A050361 -id:A050361 - cp's OEIS frontend

A050360 Abelian groups (factorizations into prime powers >1) indexed by prime signatures. A000688(A025487).

1, 1, 2, 1, 3, 2, 5, 3, 1, 7, 4, 5, 2, 11, 6, 7, 3, 15, 10, 4, 11, 1, 9, 5, 22, 14, 6, 15, 2, 15, 7, 30, 22, 10, 22, 3, 21, 8, 11, 42, 9, 30, 4, 25, 14, 30, 5, 33, 12, 15, 56, 15, 44, 1, 6, 35, 22, 42, 7, 45, 20, 22, 77, 21, 60, 2, 10, 55, 18, 30, 56, 8, 25, 11, 66, 28, 9, 30, 49, 101

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to groups

Cf. A000041, A000688, A000961, A050361-A050364.

Cf. A025487.

a050360 = a000688 . a025487  -- Reinhard Zumkeller, Aug 28 2014

More terms from David Wasserman, Feb 14 2002

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

Original entry on oeis.org

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193

Offset: 1

Gus Wiseman, Jul 25 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.

These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}

Heinz numbers of the partitions counted by A023893.
Allowing prime index 1 gives A356066.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A023894 counts partitions into prime-powers, strict A054685.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@PrimePowerQ/@DeleteCases[primeMS[#],1]&]

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Apr 22 2025

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, sum A056239.

			The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
  40: {1,1,1,3}

Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.
Includes all elements of A353833.
For distinct sums we have the complement of A381636.
For strict blocks we have the complement of A381719.
For distinct sums and strict blocks we have the complement of A381806.
The complement is A381871, counted by A381993.
These are the positions of positive terms in A381995.
Partitions of this type are counted by A383093.
Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
A001055 counts factorizations (multiset partitions of prime indices), strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A321469, A381635, A381715, A381716, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn, {ptn,IntegerPartitions[Length[y]]}];
Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]

A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A384913 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with squarefree exponents 1 and 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A073576, A209061, A384419.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384913, A384914, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Abs[MoebiusMu[d]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n]; (* Jean-François Alcover at A073576 *)
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquarefree(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A073576(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.1069024289184419840496..., where f(x) = (1-x) / Product_{k>=1} (1-x^A005117(k)).

A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A384912 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are Fibonacci numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A003107, A115063, A115975.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384914, A384915, A384916.

fib[n_] := Boole[Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]];
s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * fib[d], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*isfib(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A003107(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.05893526314055968638..., where f(x) = (1-x) / Product_{k>=2} (1-x^A000045(k)).

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.

			a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001156, A197680, A323520.
Cf. A043289, A053164, A063775, A203640, A295658, A365333.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquare(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A001156(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are <= 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A026820, A048103, A074583.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384916.

f[p_, e_] := Length[IntegerPartitions[e, p]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]));}

Multiplicative with a(p^e) = A026820(e, p).

a(n) >= A384916(n), with equality if and only if n is in A048103.

A384916 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e < p.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A298735 at n = 125.

			a(9) = 2 since 9 has 2 factorizations: 3^1 * 3^1 and 3^2, with exponents 1 and 2 that are < 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A026820, A048103, A298735.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915.

f[p_, e_] := Length[IntegerPartitions[e, p-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]-1));}

Multiplicative with a(p^e) = A026820(e, p-1).

a(n) <= A384915(n), with equality if and only if n is in A048103.

A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101

Offset: 1

Gus Wiseman, Jul 31 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 terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

The complement is A355743, counted by A023894.
The squarefree complement is A356065, counted by A054685.
Allowing prime index 1 gives A356064, complement A302492.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

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

Union of A299174 and A356064.

A382215 MM-numbers of multiset partitions into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 35, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461

Offset: 1

Gus Wiseman, Mar 21 2025

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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of 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 terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}

Twice-partitions of this type are counted by A279789.
For just constant blocks we have A302492, counted by A000688.
For sets of constant multisets we have A302496, counted by A050361.
For just common sums we have A326534, counted by A321455.
Factorizations of this type are counted by A381995.
For strict blocks and distinct sums we have A382201, counted by A381633.
Normal multiset partitions of this type are counted by A382204.
For strict instead of constant blocks we have A382304, counted by A382080.
For sets of constant multisets with distinct sums A382426, counted by A381635.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A000720, A000961, A001055, A302242, A302601, A368100, A381715, A381716, A381636, A381871.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#] && And@@SameQ@@@prix/@prix[#]&]

is(k) = my(f=factor(k)[, 1]~, k, p, v=vector(#f, i, primepi(f[i]))); for(i=1, #v, k=isprimepower(v[i], &p); if(k||v[i]==1, v[i]=k*primepi(p), return(0))); #Set(v)<2; \\ Jinyuan Wang, Apr 02 2025

Equals A326534 /\ A302492.

cp's OEIS Frontend

A050360 Abelian groups (factorizations into prime powers >1) indexed by prime signatures. A000688(A025487).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Extensions

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A384916 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e < p.

Views

Author

Keywords

Comments