    56: (4,1,1,1)
    75: (3,3,2)
    84: (4,2,1,1)
   125: (3,3,3)
   176: (5,1,1,1,1)
   210: (4,3,2,1)
   264: (5,2,1,1,1)
   294: (4,4,2,1)
   315: (4,3,2,2)
   350: (4,3,3,1)
   416: (6,1,1,1,1,1)

The second condition alone is A350944, counted by A277103.
The first condition alone is A350945, counted by A350948.
The case of all four statistics equal is A350947, counted by A351978.
These partitions are counted by A351976.
There are four other possible pairings of statistics:
- A045931: # even = # odd, ranked by A325698, strict A239241.
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
There are two other possible double-pairings of statistics:
- A350946, counted by A351977.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350849, A350941, A350942, A350950, A350951.

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[1000],Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?EvenQ]&]

Intersection of A350944 and A350945.
A257991(a(n)) = A344616(a(n)).
A257992(a(n)) = A350847(a(n)).
Closed under A122111 (conjugation).

cp's OEIS Frontend

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A045931 Number of partitions of n with equal number of even and odd parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A195017 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} c_k*((-1)^(k-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A366528 Sum of odd prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Views

Author