A053707 -id:A053707 - cp's OEIS frontend

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are zero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...

Gus Wiseman, Inflection and undulation points in the prime-powers inclusive.

The first differences were A057820, see also A053707, A376340.
These are the zeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376598.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376596 (second differences), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

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 not 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.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
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, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

Original entry on oeis.org

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Jon Perry, Mar 07 2002

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

For primes between nonsquarefree numbers we have A236575.
For composite instead of prime we have A378456.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A080101 counts prime powers between primes.
A246547 lists the non prime prime powers, differences A053707.
A246655 lists the prime powers not including 1, complement A361102.
Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023

A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, -5, 15, -34, 63, -97, 115, -54, -251, 1184, -3536, 8736, -18993, 37009, -64545, 98442, -121393, 82008, 147432, -860818, 2710023, -7110594, 17077281, -38873146, 85085287, -179965647, 367885014, -725051280, 1372311999, -2481473550, 4257624252

Offset: 0

Gus Wiseman, Oct 22 2024

sign

			The sixth differences of A000961 begin: -5, 10, -9, 1, 6, -10, 16, -18, ..., so a(6) = -5.

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree numbers we have A377041, nonsquarefree A377049.
This is the first column of the array A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For positions of first zeros we have A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A001597 lists perfect-powers, complement A007916.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A175804, A246655, A361102, A376340, A376596, A377056.

q=Select[Range[100],#==1||PrimePowerQ[#]&];
Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*q(k)

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

4, 9, 30, 327, 3512

Offset: 1

Gus Wiseman, Oct 25 2024

Is this sequence finite? For this conjecture see A053706, A080101, A366833.

Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

			Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.

Lucas A. Brown, Python program.

The interval from A008864(n) to A006093(n+1) has A046933 elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053706.
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 2 in A080101, or 3 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
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, A376596, A376597, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

prime(a(n)) = A053706(n).

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the prime-powers inclusive.

The first differences were A057820, see also A376340.
First differences are A376309.
These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376597.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
`A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, -6, 45, -50, 113, -98, 73, 274, -1159, 3563, -8707, 19024, -36977, 64582, -98401, 121436, -81961, -147383, 860871, -2709964, 7110655, -17077217, 38873213, -85085216, 179965720, -367884935, 725051361, -1372311916, 2481473639, -4257624155

Offset: 0

Gus Wiseman, Oct 22 2024

sign

These are the row-sums of the triangle-version of A377051.

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = -6.

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree numbers we have A377039, nonsquarefree A377047.
These are the antidiagonal-sums of A377051.
The unsigned version is A377053.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A093555, A174965, A246655, A361102, A376340, A376596, A376598.

nn=20;
t=Table[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

Original entry on oeis.org

1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, 206, 207, 216, 244, 245, 304, 325, 327, 402, 466, 484, 605, 801, 880, 939, 1033, 1055, 1077, 1234, 1281, 1721, 1890, 1891, 1906, 1940, 1960, 1962, 2257, 2290, 2410, 2880, 3150

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with first appearances (A376653):
  1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
These are the sorted positions of first appearances in A376596.
The exclusive version is a(n) - 1 = A376654(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.

q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, 24, 45, 80, 123, 174, 229, 382, 1219, 3591, 8849, 19288, 37899, 67442, 108323, 156054, 206733, 311525, 860955, 2710374, 7111657, 17080759, 38884849, 85124764, 180097856, 368321633, 726482493, 1377039690, 2496856437, 4306569569, 7016267449

Offset: 0

Gus Wiseman, Oct 22 2024

nonn

These are the row-sums of the absolute value of the triangle-version of A377051.

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = 24.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree numbers we have A377040, nonsquarefree A377048.
This is the antidiagonal-sums of the absolute value of A377051.
The signed version is A377052.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A069623, A093555, A174965, A246655, A361102, A376340, A376596.

nn=20;
t=Table[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],{k,0,nn}];
Total/@Abs[Table[t[[j,i-j+1]],{i,nn},{j,i}]]

A377055 Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 1, 4, 48, 61, 83, 29, 57, 290, 121, 7115, 14207, 68320, 14652, 149979, 122704, 481540, 980376, 632441, 29973, 25343678, 50577935, 7512418, 210836403, 67253056, 224083553, 910629561, 931524323, 452509699, 2880227533, 396690327, 57954538325, 77572935454, 35395016473

Offset: 0

Gus Wiseman, Oct 22 2024

nonn

			The fourth differences of A246655 begin: 1, -3, 3, 0, -2, 2, ... so a(4) = 4.

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree numbers we have A377042, nonsquarefree A377050.
These are the positions of first zeros in each row of A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For leaders we have A377054, for primes A007442 or A030016.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A246655 lists the prime-powers, differences A057820 (except first term).
Cf. A025475, A053707, A093555, A174965, A175804, A361102, A376340, A376596, A377056.

nn=10000;
u=Table[Differences[Select[Range[nn],PrimePowerQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]], {k,mnrm[Union[First/@Position[u,0]]]}]

a(12)-a(27) from Pontus von Brömssen, Oct 22 2024
a(28)-a(30) from Chai Wah Wu, Oct 23 2024
a(31)-a(35) from Lucas A. Brown, Nov 03 2024

cp's OEIS Frontend

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

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A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

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A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

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A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

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A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

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A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

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A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

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A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

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A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.