A024619 -id:A024619 - cp's OEIS frontend

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

			From _Michael De Vlieger_, Aug 11 2024: (Start)
Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
   n  a(n)  row n of A272618.
  ---------------------------
   6    1   {4}
  10    2   {4, 8}
  12    2   {8, 9}
  14    2   {4, 8}
  15    1   {9}
  18    4   {4, 8, 12*, 16}
  20    2   {8, 16}
  21    1   {9}
  22    3   {4, 8, 16}
  24    3   {9, 16, 18*}
  26    3   {4, 8, 16}
  28    2   {8, 16}
  30   10   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.

Table[Count[Range[n], ?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger, Aug 11 2024 *)

a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
a(n) = A355432(n) + A361235(n).
a(n) = A355432(n) for n in A360768.
a(n) = A361235(n) for n not in A360768.
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)

New name from David James Sycamore, Aug 11 2024

A377289 Difference between prime(n) and the previous prime-power (exclusive).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 2, 4, 2, 2, 5, 4, 2, 4, 4, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 2, 3, 6, 2, 10, 2, 6, 6, 4, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 8, 1, 6, 6, 2, 6, 4, 2, 4, 14, 4, 2, 4, 14, 6, 4, 2, 4, 6, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2

Offset: 1

Gus Wiseman, Oct 23 2024

nonn

			The twelfth prime is 37, with previous prime-power 32, so a(12) = 5.

For powers of two see A013597, A014210, A014234, A244508, A304521.
For prime instead of prime-power we have A075526.
This is the restriction of A276781 (shifted right) to the primes.
For next instead of previous prime-power we have A377281, restriction of A377282.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, complement A361102.
A031218 gives the greatest prime-power <= n.
A065514 gives the greatest prime-power < prime(n).
A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.
A246655 lists the prime-powers not including 1.
Cf. A002808, A024619, A053707, A059305, A064113, A065890, A366833, A376596, A376597, A377051, A377054.

Table[Prime[n]-NestWhile[#-1&, Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]

from sympy import prime, factorint
def A377289(n): return (p:=prime(n))-next(filter(lambda m:len(factorint(m))<=1, range(p-1,0,-1))) # Chai Wah Wu, Oct 25 2024

a(n) = prime(n) - A031218(prime(n)-1).
a(n) = prime(n) - A065514(n).
a(n) = A276781(prime(n)-1).

A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

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

			Primes 8 and 9 are 19 and 23, and the interval (20,21,22) contains no prime-powers, so 8 is in the sequence.

For powers of 2 instead of primes see A377467, A013597, A014210, A014234, A244508.
For squarefree instead of perfect-power we have A068360, see A061398, A377430, A377431.
For just squares (instead of all perfect-powers) we have A221056, primes A224363.
For prime-powers (instead of perfect-powers) we have A377286.
These are the positions of 0 in A377432.
For one instead of none we have A377434, for prime-powers A377287.
For two instead of none we have A377466, for prime-powers A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the nearest prime-power before prime(n)-1, difference A377289.
A080101 and A366833 count prime-powers between primes, see A377057, A053607, A304521.
A081676 gives the nearest perfect-power up to n.
A246655 lists the prime-powers not including 1, complement A361102.
A377468 gives the nearest perfect-power after n.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A276781, A345531, A376597, A377054, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Length[Select[Range[Prime[#]+1, Prime[#+1]-1],perpowQ]]==0&]

A377281 Difference between the n-th prime and the next prime-power (exclusive).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 4, 2, 2, 1, 4, 2, 4, 2, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 8, 1, 6, 2, 10, 2, 6, 6, 4, 2, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 5, 6, 6, 2, 6, 4, 2, 6, 14, 4, 2, 4, 14, 6, 6, 2, 4, 6, 2, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Gus Wiseman, Oct 23 2024

nonn

			The twelfth prime is 37, with next prime-power 41, so a(12) = 4.

For prime instead of prime-power we have A001223.
For powers of two instead of primes we have A013597, A014210, A014234, A244508, A304521.
This is the restriction of A377282 to the prime numbers.
For previous instead of next prime-power we have A377289, restriction of A276781.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, complement A361102.
A031218 gives the greatest prime-power <= n.
A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.
A246655 lists the prime-powers not including 1.
Cf. A001597, A024619, A053289, A053707, A059305, A064113, A366833, A376596, A376597, A376598, A377051, A377054.

Table[NestWhile[#+1&,Prime[n]+1,!PrimePowerQ[#]&]-Prime[n],{n,100}]

from itertools import count
from sympy import prime, factorint
def A377281(n): return -(p:=prime(n))+next(filter(lambda m:len(factorint(m))<=1, count(p+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000015(prime(n)) - prime(n).
a(n) = A345531(n) - prime(n).
a(n) = A377282(prime(n)).

A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 6, 11, 15, 18, 22, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457, 1523, 1715, 1821, 1877, 1900

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains only the one prime-power 64, so 18 is in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 1 in A080101, or 2 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For two prime-powers we have A377288, primes A053706.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377282.

Select[Range[100],Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==1&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377287_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if sum(1 for i in range(p+1,q) if len(factorint(i))<=1)==1:
            yield k
        p, q = q, nextprime(q)
A377287_list = list(islice(A377287_gen(),53)) # Chai Wah Wu, Oct 28 2024

A375735 First differences of non-prime-powers (inclusive).

Original entry on oeis.org

4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1

Offset: 1

Gus Wiseman, Sep 04 2024

nonn

Inclusive means 1 is a prime-power but not a non-prime-power.

Non-prime-powers (inclusive) are listed by A024619.

			The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.

For perfect powers (A001597) we have the latter terms of A053289.
For nonprime numbers (A002808) we have the latter terms of A073783.
For squarefree numbers (A005117) we have the latter terms of A076259.
First differences of A024619.
For prime-powers (A246655) we have the latter terms of A057820.
Essentially the same as the exclusive version, A375708.
Positions of 1's are A375713(n) - 1.
For runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, first differences A001223.
A000961 lists prime-powers (inclusive).
A007916 lists non-perfect-powers, first differences A375706.
A013929 lists the nonsquarefree numbers, first differences A078147.
A246655 lists prime-powers (exclusive).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power anti-runs: A373576, min A120430, max A006549, length A373671.
Non-prime-power anti-runs: A373679, min A373575, max A255346, len A373672.
Cf. A046933, A061399, A093555, A176246, A251092, A323054, A375709, A375736.

Differences[Select[Range[2,100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375735(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024

A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 4, 6, 9, 11, 15, 18, 22, 30, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 327, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053607.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of positive terms in A080101, or terms >1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For exactly two prime-powers we have A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]>=1&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377057_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if any(len(factorint(i))<=1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377057_list = list(islice(A377057_gen(),52)) # Chai Wah Wu, Oct 27 2024

prime(a(n)) = A053607(n).

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 1, 2

Offset: 1

N. J. A. Sloane, Sep 29 2016, following a suggestion from Eric Desbiaux

The definition in the video has "b < k < n-b" rather than "b <= k <= n-b", but that appears to be a typographical error.
From Antti Karttunen, Jan 21 2020: (Start)
a(n) = 1 if n is a power of prime (term of A000961), otherwise a(n) is one more than the distance to the nearest preceding prime power.
For n > 1, a(n) indicates the maximum region on the row n of Pascal's triangle (A007318) such that binomial terms C(n,a(n)) .. C(n,n-a(n)) all share a common prime factor. Because for all prime powers, p^k, the binomial terms C(p^k,1) .. C(p^k,p^k-1) have p as their prime factor, we have a(A000961(n)) = 1 for all n, while for each successive n that is not a prime power, the region of shared prime factor shrinks one step more towards the center of the triangle. From this follows that this is the ordinal transform of A025528 (equally, of A065515, or of A003418(n) from n >= 1 onward), equivalent to the simple definition given above.
(End)

			Row 6 of Pascal's triangle is 1,6,15,20,15,6,1 and [15,20,15] have a common divisor of 5. Since 15 = binomial(6,2), a(6)=2.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms for n = 2..1685 from R. J. Mathar, terms 1686..10000 from Chai Wah Wu).
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Christophe Soulé, Le triangle de Pascal et ses propriétés, Lecture, Soc. Math. de France, Feb 13 2008; SMF link (in French).

Cf. A007318, A010055, A276782 (positions of records), A000961 (positions of ones), A024619 (positions of terms > 1).

Cf. A002944, A003418, A025528, A065515, A175851.

mygcd:=proc(lis) local i,g,m;
m:=nops(lis); g:=lis[1];
for i from 2 to m do g:=gcd(g,lis[i]); od:
g; end;
f:=proc(n) local b,lis; global mygcd;
for b from floor(n/2) by -1 to 1 do
lis:=[seq(binomial(n,i),i=b..n-b)];
if mygcd(lis)=1 then break; fi; od:
b+1;
end;
[seq(f(n),n=2..120)];

Table[b = 1; While[GCD @@ Map[Binomial[n, #] &, Range[b, n - b]] == 1, b++]; b, {n, 92}] (* Michael De Vlieger, Oct 03 2016 *)

A276781(n) = if(1==n,1,forstep(k=n,1,-1,if(isprimepower(k),return(1+n-k)))); \\ Antti Karttunen, Jan 21 2020

from sympy import factorint
def A276781(n): return 1+n-next(filter(lambda m:len(factorint(m))<=1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

If A010055(n) == 1, a(n) = 1, otherwise a(n) = 1 + a(n-1). - Antti Karttunen, Jan 21 2020

Term a(1) = 1 prepended and alternative simpler definition added to the name by Antti Karttunen, Jan 20 2020

A373671 Length of the n-th maximal antirun of prime-powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 26, 27, 1007, 5558, 5734, 31209

Offset: 1

Gus Wiseman, Jun 14 2024

An antirun of a sequence (in this case A000961 without 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of prime-powers begin:
   2
   3
   4
   5   7
   8
   9  11  13  16
  17  19  23  25  27  29  31

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671 (this sequence)
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
A000961 lists the powers of primes (including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers (not including 1 A024619).
Cf. A000040, A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

Length/@Split[Select[Range[100],PrimePowerQ[#]&],#1+1!=#2&]//Most

Partial sums are A025528(A006549(n)).

cp's OEIS Frontend

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A377289 Difference between prime(n) and the previous prime-power (exclusive).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A377281 Difference between the n-th prime and the next prime-power (exclusive).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

A375735 First differences of non-prime-powers (inclusive).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica