A378456 -id:A378456 - cp's OEIS frontend

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Jon Perry, Mar 07 2002

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

For primes between nonsquarefree numbers we have A236575.
For composite instead of prime we have A378456.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A080101 counts prime powers between primes.
A246547 lists the non prime prime powers, differences A053707.
A246655 lists the prime powers not including 1, complement A361102.
Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Dec 25 2024

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 prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

A378357 Distance from n to the least non perfect power >= n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 24 2024

nonn

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

All terms are <= 2 because the only adjacent perfect powers are 8 and 9.

The version for prime numbers is A007920, subtraction of A159477 or A007918.
The version for perfect powers is A074984, subtraction of A377468.
The version for squarefree numbers is A081221, subtraction of A067535.
Subtracting from n gives A378358, opposite A378363.
The opposite version is A378364.
The version for nonsquarefree numbers is A378369, subtraction of A120327.
The version for prime powers is A378370, subtraction of A000015.
The version for non prime powers is A378371, subtraction of A378372.
The version for composite numbers is A378456, subtraction of A113646.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non perfect powers, differences A375706, seconds A376562.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A377432 counts perfect powers between primes, zeros A377436.
Cf. A013597, A013603, A013632, A031218, A045542, A052410, A081676, A131605, A188951, A216765, A375702, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&perpowQ[#]&]-n,{n,100}]

from sympy import perfect_power
def A378357(n): return 0 if n>1 and perfect_power(n)==False else 1 if perfect_power(n+1)==False else 2 # Chai Wah Wu, Nov 27 2024

a(n) = n - A378358(n).

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A378614 Number of composite numbers (A002808) between consecutive perfect powers (A001597), exclusive.

Original entry on oeis.org

0, 1, 0, 4, 5, 1, 2, 3, 8, 11, 12, 15, 15, 3, 1, 12, 19, 21, 16, 7, 12, 11, 25, 29, 16, 13, 32, 33, 35, 22, 14, 40, 39, 42, 45, 46, 47, 50, 52, 32, 19, 55, 56, 59, 60, 27, 35, 65, 64, 67, 68, 40, 30, 75, 74, 77, 19, 57, 62, 9, 9, 81, 81, 88, 89, 87, 32, 55, 94

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

The inclusive version is a(n) + 2.

			The composite numbers counted by a(n) cover A106543 with the following disjoint sets:
  .
  6
  .
  10 12 14 15
  18 20 21 22 24
  26
  28 30
  33 34 35
  38 39 40 42 44 45 46 48
  50 51 52 54 55 56 57 58 60 62 63

Michael De Vlieger, Table of n, a(n) for n = 1..12127 (between consecutive perfect powers k <= 2^27)

For prime instead of perfect power we have A046933.
For prime instead of composite we have A080769.
For nonsquarefree instead of perfect power we have A378373, for primes A236575.
For nonprime prime power instead of perfect power we have A378456.
A001597 lists the perfect powers, differences A053289.
A002808 lists the composite numbers.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A106543 lists the composite non perfect powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378365 gives the least prime > each perfect power, opposite A377283.
Cf. A045542, A052410, A065890, A076412, A081676, A216765, A276781, A377057, A377468, A378035, A378251.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
v=Select[Range[100],perpowQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

from sympy import mobius, integer_nthroot, primepi
def A378614(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return -(a:=bisection(f,n,n))+(b:=bisection(lambda x:f(x)+1,a+1,a+1))-primepi(b)+primepi(a)-1 # Chai Wah Wu, Dec 03 2024

cp's OEIS Frontend

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

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A379300 Number of prime indices of n that are composite.

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A378357 Distance from n to the least non perfect power >= n.

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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A378614 Number of composite numbers (A002808) between consecutive perfect powers (A001597), exclusive.

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