A216765 -id:A216765 - cp's OEIS frontend

A375737 Sum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

2, 8, 6, 17, 11, 12, 13, 14, 32, 18, 19, 20, 21, 22, 23, 78, 29, 30, 64, 34, 72, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 98, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 128, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 162, 83, 84, 85, 86, 87

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose sums are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For nonprime numbers we have A373404, runs A054265.
For squarefree numbers we have A373411, runs A373413.
For nonsquarefree numbers we have A373412, runs A373414.
For prime-powers we have A373576, runs A373675.
For non-prime-powers we have A373679, runs A373678.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738
- last: A375739
- sum: A375737 (this)
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A053797, A216765, A251092, A373403, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Total/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 6, 7, 10, 15, 23, 31, 41, 60, 81, 117, 165, 230, 321, 452, 634, 891, 1252, 1766, 2486, 3504, 4935, 6958, 9815, 13849, 19537, 27577, 38932, 54971, 77640, 109667, 154921, 218878, 309276, 437046, 617657, 872967, 1233895, 1744152, 2465546, 3485477

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, except for powers of 2, with n bits.

			The perfect-powers in each prescribed range (rows):
    .
    .
    .
    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
The binary expansions for n >= 3 (columns):
    1001  11001  100100  1010001  10010000  100100001
          11011  110001  1100100  10101001  101000100
                         1111001  11000100  101010111
                         1111101  11011000  101101001
                                  11100001  110010000
                                  11110011  110111001
                                            111100100

The version for squarefree numbers is A077643.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Including powers of 2 in the range gives A377435.
The version for non-perfect-powers is A377701.
The union of all numbers counted is A377702.
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, A246655, A345531.

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

from sympy import mobius, integer_nthroot
def A377467(n):
    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) = A377435(n) - 1.

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

A378356 Prime index of the next prime after the n-th perfect power.

Original entry on oeis.org

1, 3, 5, 5, 7, 10, 10, 12, 12, 16, 19, 23, 26, 31, 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

Offset: 1

Gus Wiseman, Dec 05 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

First differences are A080769.
Union is A378365.
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.
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, A377283, A378035, A378250, A378251, A378253.

Table[PrimePi[NextPrime[n]],{n,Select[Range[1000],perpowQ]}]

a(n) = A000720(A001597(n)) + 1.

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).

A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

Original entry on oeis.org

125, 216, 243, 64000, 1295029, 2535525316, 542939080312

Offset: 1

Gus Wiseman, Nov 26 2024

These are perfect-powers p such that the interval from p to the next perfect power contains a unique prime.

Is this sequence infinite? See A178700.

			We have 125 because 127 is the only prime between 125 and 128.

The next prime is A178700.
Singletons in A378035 (union A378253), restriction of A081676.
The next perfect power is A378374.
Swapping primes and perfect powers gives A379154, unique case of A377283.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the not perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378249 gives least perfect power > prime(n) (run-lengths A378251), restrict of A377468.
Cf. A000961, A031218, A045542, A052410, A067871, A076412, A080769, A131605, A188951, A216765, A378250, A378356.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]

A151800(a(n)) = A178700(n).

A378358 Least non-perfect-power >= n.

Original entry on oeis.org

2, 2, 3, 5, 5, 6, 7, 10, 10, 10, 11, 12, 13, 14, 15, 17, 17, 18, 19, 20, 21, 22, 23, 24, 26, 26, 28, 28, 29, 30, 31, 33, 33, 34, 35, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 65, 66, 67

Offset: 1

Gus Wiseman, Nov 24 2024

nonn

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

The version for prime-powers is A000015, for non-prime-powers A378372.
The union is A007916, complement A001597.
The version for nonsquarefree numbers is A067535, negative A120327 (subtract A378369).
The version for composite numbers is A113646.
The version for prime numbers is A159477.
The run-lengths are A375706.
Terms appearing only once are A375738, multiple times A375703.
The version for perfect-powers is A377468.
Subtracting from n gives A378357.
The opposite version is A378363, for perfect-powers A081676.
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.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A031218, A045542, A052410, A065514, A070321, A076412, A188951, A216765, A345531, A378033.

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

from sympy import mobius, integer_nthroot
def A378358(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = max(1,n-f(n-1))
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024

from sympy import perfect_power
def A378358(n): return n if n>1 and perfect_power(n)==False else n+1 if perfect_power(n+1)==False else n+2 # Chai Wah Wu, Nov 27 2024

a(n) = n - A378357(n).

A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 88

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose minima are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For composite numbers we have A005381, runs A008864 (except first term).
For prime-powers we have A120430, runs A373673 (except first term).
For squarefree numbers we have A373408, runs A072284.
For nonsquarefree numbers we have A373410, runs A053806.
For non-prime-powers we have A373575, runs A373676.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738 (this)
- last: A375739
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A061399, A216765, A251092, A373403, A373679, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 5, 6, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.
An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
Also non-perfect-powers x such that x + 1 is also a non-perfect-power.

			The initial anti-runs are the following, whose maxima are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For nonprime numbers we have A068780, runs A006093 with 2 removed.
For squarefree numbers we have A007674, runs A373415.
For nonsquarefree numbers we have A068781, runs A072284 minus 1 and shifted.
For prime-powers we have A006549, runs A373674.
For non-prime-powers we have A255346, runs A373677.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738
- last: A375739 (this)
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A045542, A046933, A053797, A216765, A373576, A373679, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&radQ[#+1]&]

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the perfect-powers.

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Gus Wiseman, Oct 03 2024

nonn

These are points at which the second differences (A376562) are zero.

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

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with zeros at (A376588):
  3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...

Gus Wiseman, Inflection and undulation points in the non-perfect-powers.

The version for A000002 is empty.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros in A376562, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).
For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ],2],0]

cp's OEIS Frontend

A375737 Sum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

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A378356 Prime index of the next prime after the n-th perfect power.

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

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A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

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A378358 Least non-perfect-power >= n.

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Python

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A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica