A375706 -id:A375706 - cp's OEIS frontend

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

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).

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

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

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

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]&]

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]

A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, 106, 107, 109, 110, 111, 112, 126, 127, 150, 151, 176, 177, 195, 196, 203, 204, 220, 221, 232, 233, 264, 265, 298, 299, 316, 317, 333, 334, 371, 372, 411, 412, 453, 454, 480, 481, 496

Offset: 1

Gus Wiseman, Oct 03 2024

nonn

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

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 nonzeros at (A376589):
  1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...

Gus Wiseman, Points of nonzero curvature in the non-perfect-powers.

For first differences we had A375706, ones A375740, complement A375714.
These are the positions of nonzeros in A376562, complement A376588.
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.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).
For second differences: A064113 (prime), A376602 (composite), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Sign[Differences[Select[Range[1000],radQ],2]],1|-1]

A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

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

cp's OEIS Frontend

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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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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

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Mathematica

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

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