A378615 Number of non prime powers <= prime(n).

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset

1,4

Links

Table of n, a(n) for n=1..60.

Formula

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

Example

The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Mathematica

Table[Length[Select[Range[Prime[n]], Not@*PrimePowerQ]], {n, 100}]

Prog

(Python)
from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p, k)[0]) for k in range(2, p.bit_length()))) # Chai Wah Wu, Dec 07 2024

Crossrefs

Restriction of A356068 (first-differences A143731).

First-differences are A368748.

Maxima are A378616.

Other classes of numbers (instead of non prime powers):

- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)

- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)

- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)

- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)

- composite: A065890 (diffs A046933), restriction of A065855 (diffs A005171)

A000040 lists the primes, differences A001223

A000961 and A246655 list the prime powers, differences A057820.

A024619 lists the non prime powers, differences A375735, seconds A376599.

A080101 counts prime powers between primes (exclusive), inclusive A366833.

A361102 lists the non powers of primes, differences A375708.

Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Sequence in context: A355979 A119793 A181436 * A199118 A035941 A039854

Adjacent sequences: A378612 A378613 A378614 * A378616 A378617 A378618

Keyword

nonn,new

Author

Gus Wiseman, Dec 06 2024

Status

approved