A324939 -id:A324939 - cp's OEIS frontend

A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

1, 2, 4, 6, 8, 12, 16, 18, 24, 30, 32, 36, 48, 54, 60, 64, 72, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192, 210, 216, 240, 256, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 512, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972

Offset: 1

Leroy Quet, Jul 17 2000

a(n) is also the sorted version of A057335 which is generated recursively using the formula A057335 = A057334 * A057335(repeated), where A057334 = A000040(A000120). - Alford Arnold, Nov 11 2001
Squarefree kernels of these numbers are primorial numbers. See A080404. - Labos Elemer, Mar 19 2003
If u and v are terms then so is u*v. - Reinhard Zumkeller, Nov 24 2004
Except for the initial value a(1) = 1, a(n) gives the canonical primal code of the n-th finite sequence of positive integers, where n = (prime_1)^c_1 * ... * (prime_k)^c_k is the code for the finite sequence c_1, ..., c_k. See examples of primal codes at A106177. - Jon Awbrey, Jun 22 2005
From Daniel Forgues, Jan 24 2011: (Start)
Least integer, in increasing order, of each ordered prime signature.
The least integer of each ordered prime signature are the smallest numbers with a given tuple of exponents of prime factors.
The ordered prime signature (where the order of exponents matters) of n corresponds to a given composition of Omega(n), as opposed to the prime signature of n, which corresponds to a given partition of Omega(n). (End)
Except for the initial entry 1, the entries of the sequence are the Heinz numbers of all partitions that contain all parts 1,2,...,k, where k is the largest part. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436. The number 150 (= 2*3*5*5) is in the sequence because it is the Heinz number of the partition [1,2,3,3]. - Emeric Deutsch, May 22 2015
Numbers n such that A053669(n) > A006530(n). - Anthony Browne, Jun 06 2016
From David W. Wilson, Dec 28 2018: (Start)
Numbers n such that for primes p > q, p | n => q | n.
Numbers n such that prime p | n => A034386(p) | n. (End)

			60 is included because 60 = 2^2 * 3 * 5 and 2, 3 and 5 are consecutive primes beginning at 2.
Sequence A057335 begins
1..2..4..6..8..12..18..30..16..24..36..60..54..90..150..210... which is equal to
1..2..2..3..2...3...3...5...2...3...3...5...3...5....5....7... times
1..1..2..2..4...4...6...6...8...8..12..12..18..18...30...30...

Michael De Vlieger, Table of n, a(n) for n = 1..10000, first 1001 terms from Franklin T. Adams-Watters.
Jon Awbrey, Riffs and Rotes.
Michael De Vlieger, Extended table of n, a(n) for n = 1..100000.
Robert Vajda, Computational Exploration of the Degree Sequence of the Malyshev Polynomials, Proceedings of the 11th International Conference on Applied Informatics (Eger, Hungary, 2020).
Index entries for sequences related to prime signature.

Cf. A057335 (permuted), A056808, A025487, A007947, A002110, A080404, A034386, A106177, A124829, A124830, A124831, A124833, A080259 (complement), A215366.

Cf. A005867, A324939, A345974.

[1] cat [k:k in[2..1000 by 2]|forall{i:i in [1..#PrimeDivisors(k)-1]|NextPrime(pd[i]) in pd where pd is PrimeDivisors(k)}]; // Marius A. Burtea, Feb 01 2020

isA055932 := proc(n)
    local s,p ;
    s := numtheory[factorset](n) ;
    for p in s do
        if p > 2 and not prevprime(p)  in s then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 2 to 100 do
    if isA055932(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 02 2012

Select[Range[1000], #==1||FactorInteger[ # ][[ -1, 1]]==Prime[Length[FactorInteger[ # ]]]&]
cpQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},f=={1}||f==Prime[ Range[Length[f]]]]; Select[Range[1000],cpQ] (* Harvey P. Dale, Jul 14 2012 *)

is(n)=my(f=factor(n)[,1]~);f==primes(#f) \\ Charles R Greathouse IV, Aug 22 2011

list(lim,p=2)=my(v=[1],q=nextprime(p+1),t=1);while((t*=p)<=lim,v=concat(v,t*list(lim\t,q))); vecsort(v) \\ Charles R Greathouse IV, Oct 02 2012

from itertools import count, islice
from sympy import primepi, primefactors
def A055932_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        p = list(map(primepi,primefactors(k)))
        if k==1 or (min(p)==1 and max(p)==len(p)):
            yield k
A055932_list = list(islice(A055932_gen(),40)) # Chai Wah Wu, Aug 07 2025

Sum_{n>=1} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

Edited by Daniel Forgues, Jan 24 2011

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

A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410

Offset: 0

Alford Arnold, Aug 27 2000

Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020

			From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     12: {1,1,2}
     18: {1,2,2}
     30: {1,2,3}
     16: {1,1,1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     60: {1,1,2,3}
     54: {1,2,2,2}
     90: {1,2,2,3}
    150: {1,2,3,3}
    210: {1,2,3,4}
     32: {1,1,1,1,1}
     48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000120, A057334, A055932 and A056808.
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A005867, A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220, A345974.
Related to A019565 via A122111 and to A000079 via A336321.

Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)

mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014

A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020

a(n) = A057334(n) * a (repeated).
A334032(a(n)) = n; a(A334032(n)) = A071364(n). - Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)). - Peter Munn, Jul 18 2020
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
Sum_{n>=0} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

New primary name from Antti Karttunen, Jul 20 2020

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0

Offset: 0

Alois P. Heinz, Apr 27 2021

			A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
Square array A(n,k) begins:
  1,  1,    1,     1,      1,        1,        1, ...
  0,  2,    5,    10,     17,       28,       41, ...
  0,  4,   19,    69,    188,      496,     1029, ...
  0,  8,   65,   410,   1726,     7182,    20559, ...
  0, 16,  211,  2261,  14343,    93345,   360612, ...
  0, 32,  665, 11970, 112371,  1139166,  5827122, ...
  0, 64, 2059, 61909, 848506, 13379332, 89131918, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-4 give: A000007, A000079, A001047(n+1), A016273, A025931.
Rows n=0-2 give: A000012, A007504, A357251.
Main diagonal gives A332967.
Cf. A000040, A074140, A124960, A324939, A325054.

A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1,
     If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)

A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).

A(n,k) = A124960(n+k,k) for k >= 1.

A325054 Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).

Original entry on oeis.org

1, 2, 10, 68, 640, 8372, 147820, 3321908, 90184300, 2857153892, 104146026820, 4363900557128, 209763325978480, 11462371025215112, 702793156696129600, 47649412958404240688, 3521160558576929028400, 280427910532671712997732, 23932837097476310995036900

Offset: 0

Alois P. Heinz, Sep 04 2019

nonn

			The compositions of 3 and their encodings are [1,1,1]-> 30, [1,2]-> 18, [2,1]-> 12, [3]-> 8.  The sum gives a(3) = 68.

Alois P. Heinz, Table of n, a(n) for n = 0..335
Wikipedia, Gödel's encoding

Row sums of A324939.

Cf. A000040, A074140.

b:= proc(n, j) option remember; `if`(n=0, 1,
      add(ithprime(j)^i*b(n-i, j+1), i=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 04 2019

b[n_, j_] := b[n, j] = If[n==0, 1, Sum[Prime[j]^i*b[n-i, j+1], {i, 1, n}]];
a[n_] := b[n, 1];
a /@ Range[0, 20] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)

A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

Original entry on oeis.org

2, 4, 6, 8, 30, 16, 36, 210, 32, 2310, 64, 216, 900, 30030, 128, 510510, 256, 1296, 44100, 9699690, 512, 27000, 223092870, 1024, 7776, 5336100, 6469693230, 2048, 200560490130, 4096, 46656, 810000, 9261000, 901800900, 7420738134810, 8192, 304250263527210

Offset: 1

Gus Wiseman, May 25 2025

A permutation of A100778 (powers of primorials).
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.
An integer partition is uniform iff all parts appear with the same multiplicity, and normal iff it covers an initial interval of positive integers.

			The uniform normal multisets of length 6 are: {1,1,1,1,1,1}, {1,1,1,2,2,2}, {1,1,2,2,3,3}, {1,2,3,4,5,6}, so row 6 is: 64, 216, 900, 30030.
Triangle begins:
    2
    4       6
    8      30
   16      36    210
   32    2310
   64     216    900    30030
  128  510510
  256    1296  44100  9699690

Row lengths are A000005.
Final term in each row is A002110.
The union is A100778.
Reversing rows gives A322792.
For just normal multisets we have A324939.
A047966 counts uniform partitions.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A381431 is the section-sum transform.
Cf. A000009, A001694, A056239, A106529, A112798, A116540, A122111, A258280, A325326, A325337.

Table[Table[Times@@Prime/@Range[d]^(n/d),{d,Divisors[n]}],{n,10}]

cp's OEIS Frontend

A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

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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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A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

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A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A325054 Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).

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A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

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