A052410 -id:A052410 - cp's OEIS frontend

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Nov 25 2023

Row n = 1 is empty.
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 multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.

			The first 45 rows:
     1: {}      16: {1}       31: {11}
     2: {1}     17: {7}       32: {1}
     3: {2}     18: {1,2}     33: {2,2}
     4: {1}     19: {8}       34: {1,1}
     5: {3}     20: {1,3}     35: {3,3}
     6: {1,1}   21: {2,2}     36: {1,1}
     7: {4}     22: {1,1}     37: {12}
     8: {1}     23: {9}       38: {1,1}
     9: {2}     24: {1,2}     39: {2,2}
    10: {1,1}   25: {3}       40: {1,3}
    11: {5}     26: {1,1}     41: {13}
    12: {1,2}   27: {2}       42: {1,1,1}
    13: {6}     28: {1,4}     43: {14}
    14: {1,1}   29: {10}      44: {1,5}
    15: {2,2}   30: {1,1,1}   45: {2,3}

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row minima are A055396.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Rows have Heinz numbers A367580.
Row sums are A367581.
Row maxima are A367583, opposite A367587.
Index of first row with Heinz number n is A367584.
Sorted row indices of first appearances are A367585.
Indices of rows of the form {1,1,...} are A367586.
Agrees with sorted prime signature at A367683, counted by A367682.
A submultiset of prime indices at A367685, counted by A367684.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2

Offset: 1

Gus Wiseman, Nov 28 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.

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.

Positions of 1's are A000079 without 1.
Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
The triangle A367579 has these as row sums, ranks A367580.
The triangle for this rank statistic is A367582.
For maximum instead of sum we have A367583, opposite A367587.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Total[mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[n]]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
a(n) = A056239(A367580(n)).
If n is squarefree, a(n) = A055396(n)*A001222(n).

A378035 Greatest perfect power < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 9, 16, 16, 16, 27, 27, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 125, 128, 128, 128, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 216, 225, 225, 225, 225, 225, 243, 256, 256, 256, 256

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

			The first number line below shows the perfect powers.
The second shows each positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

Restriction of A081676 to the primes.
Positions of last appearances are also A377283.
A version for squarefree numbers is A378032.
The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.
The union is A378253.
Terms appearing exactly once are A378355.
Run lengths are A378356, first differences of A377283, complement A377436.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the nonperfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A080769 counts primes between perfect powers, prime powers A067871.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]

a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024

from sympy import mobius, integer_nthroot, prime
def A378035(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = (p:=prime(n)-1)-f(p)
    return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024

A378249 Least perfect power > prime(n).

Original entry on oeis.org

4, 4, 8, 8, 16, 16, 25, 25, 25, 32, 32, 49, 49, 49, 49, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 121, 121, 121, 121, 121, 128, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 216, 216, 216, 225, 243, 243, 243, 243, 243, 256, 289, 289, 289

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Which terms appear only once? Just 128, 225, 256, 64009, 1295044?

			The first number line below shows the perfect powers. The second shows each prime.
-1-----4-------8-9------------16----------------25--27--------32------36------------------------49--
===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======

A version for prime powers (but starting with prime(k) + 1) is A345531.
Positions of last appearances are A377283, complement A377436.
Restriction of A377468 to the primes, for prime powers A000015.
The opposite is A378035, restriction of A081676.
The union is A378250.
Run lengths are A378251.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists numbers that are not perfect powers, differences A375706, seconds A376562.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]

f(p) = p++; while(!ispower(p), p++); p;
lista(nn) = apply(f, primes(nn)); \\ Michel Marcus, Dec 19 2024

A378251 Number of primes between consecutive perfect powers, zeros omitted.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 1, 3, 5, 5, 3, 1, 5, 1, 7, 5, 2, 4, 6, 7, 7, 5, 2, 6, 9, 8, 7, 8, 9, 8, 8, 6, 4, 9, 10, 9, 10, 7, 2, 9, 12, 11, 12, 6, 5, 9, 12, 11, 3, 10, 8, 2, 13, 15, 10, 11, 15, 7, 9, 12, 13, 11, 12, 17, 2, 11, 16, 16, 13, 17, 15, 14, 16, 15

Offset: 1

Gus Wiseman, Nov 23 2024

First differences of A377283 and A378365. Run-lengths of A378035 and A378249.

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

			The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==

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

Same as A080769 with 0's removed (which were at positions A274605).
First differences of A377283 and A378365 (union of A378356).
Run-lengths of A378035 (union A378253) and A378249 (union A378250).
The version for nonprime prime powers is A378373, with zeros A067871.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, run-lengths of A377468.
A007916 lists the non-perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
Cf. A007918, A023055, A045542, A052410, A076412, A081676, A188951.
Cf. A216765, A377057, A377431, A378355.

N:= 10^6: # to use perfect powers up to N
PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
PP:= sort(convert(PP,list)):
M:= map(numtheory:-pi, PP):
subs(0=NULL, M[2..-1]-M[1..-2]): # Robert Israel, Jan 23 2025

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 6, 3, 2, 2, 2, 6, 3, 2, 2, 2, 2, 10, 3, 3, 2, 2, 2, 2, 10, 5, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 26, 7, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Marc LeBrun, Jan 08 2005

All entries above main diagonal are = 1.

			The T(9,3)=5 partitions of 9 into powers of 3: 111111111, 1111113, 11133, 333, 9.
From _Gus Wiseman_, Jun 07 2019: (Start)
Triangle begins:
   2
   2  2
   4  2  2
   4  2  2  2
   6  3  2  2  2
   6  3  2  2  2  2
  10  3  3  2  2  2  2
  10  5  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2  2
  26  7  4  3  3  3  2  2  2  2  2  2  2
  26  9  4  4  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2  2
  46 12  6  4  4  3  3  3  2  2  2  2  2  2  2  2  2
Row n = 8 counts the following partitions:
  8          3311       44         5111       611        71         8
  44         311111     41111      11111111   11111111   11111111   11111111
  422        11111111   11111111
  2222
  4211
  22211
  41111
  221111
  2111111
  11111111
(End)

Alois P. Heinz, Rows n = 2..500, flattened

Cf. A102431, A102432, A102433, A102434, A001700, A018819, A062051, A008645, A008650, A008652, A008648.
Same as A308558 except for the k = 1 column.
Row sums are A102431.
First column (k = 2) is A018819.
Second column (k = 3) is A062051.
Cf. A000961, A001597, A007916, A023894, A052410, A112344.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1, k)+(p-> `if`(p>n, 0, b(n-p, i, k)))(k^i)))
    end:
T:= (n, k)-> b(n, ilog[k](n), k):
seq(seq(T(n, k), k=2..n), n=2..20);  # Alois P. Heinz, Oct 12 2019

Table[Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]],{n,2,10},{k,2,n}] (* Gus Wiseman, Jun 07 2019 *)

T(1, k) = 1, T(n, 1) = choose(2n-1, n), T(n>1, k>1) = T(n-1, k) + (T(n/k, k) if k divides n, else 0)

Corrected and rewritten by Gus Wiseman, Jun 07 2019

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

The positions of zeros in this sequence are the prime powers A000961.

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 2, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 3, 2, 6, 16, 2, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 3, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 28 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.

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.

			For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.

Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
For minimum instead of maximum element we have A055396.
Row maxima of A367579.
Greatest prime index of A367580.
Positions of 1's are A367586 (powers of even squarefree numbers).
The opposite version is A367587.
A007947 gives squarefree kernel.
A072774 lists powers of squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A363486 gives least prime index of greatest exponent.
A363487 gives greatest prime index of greatest exponent.
A364191 gives least prime index of least exponent.
A364192 gives greatest prime index of least exponent.
Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.

mmk[q_]:=With[{mts=Length/@Split[q]},Sort[Table[Min@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

a(n) = A061395(A367580(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A055396(n).

A295924 Number of twice-factorizations of n of type (R,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is the number of ways to choose an integer partition of a divisor of A052409(n).

			The a(16) = 8 twice-factorizations are (2)*(2)*(2)*(2), (2)*(2)*(2*2), (2)*(2*2*2), (2*2)*(2*2), (2*2*2*2), (4)*(4), (4*4), (16).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000041, A001055, A047968, A052409, A052410, A089723, A281113, A284639, A295923, A295931, A295935, A296134.

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsP],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295924(n) = if(1==n,n,sumdiv(A052409(n),d,numbpart(d))); \\ Antti Karttunen, Jul 29 2018

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000041(d). - Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

cp's OEIS Frontend

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

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A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

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A378035 Greatest perfect power < prime(n).

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A378249 Least perfect power > prime(n).

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A378251 Number of primes between consecutive perfect powers, zeros omitted.

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A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.

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A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

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