A088887 -id:A088887 - cp's OEIS frontend

A056239 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} k*c_k.

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 8, 6, 13, 7, 14, 7, 7, 10, 15, 6, 8, 7, 9, 8, 16, 7, 8, 7, 10, 11, 17, 7, 18, 12, 8, 6, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 7, 8, 14, 23, 8, 10, 15, 12, 8, 24, 8, 10

Offset: 1

Leroy Quet, Aug 19 2000

A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. [Cf. A248692 for example.] Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2, etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)-th prime and last with the a(n)-th power of 2. Produces triangular numbers from primorials. - Henry Bottomley, Nov 22 2001
Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.
All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012
a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

			a(12) = 1*2 + 2*1 = 4, since 12 = 2^2 *3^1 = (p_1)^2 *(p_2)^1.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Row sums of A112798.
Cf. A003963 (gives the corresponding products).
Cf. also A000041, A000217, A000720, A001221, A001222, A002110, A005940, A008475, A027748, A049084, A060437, A081401, A082090, A088314, A088318, A088850, A088880, A088881, A088887, A088902, A108951, A122111, A124010, A127668, A141128, A153734, A154351, A156552, A163517, A178503, A215366, A215369, A215501, A242422, A242423, A242424, A243070, A243353, A243354, A243503, A248692, A249336, A249337, A329605, A329902.

a056239 n = sum $ zipWith (*) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013

# To get 10000 terms. First make prime table: M:=10000; pl:=array(1..M); for i from 1 to M do pl[i]:=0; od: for i from 1 to M do if ithprime(i) > M then break; fi; pl[ithprime(i)]:=i; od:
# Decode Maple's amazing syntax for factoring integers: g:=proc(n) local e,p,t1,t2,t3,i,j,k; global pl; t1:=ifactor(n); t2:=nops(t1); if t2 = 2 and whattype(t1) <> `*` then p:=op(1,op(1,t1)); e:=op(2,t1); t3:=pl[p]*e; else
t3:=0; for i from 1 to t2 do j:=op(i,t1); if nops(j) = 1 then e:=1; p:=op(1,j); else e:=op(2,j); p:=op(1,op(1,j)); fi; t3:=t3+pl[p]*e; od: fi; t3; end; # N. J. A. Sloane, May 10 2006
A056239 := proc(n) add( numtheory[pi](op(1,p))*op(2,p), p = ifactors(n)[2]) ; end proc: # R. J. Mathar, Apr 20 2010
# alternative:
with(numtheory): a := proc (n) local B: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: add(B(n)[i], i = 1 .. nops(B(n))) end proc: seq(a(n), n = 1 .. 130); # Emeric Deutsch, May 19 2015

a[1] = 0; a[2] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p];
a[n_] := a[n] = Total[#[[2]]*a[#[[1]]] & /@ FactorInteger[n]]; a /@ Range[91] (* Jean-François Alcover, May 19 2011 *)
Table[Total[FactorInteger[n] /. {p_, c_} /; p > 0 :> PrimePi[p] c], {n, 91}] (* Michael De Vlieger, Jul 12 2017 *)

A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); \\ Antti Karttunen, Oct 26 2014, edited Jan 13 2020

from sympy import primepi, factorint
def A056239(n): return sum(primepi(p)*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 01 2023

(require 'factor) ;; Uses the function factor available in Aubrey Jaffer's SLIB Scheme library.
(define (A056239 n) (apply + (map A049084 (factor n))))
;; Antti Karttunen, Oct 26 2014

Totally additive with a(p) = PrimePi(p), where PrimePi(n) = A000720(n).
a(n) = Sum_{k=1..A001221(n)} A049084(A027748(k))*A124010(k). - Reinhard Zumkeller, Apr 27 2013
From Antti Karttunen, Oct 11 2014: (Start)
a(n) = n - A178503(n).
a(n) = A161511(A156552(n)).
a(n) = A227183(A243354(n)).
For all n >= 0:
a(A002110(n)) = A000217(n). [Cf. Henry Bottomley's comment above.]
a(A005940(n+1)) = A161511(n).
a(A243353(n)) = A227183(n).
Also, for all n >= 1:
a(A241909(n)) = A243503(n).
a(A122111(n)) = a(n).
a(A242424(n)) = a(n).
A248692(n) = 2^a(n). (End)
a(n) < A329605(n), a(n) = A001222(A108951(n)), a(A329902(n)) = A112778(n). - Antti Karttunen, Jan 14 2020

A337583 Irregular triangle read by rows: T(n, k) is the number of integer multisets (partitions) that match the multiplicity multiset of exactly k partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 3, 2, 9, 1, 7, 1, 2, 12, 2, 2, 12, 2, 2, 2, 15, 3, 3, 3, 15, 5, 3, 0, 3, 0, 1, 26, 8, 2, 1, 2, 0, 1, 1, 23, 7, 2, 4, 1, 3, 0, 1, 0, 0, 1, 28, 9, 4, 5, 2, 2, 2, 0, 0, 1, 1, 33, 11, 3, 4, 2, 3, 3, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 45, 10, 8, 4, 4, 1, 4, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 40, 18, 7, 3, 5

Offset: 0

Álvar Ibeas, Sep 02 2020

			T(5, 1) = 3, T(5, 2) = 2: The partitions of 5 present A088887(5) = 5 different multiplicity multisets. Three of them are attained by a single partition of 5 (for instance, (3, 1) comes from (2, 1, 1, 1) only), whereas (1, 1) and (2, 1) arise from two partitions of 5 each (namely, (4, 1) and (3, 2) for the first and (3, 1, 1) and (2, 2, 1) for the second).
Triangle begins:
  k:  1 2 3 4
      -------
n=0:  1
n=1:  1
n=2:  2
n=3:  3
n=4:  5
n=5:  3 2
n=6:  9 1
n=7:  7 1 2
n=8: 12 2 2
n=9: 12 2 2 2

Álvar Ibeas, Rows until n=66, flattened
Álvar Ibeas, Rows until n=19

Cf. A088887 (row sums), A337587 (row lengths).

Sum_{k >= 1} k * T(n, k) = A000041(n).

A337584 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) that match the multiplicity multiset of some partition of n (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 4, 3, 2, 1, 1, 1, 1, 3, 2, 4, 3, 2, 1, 1, 1, 2, 2, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 3, 5, 3, 2, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 2, 2, 4, 5, 7, 8, 8, 5, 5, 3

Offset: 1

Álvar Ibeas, Sep 02 2020

The relevant partitions of n have exactly k parts.

Column k is k-periodic from n = k*(k+1)/2.

			There is no partition of 5 with multiplicity multiset (3) or (1, 1, 1).
Indeed, both (2 = A008284(5, 3)) partitions of 5 into 3 parts (namely, (3, 1, 1) and (2, 2, 1)) have multiplicities (2, 1). Therefore, T(5, 3) = 1.
Triangle begins:
  k:  1 2 3 4 5 6 7 8 9 10
      --------------------
n=1:  1
n=2:  1 1
n=3:  1 1 1
n=4:  1 2 1 1
n=5:  1 1 1 1 1
n=6:  1 2 3 2 1 1
n=7:  1 1 2 2 2 1 1
n=8:  1 2 2 4 3 2 1 1
n=9:  1 1 3 2 4 3 2 1 1
n=10: 1 2 2 4 4 4 3 2 1 1

Alois P. Heinz, Rows n = 1..125 (first 71 rows from Álvar Ibeas)
Álvar Ibeas, First 30 rows

Cf. A000041, A008284, A088887 (row sums).

T(2n,n) gives A344680.

b:= proc(n,i) option remember; `if`(n=0 or i=1, `if`(n=0, {[]}, {[n]}),
     {b(n, i-1)[], seq(map(x-> sort([x[], j]), b(n-i*j, i-1))[], j=1..n/i)})
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(x^add(i, i=t), t=b(n$2))):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 17 2021

If k > (2*n+1)/3, T(n, k) = A088887(n - k).
If n >= k*(k+1)/2, T(n, k) = Sum_{d | gcd(n, k)} A000837(k/d).
T(n, k) = A000041(k) iff k|n and n >= k*(k+1)/2.

A182410 Number of length sets of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 15, 17, 24, 25, 31, 34, 45, 48, 59, 64, 77, 83, 99, 109, 131, 138, 164, 175, 204, 222, 252, 274, 317, 332, 385, 403, 466, 500, 563, 592, 674, 720, 799, 854, 957, 994, 1131, 1196, 1328, 1395, 1551, 1627, 1817, 1912, 2098, 2197

Offset: 0

Olivier Gérard, May 09 2012

nonn

For an integer partition n = c(1)*1 + c(2)*2 + ... + c(n)*n, construct the set of all positive c(i) occurring at least one time.

a(n) is the number of distinct such sets in all integer partitions of n.

			For n=8 the 11 possible sets are {1}, {2}, {4}, {8}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3} and {2, 4}.

Alois P. Heinz, Table of n, a(n) for n = 0..170

Cf. A000041 (number of partitions).
Cf. A088314 (number of different ordered lists of the c(i)).
Cf. A088887 (number of different sorted lists of the c(i)).

b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i=1, {{n}},
      {b(n, i-1)[], seq(map(x-> {x[], j}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 09 2012

Table[Length@ Union@ Map[Union@(Length /@ Split[#]) &, IntegerPartitions[n]], {n, 1, 20}]

from sympy.utilities.iterables import partitions
def A182410(n): return len({tuple(sorted(set(p.values()))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

A337585 a(n) is the number of integer multisets (partitions) for which the number of partitions of n with matching multiplicity multiset is odd.

Original entry on oeis.org

1, 1, 2, 3, 5, 3, 9, 9, 14, 14, 18, 22, 31, 27, 37, 42, 65, 61, 83, 82, 111, 110, 142, 147, 187, 190, 230, 242, 296, 319, 358, 412, 471, 505, 600, 595, 753, 781, 895, 921, 1082, 1143, 1272, 1405, 1587, 1632, 1872, 2000, 2263, 2419, 2648, 2799, 3223, 3319, 3723

Offset: 0

Álvar Ibeas, Sep 02 2020

nonn

The number of multiplicity multisets met by a positive even number of partitions of n is A088887(n) - a(n).

The parity of a(n) is A040051(n).

			The partitions of 7 exhibit 10 = A088887(7) different multisets of multiplicities. Except for (3, 1) (met by partitions (4, 1, 1, 1) and (2, 2, 2, 1)), an odd number of partitions of 7 lead to each of them, so a(7) = 9.

Cf. A088887.

a(n) = Sum_{k odd} A337583(n, k).

A344680 Number of partitions of n that are also multiplicity multiset of a partition of 2n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 8, 12, 14, 18, 21, 30, 33, 41, 49, 62, 70, 86, 98, 116, 133, 160, 181, 214, 237, 282, 311, 364, 407, 466, 522, 600, 652, 761, 815, 937, 1038, 1179, 1271, 1442, 1577, 1762, 1930, 2158, 2311, 2636, 2831, 3146, 3402, 3784, 4057, 4537, 4869, 5365, 5745, 6370, 6802, 7562, 8061, 8785, 9471, 10410

Offset: 0

Alois P. Heinz, Aug 17 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 2: [2], [1,1].
a(3) = 3: [3], [1,2], [1,1,1].
a(4) = 4: [4], [1,3], [2,2], [1,1,2].
a(5) = 4: [5], [1,4], [1,1,3], [1,2,2].
a(6) = 8: [6], [1,5], [2,4], [3,3], [1,1,4], [1,2,3], [2,2,2], [1,1,1,3].
a(7) = 8: [7], [1,6], [1,1,5], [1,2,4], [1,3,3], [2,2,3], [1,1,1,4], [1,1,2,3].

Cf. A088887, A337584.

b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n=0, {[]}, {[n]}),
     {b(n, i-1)[], seq(map(x-> sort([x[], j]), b(n-i*j, i-1))[], j=1..n/i)})
    end:
a:= n-> nops(select(l-> add(i, i=l)=n, b(2*n$2))):
seq(a(n), n=0..30);

a(n) = A337584(2n,n).

cp's OEIS Frontend

A328917 Types of signatures of partitions of n (ordered firstly by decreasing greatest parts, then by decreasing sum of parts, then by increasing number of parts). Irregular triangle read by rows, length of row n is A088887(n).

Views

Author

Keywords

Examples

Crossrefs

A056239 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} k*c_k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Formula

A337583 Irregular triangle read by rows: T(n, k) is the number of integer multisets (partitions) that match the multiplicity multiset of exactly k partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A337584 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) that match the multiplicity multiset of some partition of n (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A182410 Number of length sets of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A337585 a(n) is the number of integer multisets (partitions) for which the number of partitions of n with matching multiplicity multiset is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A344680 Number of partitions of n that are also multiplicity multiset of a partition of 2n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A140312 Number of distinct values of Product_{p is in P} (m(p,P)+1) where m(p,P) is the multiplicity of part p in partition P, when P ranges over all partitions of n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions