A327407 -id:A327407 - cp's OEIS frontend

A327534 Numbers that are 1, prime, or whose prime indices are relatively prime.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Sep 17 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 whose prime indices are relatively prime are A289509.

			91 = 7 * 13 has prime indices {4,6}, which have a common divisor of 2, so 91 is not in the sequence.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.
Complement of A327407.
Cf. A056239, A112798, A281116, A289509, A302569.

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

Equals the union of {1}, A000040, and A289509.

A327535 Maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 17 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, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

			The divisors of 63 that are 1, prime, or whose prime indices are relatively prime are {1, 3, 7}, so a(63) = 7.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A000961, A006530, A056239, A112798, A281116, A289509, A327407.

Table[Max@@Select[Divisors[n],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&],{n,100}]

If n is in A327534, then a(n) = n; otherwise a(n) = A006530(n).

A327536 Number of divisors of n that are 1, prime, or whose prime indices are relatively prime.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 5, 2, 6, 3, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 3, 8, 2, 7, 2, 6, 5, 4, 2, 10, 2, 5, 4, 6, 2, 6, 4, 8, 3, 4, 2, 12, 2, 4, 3, 7, 3, 8, 2, 6, 4, 8, 2, 11, 2, 4, 5, 6, 4, 7, 2, 10, 2, 4, 2, 11, 4

Offset: 1

Gus Wiseman, Sep 17 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, prime, or whose prime indices are relatively prime are A327534. The maximum divisor of n satisfying the same conditions is A327535(n).

			The divisors of 63 that are 1, prime, or whose prime indices are relatively prime are {1, 3, 7}, so a(63) = 3.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A056239, A112798, A281116, A289509, A327407.

Table[Length[Select[Divisors[n],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]],{n,100}]

A327685 Nonprime numbers whose prime indices have a common divisor > 1.

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393, 399

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

First differs from A322336 in lacking 2535 = prime(2)*prime(3)*prime(6)^2.

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. Heinz numbers of integer partitions with a common divisor > 1 are A318978, and the enumeration of these partitions by sum is A108572.

			The sequence of terms together with their prime indices begins:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   39: {2,6}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
   87: {2,10}
   91: {4,6}
  111: {2,12}
  115: {3,9}
  117: {2,2,6}
  121: {5,5}
  125: {3,3,3}
  129: {2,14}
  133: {4,8}
  147: {2,4,4}

Cf. A000837, A007916, A018783, A056239, A108572, A112798, A289509, A318978, A327407, A327658.

Select[Range[100],#>1&&!PrimeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Complement of A000040 in A318978.

A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

			The factorizations of 6, 12, 30, 48, 60, 180, and 210:
  (2*3)  (3*4)    (5*6)    (3*16)       (3*20)     (4*45)       (3*70)
         (2*2*3)  (2*15)   (3*4*4)      (4*15)     (5*36)       (5*42)
                  (3*10)   (2*2*2*2*3)  (5*12)     (9*20)       (6*35)
                  (2*3*5)               (3*4*5)    (4*5*9)      (7*30)
                                        (2*2*15)   (5*6*6)      (10*21)
                                        (2*2*3*5)  (2*2*45)     (14*15)
                                                   (3*3*20)     (2*105)
                                                   (2*2*5*9)    (5*6*7)
                                                   (3*3*4*5)    (2*3*35)
                                                   (2*2*3*3*5)  (2*5*21)
                                                                (2*7*15)
                                                                (3*5*14)
                                                                (3*7*10)
                                                                (2*3*5*7)

Factorizations that are constant or whose distinct parts are pairwise coprime are counted by A327399.
Numbers with pairwise coprime distinct prime indices are A304711.
Cf. A001055, A089723, A281116, A318721, A302569, A319269, A327407, A327517.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],CoprimeQ@@Union[#]&]],{n,100}]

a(n) = A327399(n) - A089723(n).

A327537 Quotient of n over the maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 3

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

All terms are odd.

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, prime, or whose prime indices are relatively prime are A327534. The maximum divisor of n satisfying the same conditions is A327535(n).

			The divisors of 63 that are 1, prime, or whose prime indices are relatively prime are {1, 3, 7}, so a(63) = 63/7 = 9.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A032742, A056239, A112798, A289509, A327407.

Table[n/Max[Select[Divisors[n],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]],{n,100}]

If n is in A327534, then a(n) = 1; otherwise a(n) = n/A006530(n) = A032742(n).

A327404 Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 32, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5

Offset: 1

Gus Wiseman, Sep 23 2019

nonn

First differs from A327395 at a(195) = 65, A327395(195) = 195.

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 divisors of 90 that are 2 or whose prime indices have a common divisor > 1 are {1, 2, 3, 5, 9}, so a(90) = 90/9 = 10.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A289509, A302569, A327407, A327656.

Table[n/Max[Select[Divisors[n],#==2||GCD@@PrimePi/@First/@FactorInteger[#]!=1&]],{n,100}]

A327538 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

The first index m such that a(m) > 1 but m is not in A322336 is m = 2335.

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, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

			We have 441 -> 63 -> 9 -> 3 -> 1, so a(441) = 4.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A001222, A006530, A056239, A112798, A289509, A327407.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]]&,n]]-2,{n,100}]

a(1) = 0; if n is prime or has relatively prime prime indices, then a(n) = 1; otherwise a(n) = Omega(n) = A001222(n).

A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 7, 1, 4, 1, 4, 3, 2, 1, 12, 3, 2, 4, 1, 4, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 2, 1, 9, 2, 2, 7, 1, 9, 4, 2, 2, 2, 19, 1

Offset: 1

Gus Wiseman, Sep 17 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, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

			The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
  (2*2*3*7)  {{1},{1},{2},{4}}
  (2*3*14)   {{1},{2},{1,4}}
  (2*6*7)    {{1},{1,2},{4}}
  (2*42)     {{1},{1,2,4}}
  (3*4*7)    {{2},{1,1},{4}}
  (3*28)     {{2},{1,1,4}}
  (6*14)     {{1,2},{1,4}}
  (7*12)     {{4},{1,1,2}}
  (84)       {{1,1,2,4}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A006530, A056239, A112798, A281116, A289509, A327407.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
y=Select[Range[nn],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

cp's OEIS Frontend

A327534 Numbers that are 1, prime, or whose prime indices are relatively prime.

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A327535 Maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.

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A327536 Number of divisors of n that are 1, prime, or whose prime indices are relatively prime.

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A327685 Nonprime numbers whose prime indices have a common divisor > 1.

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A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

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A327537 Quotient of n over the maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.

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A327404 Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1.

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A327538 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537).

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