A076610 -id:A076610 - cp's OEIS frontend

A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Gus Wiseman, Jul 18 2022

nonn

First differs from A355741 at a(169) = 3, A355741(169) = 4.

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(169) = 3 multisets are: {2,2}, {2,3}, {3,3}.
The a(507) = 3 multisets are: {2,2,2}, {2,2,3}, {2,3,3}.

Wikipedia, Axiom of choice.

Choosing from all divisors gives A355733, firsts A355734.
Counting sequences instead of multisets gives A355741.
Choosing weakly increasing sequences of divisors gives A355745.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A324850 lists numbers divisible by the product of their prime indices.
A344606 counts alternating permutations of prime indices.
Cf. A000720, A076610, A120383, A289509, A335433, A340852, A355731, A355735, A355737, A355739, A355740.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@primeMS[n]]]],{n,100}]

A302478 Products of prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 66, 67, 68, 72, 73, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 90, 93, 94

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01:  {}
02:  {{}}
03:  {{1}}
04:  {{},{}}
05:  {{2}}
06:  {{},{1}}
08:  {{},{},{}}
09:  {{1},{1}}
10:  {{},{2}}
11:  {{3}}
12:  {{},{},{1}}
13:  {{1,2}}
15:  {{1},{2}}
16:  {{},{},{},{}}
17:  {{4}}
18:  {{},{1},{1}}
20:  {{},{},{2}}
22:  {{},{3}}
24:  {{},{},{},{1}}
25:  {{2},{2}}
26:  {{},{1,2}}
27:  {{1},{1},{1}}
29:  {{1,3}}
30:  {{},{1},{2}}
31:  {{5}}
32:  {{},{},{},{},{}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A007716, A050320, A056239, A063834, A076610, A270995, A275024, A281113, A296119, A301753, A302242, A302243, A302491.

Select[Range[100],Or[#===1,And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

ok(n)={!#select(p->!issquarefree(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

Original entry on oeis.org

1, 3, 7, 9, 53, 21, 311, 27, 49, 159, 8161, 63, 38873, 933, 371, 81, 147, 477, 2177, 24483, 189, 2809, 343, 2799, 1113, 243, 57127, 16483, 441, 1431, 6531, 73449, 2597, 567, 96721, 8427, 1029, 8397, 3339, 15239, 729, 49449, 1323, 19663, 4293, 2401, 19593, 7791

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355731.
Appears to be a subset of A353397.
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:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     53: {16}
     21: {2,4}
    311: {64}
     27: {2,2,2}
     49: {4,4}
    159: {2,16}
   8161: {1024}
     63: {2,2,4}
For example, the choices for a(12) = 63 are:
  (1,1,1)  (1,2,2)  (2,1,4)
  (1,1,2)  (1,2,4)  (2,2,1)
  (1,1,4)  (2,1,1)  (2,2,2)
  (1,2,1)  (2,1,2)  (2,2,4)

Positions of first appearances in A355731.
Counting distinct sequences after sorting: A355734, firsts of A355733.
Requiring the result to be weakly increasing: A355736, firsts of A355735.
Requiring the result to be relatively prime: A355738, firsts of A355737.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Times@@Length/@Divisors/@primeMS[n],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Dec 15 2023

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.

By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Partitions of this type are counted by A239312, complement A370320.
Positions of nonzero terms in A355739.
Complement of A355740.
For just prime divisors we have A368100, complement A355529 (odd A355535).
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]),t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
  S:= map(t -> convert(t,name), [op(F),op(D)]);
  E:= map(e -> map(convert,e,name),E);
  G:= Graph(S,E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024

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

Heinz numbers of the partitions counted by A239312.

A355745 Number of ways to choose a prime factor of each prime index of n (with multiplicity, in weakly increasing order) such that the result is also weakly increasing.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Gus Wiseman, Jul 18 2022

nonn

First differs from A355741 and A355744 at n = 35.

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 prime indices of 1469 are {6,30}, and there are five valid choices: (2,2), (2,3), (2,5), (3,3), (3,5), so a(1469) = 5.

Wikipedia, Cartesian product.

Allowing all divisors gives A355735, firsts A355736, reverse A355749.
Not requiring an increasing sequence gives A355741.
Choosing a multiset instead of sequence gives A355744.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 chooses of a divisor of each prime index, firsts A355732.
A355733 chooses a multiset of divisors, firsts A355734.
Cf. A000720, A076610, A335433, A340852, A355737, A355739, A355740, A355742.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Union/@primeMS/@primeMS[n]],LessEqual@@#&]],{n,100}]

A257994 Number of prime parts in the partition having Heinz number n.

Original entry on oeis.org

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

Offset: 1

Emeric Deutsch, May 20 2015

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] 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] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
The number of nonprime parts is given by A330944, so A001222(n) = a(n) + A330944(n). - Gus Wiseman, Jan 17 2020

			a(30) = 2 because the partition with Heinz number 30 = 2*3*5 is [1,2,3], having 2 prime parts.

George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of positive terms are A331386.
Primes of prime index are A006450.
Products of primes of prime index are A076610.
The number of nonprime prime indices is A330944.
Cf. A000040, A000720, A001222, A007821, A018252, A056239, A112798, A215366, A279952, A302242, A320628, A330945.

with(numtheory): a := proc (n) local B, ct, s: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for s to nops(B(n)) do if isprime(B(n)[s]) = true then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 130);

B[n_] := Module[{nn, j, m}, nn = FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[  m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
a[n_] := Module[{ct, s}, ct = 0; For[s = 1, s <= Length[B[n]], s++, If[ PrimeQ[B[n][[s]]], ct++]]; ct];
Table[a[n], {n, 1, 130}] (* Jean-François Alcover, Apr 25 2017, translated from Maple *)
Table[Total[Cases[FactorInteger[n],{p_,k_}/;PrimeQ[PrimePi[p]]:>k]],{n,30}] (* Gus Wiseman, Jan 17 2020 *)

a(n) = my(f = factor(n)); sum(i=1, #f~, if(isprime(primepi(f[i, 1])), f[i, 2], 0)); \\ Amiram Eldar, Nov 03 2023

Additive with a(p^e) = e if primepi(p) is prime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 16 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(15) = 4 multisets are: {1,1}, {1,2}, {1,3}, {2,3}.
The a(18) = 3 multisets are: {1,1,1}, {1,1,2}, {1,2,2}.

Robert Price, Table of n, a(n) for n = 1..2000
Wikipedia, Axiom of choice.

Counting all choices of divisors gives A355731, firsts A355732.
Positions of first appearances are A355734.
Choosing weakly increasing divisors gives A355735, firsts A355736.
Choosing only prime divisors gives A355744.
The version choosing a divisor of each number from 1 to n is A355747.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A340852 lists numbers that can be factored into divisors of bigomega.
Cf. A000720, A076610, A289509, A316524, A335433, A335448, A340606, A344616, A355737, A355739, A355740, A355741, A355746.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,100}]

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

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. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A330944 Number of nonprime prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 13 2020

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.

			24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.

The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
Numbers whose prime indices are not all prime are A330945.
Cf. A000040, A000720, A001222, A007097, A018252, A056239, A112798, A302242, A320629, A320633, A330946, A330947.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020

a(n) + A257994(n) = A001222(n).

Additive with a(p^e) = e if primepi(p) is nonprime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A355737 Number of ways to choose a sequence of divisors, one of each prime index of n (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 4, 1, 1, 4, 1, 2, 4, 2, 1, 2, 3, 4, 7, 3, 1, 4, 1, 1, 4, 2, 6, 4, 1, 4, 6, 2, 1, 6, 1, 2, 8, 3, 1, 2, 5, 4, 4, 4, 1, 8, 4, 3, 5, 4, 1, 4, 1, 2, 10, 1, 6, 4, 1, 2, 6, 6, 1, 4, 1, 6, 8, 4, 6, 8, 1, 2, 15, 2, 1, 6, 4, 4

Offset: 1

Gus Wiseman, Jul 17 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(2) = 1 through a(18) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111
               12          12  13     112     12  13           112
                           21                 14  21           121
                                                  23           122

Wikipedia, Coprime integers.

Dominated by A355731, firsts A355732, primes A355741, prime-powers A355742.
For weakly increasing instead of coprime we have A355735, primes A355745.
Positions of first appearances are A355738.
For strict instead of coprime we have A355739, zeros A355740.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A289507, A296150, A302696, A302698, A355733, A355744, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}]

cp's OEIS Frontend

A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

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A302478 Products of prime numbers of squarefree index.

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A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

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Mathematica

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

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Maple

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Formula

A355745 Number of ways to choose a prime factor of each prime index of n (with multiplicity, in weakly increasing order) such that the result is also weakly increasing.

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Mathematica

A257994 Number of prime parts in the partition having Heinz number n.

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A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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