A055932 -id:A055932 - cp's OEIS frontend

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.

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

A056808 Members of A055932 which are not least prime signatures (cf. A025487).

Original entry on oeis.org

18, 54, 90, 108, 150, 162, 270, 300, 324, 450, 486, 540, 600, 630, 648, 750, 810, 972, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 1944, 2100, 2250, 2400, 2430, 2700, 2916, 2940, 3000, 3150, 3240, 3750, 3780, 3888, 4050, 4200, 4374, 4410, 4500, 4800

Offset: 1

Alford Arnold, Aug 22 2000

			18 = 2*3*3 and all prime divisors are consecutive primes but the least prime signature is 12 = 2*2*3; so a(1) = 18.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A025487, A055932, A057335, A161360, A335485, A345974.

With[{nn = 4800}, Select[Range[2, nn], And[#1 != Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#3, Greater]], Last[#2] == Prime@ Length[#2]] & @@ Apply[Join, {{#1}, Transpose@ #2}] & @@ {#, FactorInteger[#]} &] ] (* Michael De Vlieger, Feb 06 2020 *)

{a(n) : n >= 1} = {A057335(A335485(k)) : k >= 1}. - Peter Munn, Feb 02 2024

Sum_{n>=1} 1/a(n) = A345974 - A161360 = 0.15229524564163275059... . - Amiram Eldar, Jun 26 2025

More terms from Larry Reeves (larryr(AT)acm.org), Nov 28 2000

A384530 Intersection of A055932 and A014574.

Original entry on oeis.org

4, 6, 12, 18, 30, 60, 72, 108, 150, 180, 192, 240, 270, 420, 432, 600, 810, 1050, 1152, 1620, 2310, 2592, 3000, 3360, 4050, 4800, 5880, 6300, 7350, 7560, 8820, 9000, 9240, 9720, 10500, 11550, 15360, 21600, 23040, 25410, 26250, 26880, 28350, 29400, 30870, 33600

Offset: 1

Ken Clements, Jun 01 2025

nonn

Numbers that have a complete set of prime factors from 2 to their greatest prime factor (A055932) that are bracketed by twin primes (in A014574).

The density of twin prime numbers around terms appears to be enhanced, since the blocks of prime factors of a(n) "sweep" out possible low prime factors from a(n)-1 and a(n)+1.

			4 is a term since 4 = prime(1)# * 2 is in A055932 and 4-1 = 3 and 4+1 = 5 are both prime.
6 is a term since 6 = prime(2)# is in A055932 and 6-1 = 5 and 6+1 = 7 are both prime.
30 is a term since 30 = prime(3)# is in A055932 and 30-1 = 29 and 30+1 = 31 are both prime.
420 is a term since 420 = prime(4)# * 2 is in A055932 and 420-1 = 419 and 420+1 = 421 are both prime.

Alois P. Heinz, Table of n, a(n) for n = 1..3719

Supersequence of A027856.

Cf. A014574, A055932.

q:= n-> andmap(isprime, [n-1, n+1]) and (s-> nops(s)=
        numtheory[pi](max(s)))({ifactors(n)[2][.., 1][]}):
select(q, [$1..40000])[];  # Alois P. Heinz, Jun 01 2025

from sympy import factorint, prime, isprime
def is_pi_complete(n):  # n is a term of A055932
    if n <= 1:
        return False
    factors = factorint(n)
    primes = list(factors.keys())
    max_prime, r = max(primes), len(primes)
    return max_prime == prime(r)
def is_twin_prime_bracketed(n):  # n is a term of A014574
    return isprime(n-1) and isprime(n+1)
def aupto(limit):
    return [n for n in range(4, limit+1, 2) if is_twin_prime_bracketed(n) and is_pi_complete(n)]
print(aupto(40_000))

A124829 Table of exponents of prime factorizations in A055932.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 3, 1, 1, 1, 1, 5, 2, 2, 4, 1, 1, 3, 2, 1, 1, 6, 3, 2, 1, 2, 1, 5, 1, 2, 3, 3, 1, 1, 7, 4, 2, 1, 1, 2, 1, 4, 2, 2, 1, 6, 1, 1, 1, 1, 1, 3, 3, 4, 1, 1, 8, 1, 3, 1, 5, 2, 2, 1, 2, 2, 4, 3, 2, 1, 7, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 5, 1, 1, 1, 5, 9, 2, 3, 1, 6, 2, 3, 1, 2, 1, 2, 1, 1, 3, 4

Offset: 1

Franklin T. Adams-Watters, Nov 09 2006

This is an enumeration of all compositions. This sequence contains all finite sequences of positive integers.

			From _Michael De Vlieger_, Feb 06 2020: (Start)
Table begins:
   n   A055932(n+1)  row n
   ---------------------
   1    2            1;
   2    4            2;
   3    6            1, 1;
   4    8            3;
   5   12            2, 1;
   6   16            4;
   7   18            1, 2;
   8   24            3, 1;
   9   30            1, 1, 1;
  10   32            5;
  11   36            2, 2;
  12   48            4, 1;
  13   54            1, 3;
  14   60            2, 1, 1;
  15   64            6;
  ...  (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10382 (rows 1 <= n <= 2500).

Cf. A055932, A124830 (row lengths), A124831 (row sums), A124832, A066099.

Map[FactorInteger[#][[All, -1]] &, Select[Range[10^3], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] // Flatten (* Michael De Vlieger, Feb 06 2020 *)

A055932(n) = Product_k Prime(k)^T(n,k).

A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 08 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.

			The a{n} multiset partitions for n = 8, 24, 72, 96:
  {{111}}      {{1112}}      {{11122}}      {{111112}}
  {{1}{11}}    {{1}{112}}    {{1}{1122}}    {{1}{11112}}
  {{1}{1}{1}}  {{11}{12}}    {{11}{122}}    {{11}{1112}}
               {{1}{1}{12}}  {{12}{112}}    {{111}{112}}
                             {{1}{1}{122}}  {{12}{1111}}
                             {{1}{12}{12}}  {{1}{1}{1112}}
                                            {{1}{11}{112}}
                                            {{11}{11}{12}}
                                            {{1}{12}{111}}
                                            {{1}{1}{1}{112}}
                                            {{1}{1}{11}{12}}
                                            {{1}{1}{1}{1}{12}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A080259, complement A055932.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Multisets covering an initial interval are counted by A000009, A000041, A011782, ranked by A055932.
Other types: A034691, A089259, A356954, A356955.
Other conditions: A050320, A050330, A322585, A356233, A356931, A356936.
Cf. A003963, A073491, A107742, A287170, A324929, A340852, A356234.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];
Table[Length[Select[facs[n],And@@nnQ/@#&]],{n,100}]

A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 26, 27, 28, 32, 36, 37, 38, 39, 42, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 72, 74, 76, 78, 81, 84, 89, 91, 96, 98, 104, 106, 108, 111, 112, 113, 114, 117, 122, 126, 128, 131, 133, 144, 147, 148, 151, 152

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
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 multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  32: {{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Multisets covering an initial interval are ctd by A011782, rkd by A055932.
This is the initial version of A356944.
Other types: A034691, A089259, A356945, A356954.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],And@@normQ/@primeMS/@primeMS[#]&]

A080404 a(n)=A007947[A055932(n)]; the sequence consists of primorial numbers;.

Original entry on oeis.org

2, 2, 6, 2, 6, 2, 6, 6, 30, 2, 6, 6, 6, 30, 2, 6, 30, 6, 6, 30, 2, 6, 30, 6, 30, 6, 210, 6, 30, 2, 30, 6, 30, 6, 30, 6, 210, 6, 30, 30, 6, 2, 30, 6, 30, 210, 6, 30, 30, 6, 30, 210, 6, 30, 30, 6, 2, 210, 30, 6, 30, 210, 6, 30, 30, 6, 210, 30, 6, 30, 210, 6, 30, 210, 30, 6, 2, 210, 30, 30, 6

Offset: 1

Labos Elemer, Mar 19 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A002110, A055932.

With[{P = FoldList[Times, Prime@ Range@ Max@ #]}, Map[P[[#]] &, #]] &@ Map[PrimeNu@ # &, Select[Range[10^4], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] (* Michael De Vlieger, Feb 06 2020 *)

A124830 Number of distinct prime factors of A055932(n).

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 09 2006

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from G. C. Greubel.)

Cf. A055932, A001221, A124829, A124831, A061394.

PrimeNu /@ Select[Range[4000], ! MemberQ[Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #, 0] &] (* Michael De Vlieger, Feb 02 2017 *)
A055932[n_] := Module[{f = Transpose[FactorInteger[n]][[1]]}, f == {1} || f == Prime[Range[Length[f]]]]; PrimeNu[Select[Range[2000], A055932]] (* G. C. Greubel, May 11 2017 *)

from sympy import nextprime, primefactors
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def ok(n): return True if n==1 else a053669(n)>max(primefactors(n))
print([len(primefactors(n)) for n in range(1, 10001) if ok(n)]) # Indranil Ghosh, May 11 2017

a(n) = A001221(A055932(n)).

A124831 Number of prime factors of A055932(n) with repetition.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 3, 5, 4, 5, 4, 4, 6, 5, 4, 6, 5, 5, 7, 6, 4, 5, 5, 7, 4, 6, 6, 8, 5, 7, 5, 6, 6, 8, 5, 7, 5, 7, 6, 9, 6, 8, 6, 5, 7, 7, 5, 9, 6, 6, 8, 6, 8, 7, 10, 5, 7, 9, 7, 6, 8, 6, 8, 7, 5, 6, 10, 7, 7, 9, 7, 6, 9, 8, 11, 6, 8, 6, 10, 5, 8, 7, 7, 9, 7, 9, 8, 6, 7, 11, 6, 8, 8, 10, 8, 6, 7, 10

Offset: 0

Franklin T. Adams-Watters, Nov 09 2006

nonn

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

Cf. A055932, A001222, A124829, A124831, A036041.

Map[PrimeOmega, {1}~Join~Select[Range[10^4], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] (* Michael De Vlieger, Feb 06 2020 *)

a(n) = A001222(A055932(n)).

A340346 The largest divisor of n that is a term of A055932 (numbers divisible by all primes smaller than their largest prime factor).

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12

Offset: 1

Peter Munn, Jan 04 2021

			For n=2: the largest divisor of 2 is 2, and 2 qualifies as divisible by all primes smaller than its largest prime factor, 2 (since there are no smaller primes). So a(2) = 2.
For n=42: of 42's divisors, no multiples of 7 qualify as being divisible by all primes smaller than their largest prime factor (since that factor is 7 and no divisor of 42 is divisible by 5, a smaller prime). The largest of 42's other divisors is 6, which qualifies (since it is divisible by 2, the only prime smaller than 6's largest prime factor, 3). So a(42) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

A003961, A006519, A055932, A064989, A341629 are used in a definition of this sequence.
Sequences with related definitions: A327832, A328479.
Cf. A234959.

a[?OddQ] = 1; a[n] := Module[{f = FactorInteger[n]}, ind = Position[PrimePi /@ First /@ f - Range @ Length[f], ?(# > 0 &)]; If[ind == {}, n, Times @@ Power @@@ f[[1 ;; ind[[1, 1]] - 1]]]]; Array[a, 100] (* _Amiram Eldar, Jan 14 2021 *)

is(n) = my(f=factor(n)[, 1]~); f==primes(#f); \\ A055932
a(n) = vecmax(select(is, divisors(n))); \\ Michel Marcus, Jan 19 2021

A341629(n) = if(1==n,1,my(f=factor(n)[, 1]~); (primepi(f[#f])==#f));
A340346(n) = fordiv(n,d,if(A341629(n/d),return(n/d))); \\ Antti Karttunen, Feb 25 2021

For n >= 1, a(2n-1) = 1, a(2n) = A006519(2n) * A003961(a(A064989(2n))).

For n >= 1, lcm(A006519(n), A234959(n)) | a(n).

cp's OEIS Frontend

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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A056808 Members of A055932 which are not least prime signatures (cf. A025487).

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A384530 Intersection of A055932 and A014574.

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A124829 Table of exponents of prime factorizations in A055932.

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A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

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A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

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A080404 a(n)=A007947[A055932(n)]; the sequence consists of primorial numbers;.

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A124830 Number of distinct prime factors of A055932(n).

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