A327534 -id:A327534 - cp's OEIS frontend

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.

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}]

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}]

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.

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).

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).

cp's OEIS Frontend

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.

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

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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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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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