A318979 -id:A318979 - cp's OEIS frontend

A355529 Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Jul 24 2022

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.

Includes all even numbers.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}

The odd case is A355535.
The case of all divisors (not just primes) is A355740, zeros of A355739.
These choices are variously counted by A355741, A355744, A355745.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000720, A076610, A318979, A335433, A335448, A355733.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

A327657 Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 3, 2, 1, 3, 2, 3, 3, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 3, 2, 2, 6, 1, 4, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 2, 3, 4, 2, 2, 5, 2, 2, 4, 3, 2, 4, 2, 2, 4, 4, 2, 3, 2, 3, 2, 2, 3, 4, 3, 2, 3, 2, 2, 5

Offset: 1

Gus Wiseman, Sep 21 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 have a common divisor > 1 are listed in A318978.

			The divisors of 90 that are 1 or whose prime indices have a common divisor > 1 are {1, 3, 5, 9}, so a(90) = 4.

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, A281116, A289509, A318979, A327406, A327656.

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

A327657(n) = sumdiv(n, d, (1==d)||(gcd(apply(x->primepi(x), factor(d)[, 1]))>1)); \\ Antti Karttunen, Dec 05 2021

a(n) = A000005(n) - A318979(n). - Antti Karttunen, Dec 05 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 05 2021

cp's OEIS Frontend

A355529 Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).

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A327657 Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.

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