A045542 -id:A045542 - 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

A378365 Next prime index after each perfect power, duplicates removed.

Original entry on oeis.org

1, 3, 5, 7, 10, 12, 16, 19, 23, 26, 31, 32, 35, 40, 45, 48, 49, 54, 55, 62, 67, 69, 73, 79, 86, 93, 98, 100, 106, 115, 123, 130, 138, 147, 155, 163, 169, 173, 182, 192, 201, 211, 218, 220, 229, 241, 252, 264, 270, 275, 284, 296, 307, 310, 320, 328, 330, 343

Offset: 1

Gus Wiseman, Nov 26 2024

nonn

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

			The first number line below shows the perfect powers. The second shows each n at position prime(n). To get a(n), we take the first prime between each pair of consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

The opposite version is A377283.
Positions of first appearances in A378035.
First differences are A378251.
Union of A378356.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A080769 counts primes between perfect powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378249 gives the least perfect power > prime(n), restriction of A377468.
Cf. A045542, A052410, A065514, A068361, A076412, A081676, A216765, A345531, A378250, A378253, A378355.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Union[1+Table[PrimePi[n],{n,Select[Range[100],perpowQ]}]]

These are the distinct elements of the set {1 + A000720(A151800(n)), n>0}.

A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 7, 8, 11, 16, 24, 32, 42, 61, 82, 118, 166, 231, 322, 453, 635, 892, 1253, 1767, 2487, 3505, 4936, 6959, 9816, 13850, 19538, 27578, 38933, 54972, 77641, 109668, 154922, 218879, 309277, 437047, 617658, 872968, 1233896, 1744153, 2465547, 3485478

Offset: 0

Gus Wiseman, Nov 04 2024

nonn

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

Also the number of perfect-powers with n bits.

			The perfect-powers in each prescribed range (rows):
    1
    .
    4
    8    9
   16   25   27
   32   36   49
   64   81  100  121  125
  128  144  169  196  216  225  243
  256  289  324  343  361  400  441  484
  512  529  576  625  676  729  784  841  900  961 1000
Their binary expansions (columns):
  1  .  100  1000  10000  100000  1000000  10000000  100000000
             1001  11001  100100  1010001  10010000  100100001
                   11011  110001  1100100  10101001  101000100
                                  1111001  11000100  101010111
                                  1111101  11011000  101101001
                                           11100001  110010000
                                           11110011  110111001
                                                     111100100

The union of all numbers counted is A001597, without powers of two A377702.
The version for squarefree numbers is A077643.
These are the first differences of A188951.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Not counting powers of 2 gives A377467.
The version for non-perfect-powers is A377701.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A345531, A377434.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n,2^(n+1)-1],perpowQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377435(n):
    if n==0: return 1
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

For n != 1, a(n) = A377467(n) + 1.

a(26)-a(46) from Chai Wah Wu, Nov 05 2024

A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 1, 3, 6, 13, 29, 59, 121, 248, 501, 1008, 2024, 4064, 8150, 16323, 32686, 65418, 130906, 261913, 523966, 1048123, 2096517, 4193412, 8387355, 16775449, 33551945, 67105359, 134212792, 268428497, 536861096, 1073727974, 2147464110, 4294939718, 8589895659

Offset: 0

Gus Wiseman, Nov 05 2024

nonn

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

Also the number of non-perfect-powers with n bits.

			The non-perfect-powers in each range (rows):
   .
   3
   5  6  7
  10 11 12 13 14 15
  17 18 19 20 21 22 23 24 26 28 29 30 31
Their binary expansions (columns):
  .  11  101  1010  10001
         110  1011  10010
         111  1100  10011
              1101  10100
              1110  10101
              1111  10110
                    10111
                    11000
                    11010
                    11100
                    11101
                    11110
                    11111

The union of all numbers counted is A007916.
For squarefree numbers we have A077643.
For prime-powers we have A244508.
For primes instead of powers of 2 we have A377433, ones A029707.
For perfect-powers we have A377467, for primes A377432, zeros A377436.
A000225(n) counts the interval from A000051(n) to A000225(n+1).
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.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A045542, A052410, A061398, A304521, A377434, A377435, A377702.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[2^n+1, 2^(n+1)-1],radQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377701(n):
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 06 2024

a(n) = 2^n-1-A377467(n). - Pontus von Brömssen, Nov 06 2024

Offset corrected by, and a(16)-a(33) from Pontus von Brömssen, Nov 06 2024

A378364 Prime numbers such that the interval from the previous prime number contains a unique perfect power.

Original entry on oeis.org

2, 5, 17, 53, 67, 83, 101, 131, 149, 173, 197, 223, 227, 251, 257, 293, 331, 347, 367, 401, 443, 487, 521, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1009, 1031, 1091, 1163, 1229, 1297, 1361, 1373, 1447, 1523, 1601, 1693, 1733, 1777, 1861, 1949, 2027, 2053

Offset: 1

Gus Wiseman, Dec 16 2024

nonn

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

			The prime before 17 is 13, and the interval (13,14,15,16,17) contains only one perfect power 16, so 17 is in the sequence.
The prime before 29 is 23, and the interval (23,24,25,26,27,28,29) contains two perfect powers 25 and 27, so 29 is not in the sequence.

For non prime powers we have A006512.
For zero instead of one perfect power we have the prime terms of A345531.
The indices of these primes are the positions of 1 in A377432.
The indices of these primes are 1 + A377434(n-1).
For more than one perfect power see A377466.
Swapping "prime" with "perfect power" gives A378374.
For next instead of previous prime we have A379154.
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.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A052410, A053607, A069623, A376560, A377287, A377436.

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

A189045 Semiprimes which are sub-perfect powers.

Original entry on oeis.org

15, 26, 35, 143, 215, 323, 511, 899, 1727, 1763, 2047, 2186, 2743, 3599, 5183, 7999, 10403, 11663, 13823, 19043, 22499, 32399, 36863, 39203, 51983, 54871, 57599, 72899, 79523, 97343, 121103, 157463, 176399, 186623, 213443, 238327, 248831, 272483, 279935, 324899, 359999, 381923

Offset: 1

Jonathan Vos Post, Apr 15 2011

Numbers of the form p*q where p and q are primes, not necessarily distinct, such that p*q + 1 is a perfect power (squares, cubes, etc.). In one sense, this is to semiprimes as Mersenne primes A000668 are to primes.

By Catalan's conjecture (or now Mihailescu's theorem), p and q are always distinct. - T. D. Noe, Apr 15 2011

			a(9) = 12^3 - 1 = 1727 = 11 * 157.

T. D. Noe, Table of n, a(n) for n = 1..8433 (terms < 10^12)

Cf. A001358, A001597, A045542, A177955, A189047.

SemiPrimeQ[n_] := Total[FactorInteger[n]][[2]] == 2; PerfectPowerQ[n_] := GCD @@ Last /@ FactorInteger[n] > 1; Select[Range[400000], SemiPrimeQ[#] && PerfectPowerQ[# + 1] &] (* T. D. Noe, Apr 15 2011 *)

A001358 INTERSECTION A045542. A001358 INTERSECTION {A001597 - 1}.

A189047 Semiprimes which are one more than a perfect power.

Original entry on oeis.org

9, 10, 26, 33, 65, 82, 122, 129, 145, 217, 226, 362, 485, 626, 785, 842, 901, 1157, 1226, 1522, 1765, 1937, 2026, 2049, 2117, 2305, 2402, 2501, 2602, 2705, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4762, 5042, 5777, 5833, 6085, 6242, 6401, 7226, 7397, 7745, 8193, 8465, 9026, 9217

Offset: 1

Jonathan Vos Post, Apr 15 2011

Numbers of the form p*q where p and q are primes, not necessarily distinct, such that p*q - 1 is a perfect power (squares, cubes, etcetera). T. D. Noe suggested the name semiprimes which are super-perfect powers.

The number of terms <= 10^k: 2, 6, 17, 51, 131, 379, 1015, 2865, 8086, ..., . - Robert G. Wilson v, Apr 16 2011

			a(21) = 42^2 + 1 = 1765 = 5 * 353.

Robert G. Wilson v, Table of n, a(n) for n = 1..8086

Cf. A001358, A001597, A045542, A177955, A189045.

fQ[n_] := GCD @@ Last /@ FactorInteger[n - 1] > 1 && Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Apr 16 2011 *)

A001358 INTERSECTION {A001597 + 1}.

A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

Original entry on oeis.org

0, 3, 7, 8, 15, 24, 26, 31, 48, 63, 80, 120, 124, 127, 168, 242, 255, 288, 342, 360, 511, 528, 624, 728, 840, 960, 1023, 1330, 1368, 1680, 1848, 2047, 2186, 2196, 2208, 2400, 2808, 3124, 3480, 3720, 4095, 4488, 4912, 5040, 5328, 6240, 6560, 6858, 6888, 7920, 8191, 9408, 10200, 10608, 11448

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A025475.
Subsequence of A181062 and also a subsequence of A249433 (after the initial zero).
Union of sequences A000225, A024023, A024049, A024075, A024127, etc. without their term a(1).
Apart from the first term, subsequence of A045542.

list(lim)=my(v=List([0])); lim=lim\1+1; for(m=2,logint(lim,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Scheme
```
(define (A249435 n) (- (A025475 n) 1))
```

a(n) = A025475(n) - 1.

A359070 Smallest k > 1 such that k^n - 1 is the product of n distinct primes.

Original entry on oeis.org

3, 4, 15, 12, 39, 54, 79, 86, 144, 318, 1591, 144, 20131, 2014, 1764, 1308, 46656, 1296

Offset: 1

Kevin P. Thompson, Dec 15 2022

a(19) > 60000 and a(20) = 3940.
a(19) > 5 * 10^5; a(21) = 132023; a(22) = 229430; a(24) = 4842. - Daniel Suteu, Dec 16 2022
Because of the algebraic factorization of x^n-1 (via cyclotomic polynomials), there is good reason to expect (on average) that prime values of n will have larger solutions than other numbers. That is, those values of n with many factors already get a head start by having many algebraic factors. - Sean A. Irvine, Jan 06 2023

			a(3) = 15 since 15^3 - 1 = 3374 = 2*7*241 is the product of 3 distinct primes and 15 is the smallest number with this property.

Cf. A001597, A005117, A045542, A219019, A281940, A359069.

isok(k, n) = my(f=factor(k^n - 1)); issquarefree(f) && (omega(f) == n);
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Dec 15 2022

a(n) >= A219019(n). - Daniel Suteu, Dec 16 2022

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)).

cp's OEIS Frontend

A377702 Perfect-powers except for powers of 2.

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A378365 Next prime index after each perfect power, duplicates removed.

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A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

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A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

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A378364 Prime numbers such that the interval from the previous prime number contains a unique perfect power.

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A189045 Semiprimes which are sub-perfect powers.

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A189047 Semiprimes which are one more than a perfect power.

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A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

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