A366833 -id:A366833 - cp's OEIS frontend

A377702 Perfect-powers except for powers of 2.

9, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197

Offset: 1

Gus Wiseman, Nov 05 2024

nonn

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

			The terms together with their prime indices begin:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   169: {6,6}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   324: {1,1,2,2,2,2}

Including the powers of 2 gives A001597, counted by A377435.
For prime-powers we have A061345.
These terms are counted by A377467, for non-perfect-powers A377701.
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.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A188951 counts perfect-powers less than 2^n.
A377468 gives the least perfect-power > n.
Cf. A014210, A014234, A023055, A045542, A052410, A065514, A069623, A246655, A304521, A336416, A345531, A366833.

Select[Range[1000],GCD@@FactorInteger[#][[All,2]]>1&&!IntegerQ[Log[2,#]]&]

from sympy import mobius, integer_nthroot
def A377702(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-2+x+(l:=x.bit_length())+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,l)))
    return bisection(f,n+1,n+1) # Chai Wah Wu, Nov 06 2024

A377780 First differences of A000015 (smallest prime-power >= n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 13 2024

nonn

First differences of A000015, restriction to primes A345531.
The opposite is A377782, restriction to primes A377781, differences of A065514.
For squarefree instead of prime-power see A067535, A112925, A112926, A120327.
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).
A361102 lists the non-powers of primes, differences A375708.
A366833 counts prime-powers between primes.
Cf. A001597, A007916, A007949, A031218, A053289, A053607, A343249, A343250, A377282, A377289, A377468.

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

A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

Original entry on oeis.org

1, 0, 4, 5, 1, 2, 12, 11, 12, 31, 3, 1, 32, 59, 11, 25, 46, 13, 125, 14, 80, 88, 94, 103, 52, 261, 35, 267, 147, 172, 120, 9, 9, 163, 355, 279, 313, 207, 329, 347, 376, 108, 257, 805, 283, 262, 25, 917, 242, 1081, 702, 365, 752, 389, 251, 535, 1679, 877, 447

Offset: 1

Gus Wiseman, Nov 30 2024

nonn

The inclusive version is a(n) + 2.

Nonprime prime powers (A246547) begin: 4, 8, 9, 16, 25, 27, 32, 49, ...

			The initial terms count the following composite numbers:
  {6}, {}, {10,12,14,15}, {18,20,21,22,24}, {26}, {28,30}, ...
The composite numbers for a(77) = 6 together with their prime indices are the following. We have also shown the nonprime prime powers before and after:
  32761: {42,42}
  32762: {1,1900}
  32763: {2,19,38}
  32764: {1,1,1028}
  32765: {3,847}
  32766: {1,2,14,31}
  32767: {4,11,36}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

For prime instead of composite we have A067871.
For nonsquarefree numbers we have A378373, for primes A236575.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A002808 lists the composite numbers.
A031218 gives the greatest prime-power <= n.
A046933 counts composite numbers between primes.
A053707 gives first differences of nonprime prime powers.
A080101 = A366833 - 1 counts prime powers between primes.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the nearest prime power after prime(n) + 1, difference A377281.
Cf. A377286, A377287, A377288 (primes A053706).
Cf. A024619, A053607, A065890, A076259, A078147, A243348, A276781, A377057, A377282.

nn=1000;
v=Select[Range[nn],PrimePowerQ[#]&&!PrimeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A377433 Number of non-perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 5, 1, 4, 3, 1, 5, 2, 5, 7, 2, 1, 3, 1, 3, 11, 2, 5, 1, 8, 1, 5, 5, 3, 4, 5, 1, 9, 1, 2, 1, 11, 10, 2, 1, 3, 5, 1, 8, 4, 5, 5, 1, 5, 3, 1, 8, 13, 3, 1, 3, 12, 5, 8, 1, 3, 5, 6, 5, 5, 3, 5, 7, 2, 7, 9, 1, 9, 1, 5, 2

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

Positions of terms > 1 appear to be A049579.

			Between prime(4) = 7 and prime(5) = 11 the only non-perfect-power is 10, so a(4) = 1.

Positions of 1 are latter terms of A029707.
Positions of terms > 1 appear to be A049579.
For prime-powers instead of non-perfect-powers we have A080101.
For non-prime-powers instead of non-perfect-powers we have A368748.
Perfect-powers in the same range are counted by A377432.
A000040 lists the primes, differences A001223.
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.
A065514 gives the greatest prime-power < prime(n), difference A377289.
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A064113, A065890, A244508, A377282, A377434, A377436, A377466, A377467.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],radQ]],{n,100}]

a(n) + A377432(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A378366 Difference between n and the greatest non prime power <= n (allowing 1).

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we almost have A010051 (A179278).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
For prime power we have A378457 = A276781-1 (A031218).
For nonsquarefree we have (A378033).
For non perfect power we almost have A075802 (A378363).
Subtracting from n gives (A378367).
The opposite is A378371, adding n A378372.
A000015 gives the least prime power >= n (cf. A378370 = A377282 - 1).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A007920, A013632, A065514, A074984, A113646, A345531, A377051, A377054, A377281, A377289, A378357.

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

a(n) = n - A378367(n).

A362965 Number of primes <= the n-th prime power.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 9, 9, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 54, 54, 55, 56, 57, 58, 59, 60

Offset: 1

Max Alekseyev, Jun 03 2023

nonn

Also, number of distinct primes among the first n prime powers (cf. A246655).

			The 4th prime, 7, is followed by prime powers 8 and 9 before the next prime (11), accounting for three consecutive 4s in the sequence (at indices n = 5..7). Similarly, the three 9s (at n = 13..15) show that the 9th prime (23) is followed by two prime powers (25, 27) before the next prime (29). This occurs again at n = 40..42 (a(n) = 30), 358..360 (a(n) = 327) and 3588..3590 (a(n) = 3512). - _M. F. Hasler_, Oct 31 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000961, A000720, A246655, A366833 (run lengths).

A362965list[upto_]:=PrimePi[Select[Range[upto],PrimePowerQ]];A362965list[500] (* Paolo Xausa, Jun 29 2023 *)

apply(primepi, [p| p <- [1..300], isprimepower(p)]) \\ Michel Marcus, Jun 04 2023

from sympy import primepi, integer_nthroot
def A362965(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-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return int(primepi(bisection(f,n,n))) # Chai Wah Wu, Oct 28 2024

a(n) = A000720(A246655(n)).

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

Original entry on oeis.org

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

Gus Wiseman, Dec 06 2024

nonn

			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

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.

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

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

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

A379154 Prime numbers p such that the interval from p to the next prime number contains a unique perfect power.

Original entry on oeis.org

3, 13, 47, 61, 79, 97, 127, 139, 167, 193, 211, 223, 241, 251, 283, 317, 337, 359, 397, 439, 479, 509, 523, 571, 619, 673, 727, 773, 839, 887, 953, 997, 1021, 1087, 1153, 1223, 1291, 1327, 1367, 1439, 1511, 1597, 1669, 1723, 1759, 1847, 1933, 2017, 2039, 2113

Offset: 1

Gus Wiseman, Dec 18 2024

nonn

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

			The prime after 13 is 17, and the interval (13,14,15,16,17) contains only one perfect power 16, so 13 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

The indices of these primes are one plus the positions of 1 in A377432.
For zero instead of one perfect power we have the primes indexed by A377436.
The indices of these primes are A377434.
Swapping "prime" with "perfect power" gives A378355, indices A378368.
For previous instead of next prime we have A378364.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A081676 gives the greatest perfect power <= n.
A116086 gives perfect powers with no primes between them and the next perfect power.
A366833 counts prime powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A052410, A069623, A376560, A377283, A377287, A377466, A378374.

N:= 10^4: # to get all entries <= N
S:={seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}:
S:= sort(convert(S,list)):
J:= select(i -> nextprime(S[i]) < S[i+1] and prevprime(S[i]) > S[i-1], [$2..nops(S)-1]):
J:= [1,op(J)]:
map(prevprime, S[J]); # Robert Israel, Jan 19 2025

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[1000],PrimeQ[#]&&Length[Select[Range[#,NextPrime[#]],perpowQ]]==1&]

is_a379154(n) = isprime(n) && #select(x->ispower(x), [n+1..nextprime(n+1)-1])==1 \\ Hugo Pfoertner, Dec 19 2024

a(n) = A151799(A378364(n+1)).

A378252 Least prime power > 2^n.

Original entry on oeis.org

2, 3, 5, 9, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659, 4294967311, 8589934609

Offset: 0

Gus Wiseman, Nov 30 2024

nonn

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

Subtracting 2^n appears to give A013597 except at term 3.
For prime we have A014210.
For previous we have A014234.
For perfect power we have A357751.
For squarefree we have A372683.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, diffs A375708 and A375735.
A031218 gives the greatest prime power <= n.
A244508 counts prime powers between powers of 2.
Prime powers between primes are counted by A080101 and A366833.
Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Table[NestWhile[#+1&,2^n+1,!PrimePowerQ[#]&],{n,0,20}]

a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<Chai Wah Wu, Dec 02 2024

A378616 Greatest non prime power <= prime(n).

Original entry on oeis.org

1, 1, 1, 6, 10, 12, 15, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 255, 262, 268, 270

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

Conjecture: Equal to A006093(n) = prime(n) - 1 except at terms of A159611.

			The first number line below shows the non prime powers. The second shows the primes:
--1-------------6----------10----12----14-15-------18----20-21-22----24--
=====2==3====5=====7==========11====13==========17====19==========23=====

For nonprime instead of non prime power we have A156037.
Restriction of A378367.
Lengths are A378615.
For nonsquarefree: A378032 (diffs A378034), restriction of A378033 (diffs A378036).
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.
Prime powers between primes:
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A006093, A053607, A143731, A159611, A304521, A343249, A345531, A356068, A368748, A377281, A377289, A377703, A377781.

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

cp's OEIS Frontend

A377702 Perfect-powers except for powers of 2.

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A377780 First differences of A000015 (smallest prime-power >= n).

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A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

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A377433 Number of non-perfect-powers x in the range prime(n) < x < prime(n+1).

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A378366 Difference between n and the greatest non prime power <= n (allowing 1).

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A362965 Number of primes <= the n-th prime power.

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

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A379154 Prime numbers p such that the interval from p to the next prime number contains a unique perfect power.

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