A302696 -id:A302696 - cp's OEIS frontend

A327513 Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.

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

Offset: 1

Gus Wiseman, Sep 19 2019

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. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.
Note that the maximum odd divisor of any entry must be squarefree.
Number of terms of A302696 that divide n. Put in other words, this sequence is the inverse Möbius transform of the characteristic function of A302696. - Antti Karttunen, Dec 06 2021

			The divisors of 72 that are 1, 2, or nonprime numbers whose prime indices are pairwise coprime are: {1, 2, 4, 6, 8, 12, 24}, so a(72) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

See link for additional cross-references.

Cf. A000005, A056239, A112798, A302569, A302696, A304711.

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

isA302696(n) = if(isprimepower(n),!(n%2), if(!issquarefree(n>>valuation(n,2)), 0, my(pis=apply(primepi,factor(n)[,1])); (lcm(pis)==factorback(pis))));
A327513(n) = sumdiv(n,d,isA302696(d)); \\ Antti Karttunen, Dec 06 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 95, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 131, 135, 137, 139, 141, 143, 145, 149, 151

Offset: 1

Gus Wiseman, Oct 24 2020

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. a(n) gives the n-th Heinz number of an integer partition with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime (A338317).

			The sequence of terms together with their prime indices begins:
      1: {}          33: {2,5}       71: {20}
      3: {2}         35: {3,4}       73: {21}
      5: {3}         37: {12}        75: {2,3,3}
      7: {4}         41: {13}        77: {4,5}
      9: {2,2}       43: {14}        79: {22}
     11: {5}         45: {2,2,3}     81: {2,2,2,2}
     13: {6}         47: {15}        83: {23}
     15: {2,3}       49: {4,4}       85: {3,7}
     17: {7}         51: {2,7}       89: {24}
     19: {8}         53: {16}        93: {2,11}
     23: {9}         55: {3,5}       95: {3,8}
     25: {3,3}       59: {17}        97: {25}
     27: {2,2,2}     61: {18}        99: {2,2,5}
     29: {10}        67: {19}       101: {26}
     31: {11}        69: {2,9}      103: {27}

A338315 does not consider singletons coprime, with Heinz numbers A337987.
A338317 counts the partitions with these Heinz numbers.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A007359 counts singleton or pairwise coprime partitions with no 1's, with Heinz numbers A302568.
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
A302797 lists squarefree numbers whose distinct parts are pairwise coprime.
A304709 counts partitions whose distinct parts are pairwise coprime, with Heinz numbers A304711.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337485 counts pairwise coprime partitions with no 1's, with Heinz numbers A337984.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A056239, A112798, A220377, A289509, A302569, A303282, A318719, A328673, A328867.

Select[Range[1,100,2],#==1||PrimePowerQ[#]||CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376

Offset: 0

Gus Wiseman, Oct 24 2020

nonn

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222

A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Cf. A051424, A289508, A302569, A303140, A337694.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

Original entry on oeis.org

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 205, 209, 215, 217, 219, 221, 231, 249, 253, 255, 265, 285, 287, 291, 295, 309, 323, 327, 329, 335, 341, 345, 355, 357, 381, 385, 391, 395, 403, 407, 411

Offset: 1

Gus Wiseman, Oct 29 2020

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.

Also Heinz numbers of relatively prime strict integer partitions with no 1's (A337452). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}      145: {3,10}     249: {2,23}     355: {3,20}
     33: {2,5}      155: {3,11}     253: {5,9}      357: {2,4,7}
     35: {3,4}      161: {4,9}      255: {2,3,7}    381: {2,31}
     51: {2,7}      165: {2,3,5}    265: {3,16}     385: {3,4,5}
     55: {3,5}      177: {2,17}     285: {2,3,8}    391: {7,9}
     69: {2,9}      187: {5,7}      287: {4,13}     395: {3,22}
     77: {4,5}      195: {2,3,6}    291: {2,25}     403: {6,11}
     85: {3,7}      201: {2,19}     295: {3,17}     407: {5,12}
     93: {2,11}     205: {3,13}     309: {2,27}     411: {2,33}
     95: {3,8}      209: {5,8}      323: {7,8}      413: {4,17}
    105: {2,3,4}    215: {3,14}     327: {2,29}     415: {3,23}
    119: {4,7}      217: {4,11}     329: {4,15}     429: {2,5,6}
    123: {2,13}     219: {2,21}     335: {3,19}     435: {2,3,10}
    141: {2,15}     221: {6,7}      341: {5,11}     437: {8,9}
    143: {5,6}      231: {2,4,5}    345: {2,3,9}    447: {2,35}

A302568 is the prime or pairwise coprime version, counted by A007359.
A302697 is not required to be squarefree, counted by A302698 (ordered version: A337450).
A302796 allows evens, counted by A078374 (ordered version: A332004).
A337452 counts partitions with these Heinz numbers (ordered version: A337451).
A337984 is the pairwise coprime version, counted by A337485 (ordered version: A337697).
A005117 lists squarefree numbers.
A005408 lists odd numbers.
A056911 lists odd squarefree numbers.
A289509 lists Heinz numbers of relatively prime partitions, counted by A000837 (ordered version: A000740).
Cf. A000010, A007359, A051424, A055684, A056239, A101268, A289508, A302505, A302569, A302696, A302798, A337694.

Select[Range[1,100,2],SquareFreeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]==1&]

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of partitions either constant or relatively prime (A338553). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

A327534 uses primes instead of prime powers.
A338331 is the pairwise coprime version, with complement A338330.
A338552 is the complement.
A338553 counts the partitions with these Heinz numbers.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A000961 lists powers of primes.
A018783 counts partitions whose prime indices are not relatively prime, with Heinz numbers A318978.
A051424 counts pairwise coprime or singleton partitions.
A291166 ranks relatively prime compositions, with complement A291165.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000740, A056239, A108572, A112798, A302569, A302796, A327685, A328677.

Select[Range[100],#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]

Equals A000961 \/ A289509.

Complement of A024619 /\ A318978.

A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A000009, A000837, A051424, A078374, A289509, A302696, A302796, A302797, A303280, A305713.

Select[Range[200],And[SquareFreeQ[#],GCD@@PrimePi/@FactorInteger[#][[All,1]]>1]&]

A319328 Heinz numbers of integer partitions such that not every distinct submultiset has a different GCD but every distinct submultiset has a different LCM.

Original entry on oeis.org

165, 255, 385, 465, 561, 595, 615, 759, 885, 935, 1001, 1005, 1015, 1023, 1045, 1085, 1173, 1245, 1309, 1353, 1435, 1455, 1505, 1547, 1581, 1615, 1635, 1705, 1771, 1905, 1947, 2065, 2091, 2139, 2211, 2235, 2255, 2345, 2355, 2365, 2387, 2397, 2409, 2431, 2465

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The first term of this sequence absent from A302696 (numbers whose prime indices are pairwise coprime) is 1001 with prime indices {4,5,6}.

			The sequence of partitions whose Heinz numbers belong to this sequence begins (5,3,2), (7,3,2), (5,4,3), (11,3,2), (7,5,2), (7,4,3), (13,3,2), (9,5,2), (17,3,2), (7,5,3).

Cf. A056239, A275972, A299702, A301899, A301900, A319315, A319319, A319327.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],UnsameQ@@primeMS[#]&&And[!UnsameQ@@GCD@@@Union[Rest[Subsets[primeMS[#]]]],UnsameQ@@LCM@@@Union[Rest[Subsets[primeMS[#]]]]]&]

A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 08 2019

nonn

Also the number of multiset partitions of the multiset of prime indices of n using multisets each of which is relatively prime.
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.
Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

			The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of 0's are A318978.

Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facsrp[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsrp[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],GCD@@primeMS[#]==1&]}]];
Table[Length[facsrp[n]],{n,100}]

A320439(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d<=m)&&(1==gcd(apply(x->primepi(x), factor(d)[, 1]))), s += A320439(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2021

A327515 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, 2, or a nonprime number whose prime indices are pairwise coprime (A327512, A327514).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Sep 19 2019

nonn

Positions of zeros are A289509.
First term > 1 is a(225) = 2.
First zero not in A318978 is a(17719) = 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. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.

			We have 50625 -> 3375 -> 225 ->  15 -> 1, so a(50625) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences related to prime indices in the factorization of n.

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A289509, A302569, A302696, A304711.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[FixedPointList[#/Max[Select[Divisors[#],#==1||CoprimeQ@@primeMS[#]&]]&,n]]-2,{n,100}]

isA302696(n) = if(isprimepower(n), !(n%2), if(!issquarefree(n>>valuation(n, 2)), 0, my(pis=apply(primepi, factor(n)[, 1])); (lcm(pis)==factorback(pis))));
A327512(n) = vecmax(select(isA302696, divisors(n)));
A327515(n) = for(k=0,oo,my(nextn=n/A327512(n)); if(nextn==n,return(k)); n = nextn); \\ Antti Karttunen, Jan 28 2025

a(15^n) = n.

Data section extended to a(105) and secondary offset added by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A327513 Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.

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A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

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A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

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A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

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A338552 Non-powers of primes whose prime indices have a common divisor > 1.

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A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

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A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

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A319328 Heinz numbers of integer partitions such that not every distinct submultiset has a different GCD but every distinct submultiset has a different LCM.

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A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

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