A356064 -id:A356064 - cp's OEIS frontend

A355743 Numbers whose prime indices are all prime-powers.

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 41, 45, 49, 51, 53, 55, 57, 59, 63, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 105, 109, 115, 119, 121, 123, 125, 127, 131, 133, 135, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 179, 187

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.

Also MM-numbers of multiset partitions into constant multisets, where the multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  35: {3,4}
  41: {13}
  45: {2,2,3}

Robert Price, Table of n, a(n) for n = 1..1410

The multiplicative version is A000688, strict A050361, coprime A354911.
The case of only primes (not all prime-powers) is A076610, strict A302590.
Allowing prime index 1 gives A302492.
These are the products of elements of A302493.
Requiring n to be a prime-power gives A302601.
These are the positions of 1's in A355741.
The squarefree case is A356065.
The complement is A356066.
A001222 counts prime-power divisors.
A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A085970, A106244, A279784, A295935, A355731, A356064.

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

A085970 Number of integers ranging from 2 to n that are not prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43

Offset: 1

Reinhard Zumkeller, Jul 06 2003

nonn

For n > 2, a(n) gives the number of duplicate eliminations performed by the Sieve of Eratosthenes when sieving the interval [2, n]. - Felix Fröhlich, Dec 10 2016
Number of terms of A024619 <= n. - Felix Fröhlich, Dec 10 2016
First differs from A082997 at n = 30. - Gus Wiseman, Jul 28 2022

			The a(30) = 13 numbers: 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30. - _Gus Wiseman_, Jul 28 2022

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A000720, A024619, A085972.
The complement is counted by A065515, without 1's A025528.
For primes instead of prime-powers we have A065855, with 1's A062298.
Partial sums of A143731.
The version not treating 1 as a prime-power is A356068.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A023893, A023894, A036234, A069513, A207375, A354911, A355743, A356064, A356065.

With[{nn = 75}, Table[n - Count[#, k_ /; k < n] - 1, {n, nn}] &@ Join[{1}, Select[Range@ nn, PrimePowerQ]]] (* Michael De Vlieger, Dec 11 2016 *)

a(n) = my(i=0); forcomposite(c=4, n, if(!isprimepower(c), i++)); i \\ Felix Fröhlich, Dec 10 2016

from sympy import primepi, integer_nthroot
def A085970(n): return n-1-sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Aug 20 2024

a(n) = Max{A024619(k)<=n} k;

a(n) = n - A065515(n) = A085972(n) - A000720(n).

Name modified by Gus Wiseman, Jul 28 2022. Normally 1 is not considered a prime-power, cf. A000961, A246655.

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

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

Intersection of A005117 and A355743.

A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 31 2022

nonn

			The a(30) = 14 numbers: 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30.

The complement is counted by A025528, with 1's A065515.
For primes instead of prime-powers we have A062298, with 1's A065855.
The version treating 1 as a prime-power is A085970.
One more than the partial sums of A143731.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A000720, A023894, A036234, A050361, A054685, A069513, A355743, A356064, A356065, A356066.

Table[Length[Select[Range[n],!PrimePowerQ[#]&]],{n,100}]

a(n) = A085970(n) + 1.

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101

Offset: 1

Gus Wiseman, Jul 31 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.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

The complement is A355743, counted by A023894.
The squarefree complement is A356065, counted by A054685.
Allowing prime index 1 gives A356064, complement A302492.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

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

Union of A299174 and A356064.

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A355743 Numbers whose prime indices are all prime-powers.

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A085970 Number of integers ranging from 2 to n that are not prime-powers.

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A356065 Squarefree numbers whose prime indices are all prime-powers.

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A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

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A354911 Number of factorizations of n into relatively prime prime-powers.

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A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

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