A266477 -id:A266477 - cp's OEIS frontend

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

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

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

A077285 Number of partitions of n with designated summands.

Original entry on oeis.org

1, 1, 3, 5, 10, 15, 28, 41, 69, 102, 160, 231, 352, 498, 732, 1027, 1470, 2031, 2856, 3896, 5382, 7272, 9896, 13233, 17800, 23579, 31362, 41219, 54288, 70791, 92456, 119698, 155097, 199512, 256664, 328134, 419436, 533162, 677412, 856573, 1082284, 1361679

Offset: 0

Jorn B. Olsson (olsson(AT)math.ku.dk), Nov 26 2003

nonn

Sum of products of multiplicities of parts in all partitions of n. The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 1,1,2,2,4 and their sum is a(4) = 10. - Vladeta Jovovic, Feb 16 2005

			a(3)=5 because the partitions of 3 with designated summands are 3', 2'1', 1'11, 11'1, 111'.
1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 28*x^6 + 41*x^7 + 69*x^8 + 102*x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.
William Y. C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y. Y. Shen, On the Number of Partitions with Designated Summands, arXiv:1208.2210 [math.CO], 2012.
Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3, arXiv:2101.04058 [math.CO], 2021. See also Integers (2023) Vol. 23, Art. No. A43.
N. J. A. Sloane, Transforms

Cf. A000041, A091601.
Cf. A102186 (partitions into odd parts with designated summands).
Cf. A258210, A266477.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

max = 50; f = Product[(1-x^i+x^(2*i))/(1-x^i)^2, {i, 1, max}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014, after Vladeta Jovovic *)
nmax=100; CoefficientList[Series[Product[(1+x^(3*k)) / ((1-x^k) * (1-x^(2*k))), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
QP = QPochhammer; s = QP[q^6]/(QP[q]*QP[q^2]*QP[q^3]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
Table[Total[l = Tally /@ IntegerPartitions@n;
Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 41}] (* Robert Price, Jun 06 2020 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))} /* Michael Somos, Feb 05 2004 */

Expansion of eta(q^6) / (eta(q) * eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Feb 05 2004
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 2, ...]. - Michael Somos, Feb 05 2004
G.f.: P(x)*P(x^2)*P(x^3)/P(x^6), where P(x)=Product_{k>0} 1/(1-x^k) is the partition generating function (A000041).
Equals EULER(DCONV(A000012, iEULER(A000027))).
G.f.: Product_{i>=1} (1-x^i+x^(2*i)) / (1-x^i)^2. - Vladeta Jovovic, Jan 16 2005
a(n) ~ 5^(3/4) * exp(Pi*sqrt(10*n)/3) / (2^(11/4) * 3^(3/2) * n^(5/4)). - Vaclav Kotesovec, Nov 28 2015
a(n) = Sum_{k>=1} k*A266477(n,k). - Alois P. Heinz, Dec 29 2015
G.f.: Product_{i>0} (1 + Sum_{j>0} j*x^(j*i)). - Seiichi Manyama, Oct 08 2017

Edited and extended by Christian G. Bower, Jan 23 2004

A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 19 2022

nonn

Warning: If the prime multiplicities of n are a multiset y, this sequence gives the product of multiplicities in y, not the product of y.
Differs from A351946 at A351946(1260) = 4, a(1260) = 2.
Differs from A327500 at A327500(450) = 3, a(450) = 2.
We set a(1) = 0 so that the positions of first appearances are the primorials A002110.
Also the product of the prime metasignature of n (row n of A238747).

			The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of first appearances are A002110.
The prime indices themselves have product A003963, counted by A339095.
The prime signature itself has product A005361, counted by A266477.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A071625 counts distinct prime exponents (third omega).
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A323022 gives fourth omega.
Cf. A085629, A097318, A182850, A304678, A353394, A353399, A353500, A353503.

f:= proc(n) local M,s;
  M:= ifactors(n)[2][..,2];
  mul(numboccur(s,M),s=convert(M,set));
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, May 19 2023

Table[If[n==1,0,Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]],{n,100}]
Join[{0},Table[Times@@(Length/@Split[FactorInteger[n][[;;,2]]]),{n,2,100}]] (* Harvey P. Dale, Oct 20 2024 *)

from math import prod
from itertools import groupby
from sympy import factorint
def A353507(n): return 0 if n == 1 else prod(len(list(g)) for k, g in groupby(factorint(n).values())) # Chai Wah Wu, May 20 2022

a(n) = A005361(A181819(n)) = A003963(A181819(A181819(n))).

A353394 Product of prime shadows of prime indices of n (with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Gus Wiseman, May 17 2022

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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

Positions of first appearances are A353397.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Times@@red/@primeMS[n],{n,100}]

a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.

A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

Original entry on oeis.org

1, 2, 12, 20, 36, 44, 56, 68, 100, 124, 164, 184, 208, 236, 240, 268, 332, 436, 464, 484, 508, 528, 608, 628, 688, 716, 720, 752, 764, 776, 816, 844, 880, 964, 1108, 1132, 1156, 1168, 1200, 1264, 1296, 1324, 1344, 1360, 1412, 1468, 1488, 1584, 1604, 1616, 1724

Offset: 1

Gus Wiseman, May 17 2022

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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    20: {1,1,3}
    36: {1,1,2,2}
    44: {1,1,5}
    56: {1,1,1,4}
    68: {1,1,7}
   100: {1,1,3,3}
   124: {1,1,11}
   164: {1,1,13}
   184: {1,1,1,9}
   208: {1,1,1,1,6}
   236: {1,1,17}
   240: {1,1,1,1,2,3}

Product of prime indices is A003963, counted by A339095.
The LHS (product of exponents) is A005361, counted by A266477.
The RHS (product of shadows) is A353394, first appearances A353397.
A related comparison is A353395, counted by A353396.
The partitions are counted by A353398.
Taking indices instead of exponents on the LHS gives A353503.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393
- recursive version counted by A353426
Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==Times@@Last/@FactorInteger[#]&]

A005361(a(n)) = A353394(a(n)).

A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

Original entry on oeis.org

1, 2, 12, 36, 40, 112, 352, 832, 960, 1296, 2176, 2880, 4864, 5376, 11776, 12544, 16128, 29696, 33792, 34560, 38400, 63488, 64000, 101376, 115200, 143360, 151552, 159744, 335872, 479232, 704512, 835584, 1540096, 1658880, 1802240

Offset: 1

Gus Wiseman, May 17 2022

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. A number's prime signature (row n A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   352: {1,1,1,1,1,5}
   832: {1,1,1,1,1,1,6}
   960: {1,1,1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  2176: {1,1,1,1,1,1,1,7}
  2880: {1,1,1,1,1,1,2,2,3}
  4864: {1,1,1,1,1,1,1,1,8}
  5376: {1,1,1,1,1,1,1,1,2,4}

For shadows instead of exponents we get A003586, counted by A008619.
The LHS (product of prime indices) is A003963, counted by A339095.
The RHS (product of prime exponents) is A005361, counted by A266477.
The version for shadows instead of indices is A353399, counted by A353398.
These partitions are counted by A353506.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

from itertools import count, islice
from math import prod
from sympy import primepi, factorint
def A353503_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p,e in f.items()), count(max(startvalue,1)))
A353503_list = list(islice(A353503_gen(),20)) # Chai Wah Wu, May 20 2022

A003963(a(n)) = A005361(a(n)).

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156

Offset: 0

Gus Wiseman, May 17 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
  3311  711     61111  521111   5511      B11       A1111
        321111         3221111  9111      721111    731111
                                531111    811111    33221111
                                3321111   5221111   422111111
                                22221111  43111111
                                42111111

The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]

A353506 Number of integer partitions of n whose parts have the same product as their multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 3, 3, 2, 3, 2, 0, 2, 3, 2, 1, 3, 1, 6, 3, 2, 3, 3, 2, 3, 4, 1, 2, 3, 6, 3, 2, 2, 3, 3, 1, 2, 6, 6, 4, 7, 2, 3, 6, 4, 3, 3, 0, 4, 5, 3, 5, 5, 6, 5, 3, 3, 3, 6, 5, 5, 6, 6, 3, 3, 3, 4, 4, 4, 6, 7, 2, 5, 7, 6, 2, 3, 4, 6, 11, 9, 4, 4, 1, 5, 6, 4, 7, 9, 6, 4

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(18) = 2 partitions:
  n= 0: ()
  n= 1: (1)
  n= 2:
  n= 3:
  n= 4: (211)
  n= 5:
  n= 6: (3111) (2211)
  n= 7:
  n= 8: (41111)
  n= 9:
  n=10: (511111)
  n=11: (32111111)
  n=12: (6111111) (22221111)
  n=13: (322111111)
  n=14: (71111111) (4211111111)
  n=15:
  n=16: (811111111) (4411111111) (42211111111)
  n=17: (521111111111) (332111111111) (322211111111)
  n=18: (9111111111) (333111111111)
For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).

LHS (product of parts) is ranked by A003963, counted by A339095.
RHS (product of multiplicities) is ranked by A005361, counted by A266477.
For shadows instead of prime exponents we have A008619, ranked by A003586.
Taking sum instead of product of parts gives A266499.
For shadows instead of prime indices we have A353398, ranked by A353399.
These partitions are ranked by A353503.
Taking sum instead of product of multiplicities gives A353698.
A008284 counts partitions by length.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A353507 gives product of multiplicities (of exponents) in prime signature.
Cf. A085629, A114640, A116608, A118914, A124010, A319000, A325702, A353394, A353500.
Cf. A000792, A266480.

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

a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++);); nb;} \\ Michel Marcus, May 20 2022

a(71)-a(100) from Alois P. Heinz, May 20 2022

A266499 Number of partitions of n with product of multiplicities of parts equal to n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 8, 1, 5, 1, 11, 6, 5, 1, 48, 7, 9, 21, 39, 1, 104, 1, 143, 27, 20, 45, 457, 1, 32, 58, 620, 1, 549, 1, 363, 514, 65, 1, 4302, 118, 858, 207, 926, 1, 4080, 437, 5171, 382, 181, 1, 20398, 1, 251, 4287, 20582, 1212

Offset: 0

Emeric Deutsch and Alois P. Heinz, Dec 30 2015

nonn

			a(8) = 2 because among the 22 (= A000041(8)) partitions of 8 only [1,1,1,1,1,1,1,1] and [1,1,1,1,2,2] have product of multiplicities of parts equal to 8.

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

Main diagonal of A266477.

Cf. A000040, A000041.

b:= proc(n, i, p) option remember; `if`(p=1 and i*(i+1)/2 `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=0..70);

b[n_, i_, p_] := b[n, i, p] = If[p == 1 && i*(i + 1)/2 < n, 0, If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, Min[p, n/i]}]]]]; a[n_] := If[PrimeQ[n], 1, b[n, n, n]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

a(n) = A266477(n,n).

p in primes => a(p) = 1.

cp's OEIS Frontend

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

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A077285 Number of partitions of n with designated summands.

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A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

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A353394 Product of prime shadows of prime indices of n (with multiplicity).

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A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

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A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

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A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

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