A063008 -id:A063008 - cp's OEIS frontend

A238963 Number of divisors of A063008(n,k).

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64, 8, 14, 18, 24, 20, 30, 40, 32, 36, 48, 64, 54, 72, 96, 128, 9, 16, 21, 28, 24, 36, 48, 25, 40, 45, 60, 80, 48, 64, 72, 96, 128, 81, 108, 144, 192, 256, 10, 18, 24, 32, 28, 42, 56, 30, 48, 54, 72, 96, 50, 60, 80, 90, 120, 160, 64, 96, 128, 108, 144, 192, 256, 162, 216, 288, 384, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

Equivalent to A074139 but using canonical order.

			Triangle begins:
  1;
  2;
  3,  4;
  4,  6,  8;
  5,  8,  9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283

Row sums are A074141.

Cf. A000005, A000041, A063008, A074139.

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> numtheory[tau](mul(ithprime(i)
        ^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 24 2020

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := DivisorSigma[0, #]&[Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n]];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

\\ here b(n) is A000005.
b(n)={numdiv(n)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 24 2020

def A238963row(n):
    return list(product(t + 1 for t in p) for p in Partitions(n))
print([A238963row(n) for n in range(10)])  # Peter Luschny, Dec 11 2023

T(n, k) = A000005(A063008(n,k)).

Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # Peter Luschny, Dec 11 2023

Offset corrected by Andrew Howroyd, Mar 24 2020

A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

Original entry on oeis.org

1, 6, 30, 36, 210, 180, 2310, 216, 900, 1260, 30030, 1080, 6300, 13860, 510510, 1296, 5400, 7560, 44100, 69300, 180180, 9699690, 6480, 27000, 37800, 83160, 485100, 900900, 3063060, 223092870, 7776, 32400, 45360, 189000, 264600, 415800, 1081080, 5336100

Offset: 0

Jack W Grahl, Jul 06 2018

The sequence A063008 gives the least number with each prime signature, ordered by the underlying partition. This sequence is a subsequence which only includes those prime signatures M for which M/2 is not a prime signature, the so-called 'leading' least prime signatures.
This sequence is therefore constructed by taking the partitions first in increasing order of their sum, then in decreasing order of the first term, then decreasing order of the second term, etc. We drop all partitions, except the empty partition, where the first term and the second term are different. Then we map (m1, m2, m3, ..., mk) to 2^m1 * 3^m2 * ... * pk^mk to give the terms of this sequence.
The sequence A062515 had a description which suggested that it had been confused with this sequence. They are the same leading least prime signatures, but in a different order, given by a different construction using integer partitions.

			The first few partitions are [], [1,1], [1,1,1], [2,2], [1,1,1,1]. So the first few terms are 1, 2 * 3 = 6, 2 * 3 * 5 = 30, 2^2 * 3^2 = 36, 2 * 3 * 5 * 7 = 210.

Subsequence of A063008. A re-ordering of A062515, also of A056153. Cf A025487.

primes :: [Integer]
primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
divides :: Integer -> Integer -> Bool
divides a b = a `mod` b == 0
partitions :: [[Integer]]
partitions = concat $ map (partitions_of_n) [0..]
partitions_of_n :: Integer -> [[Integer]]
partitions_of_n n = partitions_at_most n n
partitions_at_most :: Integer -> Integer -> [[Integer]]
partitions_at_most _ 0 = [[]]
partitions_at_most 0 _ = []
partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
prime_signature :: [Integer] -> Integer
prime_signature p = product $ zipWith (^) primes p
seq :: [Integer]
seq = map prime_signature $ filter compare_first_second partitions
    where
  compare_first_second p
        | length p == 0 = True
        | length p == 1 = False
        | otherwise = p!!0 == p!!1

A360440 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, is the number of ways of choosing nonnegative numbers for k indistinguishable A063008(n)-sided dice so that it is possible to roll every number from 0 to (A063008(n))^k-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 15, 1, 1, 1, 1, 10, 71, 105, 1, 1, 1, 1, 42, 280, 1001, 945, 1, 1, 1, 1, 115, 3660, 15400, 18089, 10395, 1, 1, 1, 1, 35, 20365, 614040, 1401400, 398959, 135135, 1, 1

Offset: 0

William P. Orrick, Feb 19 2023

The number of configurations depends on the number of sides on the dice only through its prime signature. A063008 provides a canonical representative of each prime signature.

Also the number of Krasner factorizations of (x^(A063008(n))^k)-1) / (x-1) into k polynomials each having A063008(n) nonzero terms all with coefficient +1. (Krasner and Ranulac, 1937)

			A063008(2) = 4. There are 3 ways to assign numbers to two 4-sided dice:
 {{0, 1, 2, 3}, {0, 4, 8, 12}},
 {{0, 1, 8, 9}, {0, 2, 4,  6}},
 {{0, 1, 4, 5}, {0, 2, 8, 10}}.
The table begins:
  1  1    1       1          1              1                 1  ...
  1  1    1       1          1              1                 1  ...
  1  1    3      15        105            945             10395  ...
  1  1    7      71       1001          18089            398959  ...
  1  1   10     280      15400        1401400         190590400  ...
  1  1   42    3660     614040      169200360       69444920160  ...
  1  1  115   20365    6891361     3815893741     3141782433931  ...
  1  1   35    5775    2627625     2546168625     4509264634875  ...
  1  1  230  160440  299145000  1175153779800  8396156461492800  ...
  ...
The rows shown enumerate configurations for dice of 1, 2, 4, 6, 8, 12, 30, 16, and 24 sides, which represent the prime signatures {}, {1}, {2}, {1,1}, {3}, {2,1}, {1,1,1}, {4}, and {3,1}.

M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.

The concatenation of all prime signatures, listed in the order that corresponds to the rows of T(n,k), is A080577.
T(3,k) is |A002119(k)|. Starting with k = 1, T(1,k), T(2,k), T(4,k), and T(7,k) are given by columns 1-4 of A060540.
Row n is row A063008(n) of A360098.
Cf. A273013, A131514.

@cached_function
def r(i,M):
    kminus1 = len(M)
    u = tuple([1 for j in range(kminus1)])
    if i > 1 and M == u:
        return(1)
    elif M != u:
        divList = divisors(i)[:-1]
        return(sum(r(M[j],tuple(sorted(M[:j]+tuple([d])+M[j+1:])))\
         for d in divList for j in range(kminus1)))
def f(n,k):
    if n == 1 or k == 0:
        return(1)
    else:
        return(r(n,tuple([n for j in range(k-1)]))) / factorial(k-1)
# The following function produces the top left corner of the table:
def TArray(maxn,maxk):
    retArray = []
    primesList = []
    ptnSum = 0
    ptnItr = Partitions(ptnSum)
    ptn = ptnItr.first()
    n = 0
    while n <= maxn:
        if ptn == None:
            primesList.append(Primes()[ptnSum])
            ptnSum = ptnSum + 1
            ptnItr = Partitions(ptnSum)
            ptn = ptnItr.first()
        prdct = prod(primesList[j]^ptn[j] for j in range(len(ptn)))
        retArray.append([f(prdct,k) for k in range(maxk+1)])
        n = n + 1
        ptn = ptnItr.next(ptn)
    return(retArray)

T(n,k) = f(A063008(n),k), where f(n,k) is the table given by A360098.

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 52, 55, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256

Offset: 0

Alois P. Heinz, Aug 08 2012

The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-6, 9, 10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501.
Number m is positioned in row n = A056239(m).  The number of different values m, such that both m and m+1 occur in row n is A088850(n).  A215369 lists all values m, such that both m and m+1 are in the same row.
The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }.
Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively.
This triangle is related to the triangle A145518, see in both triangles the first column, the right border, the second right border and the row sums. - Omar E. Pol, May 18 2015

			The partitions of n=3 are {[3], [2,1], [1,1,1]}, encodings give {prime(3), prime(2)*prime(1), prime(1)^3} = {5, 3*2, 2^3} => row 3 = [5, 6, 8].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8;
   7,  9, 10, 12, 16;
  11, 14, 15, 18, 20, 24, 32;
  13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64;
  17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128;
  ...
Corresponding triangle of integer partitions begins:
  ();
  1;
  2, 11;
  3, 21, 111;
  4, 22, 31, 211, 1111;
  5, 41, 32, 221, 311, 2111, 11111;
  6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111;
  7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111;  - _Gus Wiseman_, Dec 12 2016

Column k=1 gives: A008578(n+1).
Last elements of rows give: A000079.
Second to last elements of rows give: A007283(n-2) for n>1.
Row sums give: A145519.
Row lengths are: A000041.
Cf. A129129 (with row elements using order of A080577).
LCM of terms in row n gives A138534(n).
Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501.
Cf. A112798, A246867 (the same for partitions into distinct parts).
Cf. A324939, A377852, A378175.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
       [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> sort(b(n, n))[]:
seq(T(n), n=0..10);
# (2nd Maple program)
with(combinat): A := proc (n) local P, A, i: P := partition(n): A := {}; for i to nops(P) do A := `union`(A, {mul(ithprime(P[i][j]), j = 1 .. nops(P[i]))}) end do: A end proc; # the command A(m) yields row m. # Emeric Deutsch, Jan 23 2016
# (3rd Maple program)
q:= 7: S[0] := {1}: for m to q do S[m] := `union`(seq(map(proc (f) options operator, arrow: ithprime(j)*f end proc, S[m-j]), j = 1 .. m)) end do; # for a given positive integer q, the program yields rows 0, 1, 2,...,q. # Emeric Deutsch, Jan 23 2016

b[n_, i_] := b[n, i] = If[n == 0 || i<2, {2^n}, Table[Function[#*Prime[i]^j] /@ b[n - i*j, i-1], {j, 0, n/i}] // Flatten]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)
nn=7;HeinzPartition[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]//Reverse];
Take[GatherBy[Range[2^nn],Composition[Total,HeinzPartition]],nn+1] (* Gus Wiseman, Dec 12 2016 *)
Table[Map[Times @@ Prime@ # &, IntegerPartitions[n]], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

\\ From M. F. Hasler, Dec 06 2016 (Start)
A215366_row(n)=vecsort([vecprod([prime(p)|p<-P])|P<-partitions(n)]) \\ bug fix & syntax update by M. F. Hasler, Oct 20 2023
A215366_vec(N)=concat(apply(A215366_row,[0..N])) \\ "flattened" rows 0..N (End)

Recurrence relation, explained for the set S(4) of entries in row 4: multiply the entries of S(3) by 2 (= 1st prime), multiply the entries of S(2) by 3 (= 2nd prime), multiply the entries of S(1) by 5 (= 3rd prime), multiply the entries of S(0) by 7 (= 4th prime); take the union of all the obtained products. The 3rd Maple program is based on this recurrence relation. - Emeric Deutsch, Jan 23 2016

A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 36, 24, 32, 60, 64, 48, 72, 210, 128, 180, 256, 120, 144, 96, 512, 420, 216, 192, 900, 240, 1024, 360, 2048, 2310, 288, 384, 432, 1260, 4096, 768, 576, 840, 8192, 720, 16384, 480, 1800, 1536, 32768, 4620, 1296, 1080, 1152, 960, 65536

Offset: 1

Matthew Vandermast, Dec 07 2010

nonn

A permutation of A025487. a(n) is the member m of A025487 such that A181819(m) = n. a(n) is also the member of A025487 whose prime signature is conjugate to the prime signature of A108951(n).

If n = Product_i prime(e(i)) with the e(i) weakly decreasing, then a(n) = Product_i prime(i)^e(i). For example, 90 = prime(3) * prime(2) * prime(2) * prime(1), so a(90) = prime(1)^3 * prime(2)^2 * prime(3)^2 * prime(4)^1 = 12600. - Gus Wiseman, Jan 02 2019

			The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181822.
Cf. A046523, A181819, A181820.
Cf. A000040, A001221, A001222, A002110, A056239, A071625, A112798, A118914, A122111, A124859, A182850, A305936, A361808.

a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
             numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 05 2018

With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* Michael De Vlieger, Dec 17 2018 *)
Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,30}] (* Gus Wiseman, Jan 02 2019 *)

A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); }; \\ Antti Karttunen, Dec 10 2018

from math import prod
from sympy import prime, primepi, factorint
def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi,factorint(n,multiple=True)),reverse=True),1)) # Chai Wah Wu, Sep 15 2023

If A108951(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A108951(n))).
a(A181819(n)) = A046523(n). - [See also A124859]. Antti Karttunen, Dec 10 2018
a(n) = A025487(A361808(n)). - Pontus von Brömssen, Mar 25 2023
a(n) = A108951(A122111(n)). - Antti Karttunen, Sep 15 2023

Definition corrected by Gus Wiseman, Jan 02 2019

A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 4, 1, 1, 1, 3, 3, 1, 3, 2

Offset: 1

N. J. A. Sloane, Mar 23 2003

This is the "Mathematica" ordering of the partitions, referenced in numerous other sequences. The partitions of each integer are in reverse order of the conjugates of the partitions in Abramowitz and Stegun order (A036036). They are in the reverse of the order of the partitions in Maple order (A080576). - Franklin T. Adams-Watters, Oct 18 2006
The graded reverse lexicographic ordering of the partitions is often referred to as the "Canonical" ordering of the partitions. - Daniel Forgues, Jan 21 2011
Also the "MAGMA" ordering of the partitions. - Jason Kimberley, Oct 28 2011
Also an intuitive ordering described but not formalized in [Hardy and Wright] the first four editions of which precede [Abramowitz and Stegun]. - L. Edson Jeffery, Aug 03 2013
Also the "Sage" ordering of the partitions. - Peter Luschny, Aug 12 2013
While this is the order used for the constructive function "IntegerPartitions", it is different from Mathematica's canonical ordering of finite expressions, the latter giving A036036 if parts of partitions are read in reversed (weakly increasing) order, or A334301 if in the usual (weakly decreasing) order. - Gus Wiseman, May 08 2020

			First five rows are:
  {{1}}
  {{2}, {1, 1}}
  {{3}, {2, 1}, {1, 1, 1}}
  {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
  {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
Up to the fifth row, this is exactly the same as the colexicographic ordering A036037. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)). - _M. F. Hasler_, Jan 23 2020
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (2,1,1,1)    (6,1)          (8)
  (3)        (1,1,1,1,1)  (5,2)          (7,1)
  (2,1)      (6)          (5,1,1)        (6,2)
  (1,1,1)    (5,1)        (4,3)          (6,1,1)
  (4)        (4,2)        (4,2,1)        (5,3)
  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
The triangle with partitions shown as Heinz numbers (A129129) begins:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  28  25  30  40  27  36  48  64
  17  26  33  44  35  42  56  50  45  60  80  54  72  96 128
(End)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 273.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 287.

Franklin T. Adams-Watters, First 20 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831.
OEIS Wiki, Orderings of partitions (a comparison).
Sergei Viznyuk, C Program
Wikiversity, Lexicographic and colexicographic order

See A080576 Maple (graded reflected lexicographic) ordering.
See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for graded colexicographic ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.
Differs from A036037 at a(48).
See A322761 for a compressed version.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Compositions under this ordering are A066099.
Distinct parts of these partitions are counted by A115623.
Taking Heinz numbers gives A129129.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Reading partitions in reverse (weakly increasing) order gives A228531.
Lengths of these partitions are A238966.
Sorting partitions by Heinz number gives A296150.
The maxima of these partitions are A331581.
The length-sensitive version is A334439.
Cf. A000041, A048793, A063008, A185974, A334301, A334434, A334436, A334438.

&cat[&cat Partitions(n):n in[1..7]]; // Jason Kimberley, Oct 28 2011

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> x[], b(n$2))[]:
seq(T(n), n=1..8);  # Alois P. Heinz, Jan 29 2020

<Jean-François Alcover, Dec 10 2012 *)
revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 08 2020 *)

A080577_row(n)={vecsort(apply(t->Vecrev(t),partitions(n)),,4)} \\ M. F. Hasler, Jan 21 2020

L = []
for n in range(8): L += list(Partitions(n))
flatten(L)   # Peter Luschny, Aug 12 2013

A078760 Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720, 1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040, 1, 8, 28, 56, 56, 168, 336, 70, 280, 420, 840, 1680, 560, 1120, 1680, 3360, 6720, 2520

Offset: 0

Franklin T. Adams-Watters, Jan 08 2003

This is a function of the individual partitions of an integer. The number of values in each line is given by A000041; thus lines 0 to 5 of the sequence are (1), (1), (1,2), (1,3,6), (1,4,6,12,24). The partitions in each line are ordered with the largest part sizes first, so the line 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1]. Note that exponents are often used to represent repeated values in a partition, so the last index could instead be written [1^4]. The combination function (sequence A007318) C(n,m) = C([m,n-m]).

This sequence is also the sequence of multinomial coefficients for partitions ordered lexicographically, matching partition sequence A080577. This is different ordering than in sequence A036038 of multinomial coefficients. - Sergei Viznyuk, Mar 15 2012

			The irregular table starts:
  [0] {1},
  [1] {1},
  [2] {1, 2},
  [3] {1, 3,  6},
  [4] {1, 4,  6, 12, 24},
  [5] {1, 5, 10, 20, 30, 60, 120},
  [6] {1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720}
  ...
C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).

T. D. Noe, Rows n=0..25 of triangle, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.
Sergei Viznyuk, C Program
Index entries for triangles and arrays related to Pascal's triangle.

Different from A036038.

Cf. A000041, A008480, A063008, A080577.

g:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 25 2020

Flatten[Table[Apply[Multinomial, IntegerPartitions[i], {1}], {i,0,25}]] (* T. D. Noe, Oct 14 2007 *)
Flatten[ Multinomial @@@ IntegerPartitions @ # & /@ Range[ 0, 8]] (* Michael Somos, Feb 05 2011 *)
g[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times@@(ee!)];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i] & /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
row[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}] & /@ b[n, n];
T[n_] := g /@ row[n];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

C(sig)={vecsum(sig)!/vecprod(apply(k->k!, sig))}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) }  \\ Andrew Howroyd, Mar 25 2020

def A070289_row(n): return [multinomial(x) for x in Partitions(n)]
print(flatten([A070289_row(n) for n in range(8)]))  # Peter Luschny, Jun 24 2025

C([]) = (Sum a(i))! / Product a(i) !.

T(n,k) = A008480(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 60, 8, 2310, 36, 420, 24, 30030, 180, 4620, 120, 510510, 1260, 72, 60060, 16, 900, 840, 9699690, 13860, 360, 1021020, 48, 6300, 9240, 223092870, 180180, 2520, 19399380, 240, 69300, 216, 120120, 6469693230, 1800, 3063060, 144, 44100, 27720, 446185740, 1680, 900900, 1080, 2042040, 200560490130, 12600, 58198140, 32, 720

Offset: 1

Matthew Vandermast, Dec 07 2010

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

			A025487(5) = 8 = 2^3 has a prime signature of (3). The partition that is conjugate to (3) is (1,1,1), and the member of A025487 with that prime signature is 30 = 2*3*5 (or 2^1*3^1*5^1).  Therefore, a(5) = 30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821.

A181825 lists members of A025487 with self-conjugate prime signatures. See also A181823-A181824, A181826-A181827.

f[n_] := Block[{ww, dec}, dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 + Length@ NestWhileList[NextPrime@ # &, 1, Times @@ {##} <= n &, All] ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= n, Sow[w]; k = 1, If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; Sort[Map[{Times @@ MapIndexed[Prime[First@ #2]^#1 &, #], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Table[LengthWhile[#1, # >= j &], {j, #2}]] & @@ {#, Max[#]}} &, Join @@ f[2310]]][[All, -1]] (* Michael De Vlieger, Oct 16 2018 *)

partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))
primeSignature(n)=vecsort(factor(n)[,2]~,,4)
f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))
A025487=[2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768];
concat(1, apply(f, A025487)) \\ Charles R Greathouse IV, Jun 02 2016

If A025487(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A025487(n))). a(n) also equals A108951(A181820(n)).

A096443 Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 11, 15, 7, 12, 16, 21, 26, 36, 52, 11, 19, 29, 38, 31, 52, 74, 66, 92, 135, 203, 15, 30, 47, 64, 57, 98, 141, 109, 137, 198, 296, 249, 371, 566, 877, 22, 45, 77, 105, 97, 171, 250, 109, 212, 269, 392, 592, 300, 444, 560, 850, 1315, 712, 1075

Offset: 0

Jon Wild, Aug 11 2004

nonn

The signature of a multiset is the partition consisting of the multiplicities of its elements; e.g., {a,a,a,b,c} is represented by [3,1,1]. The Mathematica order for partitions orders by ascending number of total elements, then by descending numerical order of its representation. The list begins:
n.....#elements.....n-th partition
....0 elements:....[]
....1 element:.....[1]
....2 elements:....[2]
...................[1,1]
....3 elements:....[3]
...................[2,1]
...................[1,1,1]
....4 elements:....[4]
...................[3,1]
...................[2,2]
..................[2,1,1]
..................[1,1,1,1]
...5 elements:....[5]
..................[4,1]
A000041 and A000110 are subsequences for conjugate partitions. A000070 and A035098 are also subsequences for conjugate partitions. - Alford Arnold, Dec 31 2005
A002774 and A020555 is another pair of subsequences for conjugate partitions. - Franklin T. Adams-Watters, May 16 2006

			The 10th partition is [2,1,1]. The partitions of a multiset whose elements have multiplicities 2,1,1 - for example, {a,a,b,c} - are:
{{a,a,b,c}}
{{a,a,b},{c}}
{{a,a,c},{b}}
{{a,b,c},{a}}
{{a,a},{b,c}}
{{a,b},{a,c}}
{{a,a},{b},{c}}
{{a,b},{a},{c}}
{{a,c},{a},{b}}
{{b,c},{a},{a}}
{{a},{a},{b},{c}}
We see there are 11 partitions of this multiset, so a(10)=11.
Also, a(n) is the number of distinct factorizations of A063008(n). For example, A063008(10) = 60 and 60 has 11 factorizations: 60, 30*2, 20*3, 15*4, 15*2*2, 12*5, 10*6, 10*3*2, 6*5*2, 5*4*3, 5*3*2*2 which confirms that a(10) = 11.

Jun Kyo Kim and Sang Guen Hahn, Recursive Formulae for the Multiplicative Partition Function, Internat. J. Math. & Math. Sci., 22(1) (1999), 213-216.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See P(n) page 3.

Cf. A035098, A035310.

MultiPartiteP[n : {___Integer?NonNegative}] :=
Block[{p, $RecursionLimit = 1024, firstPositive},
  firstPositive =
   Compile[{{vv, _Integer, 1}},
    Module[{k = 1}, Do[If[el == 0, k++, Break[]], {el, vv}]; k]];
  p[{0 ...}] := 1;
  p[v_] :=
   p[v] = Module[{len = Length[v], it, k, zeros, sum, pos, gcd},
     it = Array[k, len];
     pos = firstPositive[v];
     zeros = ConstantArray[0, len];
     sum = 0;
     Do[If[it == zeros, Continue[]];
      gcd = GCD @@ it;
      sum += it[[pos]] DivisorSigma[-1, gcd] p[v - it];,
      Evaluate[Sequence @@ Thread[{it, 0, v}]]];
     sum/v[[pos]]];
  p[n]];
ParallelMap[MultiPartiteP,
Flatten[Table[IntegerPartitions[k], {k, 0, 8}], 1]]
(* Oleksandr Pavlyk, Jan 23 2011 *)

Edited by Franklin T. Adams-Watters, May 16 2006

cp's OEIS Frontend

A238963 Number of divisors of A063008(n,k).

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A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

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A360440 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, is the number of ways of choosing nonnegative numbers for k indistinguishable A063008(n)-sided dice so that it is possible to roll every number from 0 to (A063008(n))^k-1.

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A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

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A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

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A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

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