A181821 -id:A181821 - cp's OEIS frontend

A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

This is the sorted list of positions of first appearances in A318284 of each element of the range A045782.

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 inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			Factorizations of the inverse prime shadows of the initial terms:
    4    8      12     16       36       24       60       48
    2*2  2*4    2*6    2*8      4*9      3*8      2*30     6*8
         2*2*2  3*4    4*4      6*6      4*6      3*20     2*24
                2*2*3  2*2*4    2*18     2*12     4*15     3*16
                       2*2*2*2  3*12     2*2*6    5*12     4*12
                                2*2*9    2*3*4    6*10     2*3*8
                                2*3*6    2*2*2*3  2*5*6    2*4*6
                                3*3*4             3*4*5    3*4*4
                                2*2*3*3           2*2*15   2*2*12
                                                  2*3*10   2*2*2*6
                                                  2*2*3*5  2*2*3*4
                                                           2*2*2*2*3
The corresponding multiset partitions:
    {11}    {111}      {112}      {1111}        {1122}        {1112}
    {1}{1}  {1}{11}    {1}{12}    {1}{111}      {1}{122}      {1}{112}
            {1}{1}{1}  {2}{11}    {11}{11}      {11}{22}      {11}{12}
                       {1}{1}{2}  {1}{1}{11}    {12}{12}      {2}{111}
                                  {1}{1}{1}{1}  {2}{112}      {1}{1}{12}
                                                {1}{1}{22}    {1}{2}{11}
                                                {1}{2}{12}    {1}{1}{1}{2}
                                                {2}{2}{11}
                                                {1}{1}{2}{2}

Taking n instead of the inverse prime shadow of n gives A330972.
Factorizations are A001055, with image A045782, with complement A330976.
Factorizations of inverse prime shadows are A318284.
Cf. A025487, A033833, A045778, A045783, A181819, A181821, A318286, A325238, A330973, A330989, A330990, A330993, A330997.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
nds=Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,50}];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

Original entry on oeis.org

3, 4, 6, 18, 450, 205439850, 241382525361273331926149714645357743772646450

Offset: 0

Gus Wiseman, May 16 2018

nonn

The first entry 3 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is A014643. The number of k's in row n + 1 is equal to the k-th term of row n. The length of row n is A014644(n).
        3: {2}
        4: {1,1}
        6: {1,2}
       18: {1,2,2}
      450: {1,2,2,3,3}
205439850: {1,2,2,3,3,4,4,4,5,5,5}

Cf. A001462, A014643, A014644, A055932, A056239, A112798, A130091, A181819, A181821, A182850-A182858, A296150, A275870, A304455, A304678.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},6]

A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

Original entry on oeis.org

1, 2, 3, 4, 6, 15, 44

Offset: 1

Gus Wiseman, Jan 07 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. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
  ()  (2)  (4)    (6)    (12)     (72)
           (2*2)  (2*3)  (2*6)    (8*9)
                         (3*4)    (2*36)
                         (2*2*3)  (3*24)
                                  (4*18)
                                  (6*12)
                                  (2*4*9)
                                  (2*6*6)
                                  (3*3*8)
                                  (3*4*6)
                                  (2*2*18)
                                  (2*3*12)
                                  (2*2*2*9)
                                  (2*2*3*6)
                                  (2*3*3*4)
                                  (2*2*2*3*3)

The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]

A001055(A181821(a(n))) = 2^k for some k >= 0.

A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 5, 2, 1, 3, 0, 0, 0, 1, 0, 6, 1, 0, 2, 4, 0, 0, 1, 0, 0, 1, 0, 9, 3, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0, 1, 1, 6, 15, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 6, 2, 0, 0, 1, 0, 17, 1, 0, 7, 2, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(84) = 7 factorizations into semiprimes:
  84 = (4*4*9*35)
  84 = (4*4*15*21)
  84 = (4*6*6*35)
  84 = (4*6*10*21)
  84 = (4*6*14*15)
  84 = (4*9*10*14)
  84 = (6*6*10*14)
The a(84) = 7 multiset partitions into pairs:
  {{1,1},{1,1},{2,2},{3,4}}
  {{1,1},{1,1},{2,3},{2,4}}
  {{1,1},{1,2},{1,2},{3,4}}
  {{1,1},{1,2},{1,3},{2,4}}
  {{1,1},{1,2},{1,4},{2,3}}
  {{1,1},{2,2},{1,3},{1,4}}
  {{1,2},{1,2},{1,3},{1,4}}

Cf. A001221, A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320659.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
bepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[bepfacs[Times@@Prime/@nrmptn[n]]],{n,100}]

A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(48) = 6 factorizations:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(48) = 6 multiset partitions:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}

A001147(n) = a(2^(2n)).

Cf. A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320658.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
qepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[qepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[qepfacs[Times@@Prime/@nrmptn[n]]],{n,100}]

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, -1, 0, 2, 1, 1, 1, -2, 0, 1, 0, 0, 0, 2, 0, -2, -2, -1, 1, -1, -2, 3, -1, 1, -2, -3, -2, 3, 0, -3, 1, -4, -5, 1, -1, -2, -1, 5, -5, 1, -3, 1, -1, -5, -4, 5, 1, -1, -9, -2, -1, -6, -1, -3, -2, 7, -7, -8, -2, -2

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^(Length[m]-1),{m,mps[nrmptn[n]]}],{n,30}]

a(n) = A316441(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -2, -1, 0, -2, 0, -2, -1, -1, -1, -4, -1, -1, -1, -3, 0, -3, 0, -2, -4, -1, -1, -6, -2, -3, -2, -2, 0, -6, -2, -4, -1, -1, 0, -5, 0, -1, -3, -9, -2, -3, 0, -2, -1, -3, 0, -7, 0

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]

a(n) = A114592(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A361668 Numbers k such that A361662(k) != A181821(A361666(k)).

Original entry on oeis.org

30, 51, 60, 89, 102, 105, 113, 119, 120, 128, 135, 145, 149, 150, 153, 168, 178, 179, 181, 191, 200, 204, 210, 215, 219, 221, 224, 226, 238, 240, 245, 248, 256, 257, 267, 270, 277, 281, 290, 298, 299, 300, 305, 306, 313, 317, 323, 336, 343, 345, 349, 356, 357

Offset: 1

Pontus von Brömssen, Mar 20 2023

nonn

Equivalently, numbers k such that A361666(k) != A181820(A361663(k)).

Cf. A181820, A181821, A361662, A361663, A361666.

A365460 Number of distinct primorials in the greedy sum of primorials adding to A181821(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Cf. A122111, A181821, A267263, A329040.

Cf. also A365461.

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); };
A267263(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += !!d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A365460(n) = A267263(A181821(n));

a(n) = A267263(A181821(n)).
a(n) = A329040(A122111(n)).
a(n) <= A365461(n).

A365461 Sum of digits when A181821(n) is written in primorial base (A049345).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 2, 4, 2, 2, 4, 4, 4, 1, 6, 6, 6, 4, 8, 4, 6, 2, 2, 8, 6, 2, 12, 6, 16, 1, 6, 10, 4, 6, 16, 10, 8, 4, 10, 6, 10, 4, 12, 10, 6, 2, 8, 6, 10, 8, 12, 10, 8, 8, 20, 10, 18, 2, 26, 14, 8, 1, 12, 12, 22, 10, 20, 12, 28, 6, 28, 14, 10, 10, 6, 8, 34, 4, 8, 18, 38, 4, 10, 14, 20, 6, 52, 10, 12, 10, 24

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Minimal number of primorials (A002110) that add to A181821(n).

Cf. A002110, A049345, A122111, A181821, A276150, A324888, A365460.

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); };
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A365461(n) = A276150(A181821(n));

a(n) = A276150(A181821(n)).
a(n) = A324888(A122111(n)).
a(n) >= A365460(n).

cp's OEIS Frontend

A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

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A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

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A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

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A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.

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A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

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A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

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A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

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Maple

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A361668 Numbers k such that A361662(k) != A181821(A361666(k)).

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A365460 Number of distinct primorials in the greedy sum of primorials adding to A181821(n).

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