A366833 -id:A366833 - cp's OEIS frontend

A366835 In the pair (A246655(n), A246655(n+1)), how many primes are there?

2, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

First 0 terms appear at n = 6, 14, 41, 359, 3589, corresponding to consecutive prime powers (8,9), (25,27), (121,125), (2187,2197) and (32761,32768), respectively (cf. A068315 and A068435).

There cannot be primes strictly between consecutive prime powers, so we get the same result considering the whole interval (not just the pair). - Gus Wiseman, Dec 25 2024

			a(1) = 2 because in the first prime power pair (2 and 3) there are two primes.
a(14) = 0 because in the 14th prime power pair (25 and 27) there are no primes.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1038 X 1038 raster of a(n), n = 1..1077444, read left to right in rows, then top to bottom, showing a(n) = 0 in white, a(n) = 1 in red, and a(n) = 2 in dark blue.

For perfect powers instead of prime powers we have A080769.
Positions of 1 are A379155, indices of A379157.
Positions of 0 are A379156, indices of A068315.
Positions of 2 are A379158, indices of A379541.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A031218 gives the greatest prime power <= n.
A065514 gives the greatest prime power < prime(n), difference A377289.
A080101 and A366833 count prime powers between primes, see A053607, A304521.
A246655 lists the prime powers, differences A057820.
Cf. A000961, A025474, A046933, A067871, A068435, A175106, A178700, A345531, A362965, A377281.

With[{upto=500},Map[Count[#,_?PrimeQ]&,Partition[Select[Range[upto],PrimePowerQ],2,1]]] (* Considers prime powers up to 500 *)

lista(nn) = my(v=[p| p <- [1..nn], isprimepower(p)]); vector(#v-1, k, isprime(v[k]) + isprime(v[k+1])); \\ Michel Marcus, Oct 26 2023

A378370 Distance between n and the least prime power >= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 27 2024

nonn

Prime powers allowing 1 are listed by A000961.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime instead of prime power we have A007920 (A007918), strict A013632.
For perfect power we have A074984 (A377468), opposite A069584 (A081676).
For squarefree we have A081221 (A067535).
The restriction to the prime numbers is A377281 (A345531).
The strict version is A377282 = a(n) + 1.
For non prime power instead of prime power we have A378371 (A378372).
The opposite version is A378457, strict A276781.
A000015 gives the least prime power >= n, opposite A031218.
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.
Prime-powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A053707, A065514, A065890, A343249, A376596, A376597, A377051, A377054, A377289, A377781.

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

a(n) = A000015(n) - n.

a(n) = A377282(n - 1) - 1 for n > 1.

A379155 Numbers k such that there is a unique prime between the k-th and (k+1)-th prime powers (A246655).

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 13, 15, 17, 18, 22, 23, 26, 27, 31, 32, 40, 42, 43, 44, 52, 53, 67, 68, 69, 70, 77, 78, 85, 86, 90, 91, 116, 117, 119, 120, 135, 136, 151, 152, 169, 170, 186, 187, 197, 198, 243, 244, 246, 247, 291, 292, 312, 313, 339, 340, 358, 360, 362

Offset: 1

Gus Wiseman, Dec 22 2024

nonn

Numbers k such that exactly one of A246655(k) and A246655(k+1) is prime. - Robert Israel, Jan 22 2025

The prime powers themselves are: 3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, ...

			The 4th and 5th prime powers are 5 and 7, with interval (5,6,7) containing two primes, so 4 is not in the sequence.
The 13th and 14th prime powers are 23 and 25, with interval (23,24,25) containing only one prime, so 13 is in the sequence.
The 18th and 19th prime powers are 32 and 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 18 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..2112

These are the positions of 1 in A366835, for perfect powers A080769.
For perfect powers instead of prime powers we have A378368.
For no primes we have A379156, for perfect powers A274605.
The prime powers themselves are A379157, for previous A175106.
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.
A065514 gives the greatest prime power < prime(n), difference A377289.
A246655 lists the prime powers.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A067871, A068315, A080101, A178700, A345531, A377281, A377283, A377287, A377434, A378374.

N:= 1000: # for terms k where A246655(k+1) <+ N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
S:= convert(P,set):
for p in P while p^2 <= N do
  S:= S union {seq(p^j,j=2..ilog[p](N))}
od:
PP:= sort(convert(S,list)):
state:= 1: Res:= NULL:
ip:= 2:
for i from 2 to nops(PP) do
  if PP[i] = P[ip] then
    if state = 0 then Res:= Res,i-1 fi;
    state:= 1;
    ip:= ip+1;
  else
    if state = 1 then Res:= Res,i-1 fi;
    state:= 0;
  fi
od:
Res; # Robert Israel, Jan 22 2025

v=Select[Range[100],PrimePowerQ];
Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]

A246655(a(n)) = A379157(n).

A068315 For numbers k such that A025474(k) > 1 and A025474(k+1) > 1, sequence gives A000961(k).

Original entry on oeis.org

8, 25, 121, 2187, 32761

Offset: 1

Naohiro Nomoto, Mar 08 2002

Equivalently, prime powers (either A000961 or A246655) q such that q and the next prime power are both composite numbers. - Paolo Xausa, Oct 25 2023

			The interval (121,122,123,124,125) contains no primes, so 121 is in the sequence. - _Gus Wiseman_, Dec 24 2024

Bisection of A068435.
For perfect powers instead of prime powers we have A116086, indices A274605.
The position of a(k) in the prime powers A246655 is A379156(k).
For just one prime we have A379157, indices A379155.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A031218 gives the greatest prime power <= n.
A046933 gives run-lengths of composites between primes.
A065514 gives the greatest prime power < prime(n), difference A377289.
A246655 lists the prime powers, differences A057820.
A366833 counts prime powers between primes, see A053607, A304521.
A366835 counts primes between prime powers.
Cf. A025474, A067871, A080101, A080769, A175106, A345531, A377281, A377287, A378368.

With[{upto=33000},Map[First,Select[Partition[Select[Range[upto],PrimePowerQ],2,1],NoneTrue[#,PrimeQ]&]]] (* Paolo Xausa, Oct 25 2023 *)

a(n) = A246655(A379156(n)). - Gus Wiseman, Dec 24 2024

Definition corrected by Jinyuan Wang, Sep 05 2020

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}]]

A378367 Greatest non prime power <= n, allowing 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 6, 6, 6, 10, 10, 12, 12, 14, 15, 15, 15, 18, 18, 20, 21, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 48, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 63, 65, 66, 66

Offset: 1

Gus Wiseman, Nov 29 2024

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, ...

			The greatest non prime power <= 7 is 6, so a(7) = 6.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For prime we have A007917 (A064722).
For nonprime we have A179278 (A010051 almost).
For perfect power we have A081676 (A069584).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For non perfect power we have A378363.
The opposite is A378372, subtracting n A378371.
For prime power we have A031218 (A276781 - 1).
Subtracting from n gives (A378366).
A000015 gives the least prime power >= n (A378370).
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 (A013632), weak version A007918 (A007920).
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A065514, A113646, A345531 (A377281), A377051, A377054, A377282, A377468 (A074984), A378358, A378457.
Cf. A356068.

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

a(n) = n - A378366(n).

a(n) = A361102(A356068(n)). - Ridouane Oudra, Aug 22 2025

A379157 Prime powers p such that the interval from p to the next prime power contains a unique prime number.

Original entry on oeis.org

3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, 79, 81, 113, 125, 127, 128, 167, 169, 241, 243, 251, 256, 283, 289, 337, 343, 359, 361, 509, 512, 523, 529, 619, 625, 727, 729, 839, 841, 953, 961, 1021, 1024, 1327, 1331, 1367, 1369, 1669, 1681, 1847, 1849

Offset: 1

Gus Wiseman, Dec 22 2024

nonn

			The next prime power after 32 is 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 32 is in the sequence.

For no primes we have A068315/A379156, for perfect powers A116086/A274605.
The previous instead of next prime power we have A175106.
For perfect powers instead of prime powers we have A378355.
The positions of these prime powers (in A246655) are A379155.
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.
A065514 gives the greatest prime power < prime(n), difference A377289.
A246655 lists the prime powers.
A366833 counts prime powers between primes, see A053607, A304521.
A366835 counts primes between prime powers, for perfect powers A080769.
Cf. A046933, A067871, A080101, A178700, A345531, A377281, A377287, A377434, A378374.

v=Select[Range[100],PrimePowerQ]
nextpripow[n_]:=NestWhile[#+1&,n+1,!PrimePowerQ[#]&]
Select[v,Length[Select[Range[#,nextpripow[#]],PrimeQ]]==1&]

a(n) = A246655(A379155(n)).

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A378457 Difference between n and the greatest prime power <= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Prime powers allowing 1 are listed by A000961.

			The greatest prime power <= 6 is 5, so a(6) = 1.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we have A010051 (almost) (A179278).
Subtracting from n gives (A031218).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
Adding one gives A276781.
For nonsquarefree we have (A378033).
For non perfect power we have (A378363).
For non prime power we have A378366 (A378367).
The opposite is A378370 = A377282-1.
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, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007920, A013632, A065514, A074984, A377051, A377054, A377281, A377289, A377468, A378357, A378371.

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

a(n) = n - A031218(n).

a(n) = A276781(n) - 1.

A379156 Positions in A246655 (prime powers) of terms q such that there is no prime between q and the next prime power.

Original entry on oeis.org

6, 14, 41, 359, 3589

Offset: 1

Gus Wiseman, Dec 22 2024

The powers of primes themselves are 8, 25, 121, 2187, 32761, ... (A068315).

The prime powers themselves are A068315, for just one prime A379157.
For perfect powers instead of prime powers we have A274605.
Positions of 0 in A366835.
For just one prime we have A379155, for perfect powers A378368.
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.
A065514 gives the greatest prime power < prime(n), difference A377289.
A131605 finds perfect powers that are not prime powers.
A246655 lists the prime powers.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A046933, A067871, A080101, A080769, A175106, A178700, A345531, A377281, A377287, A377432, A377434.

v=Select[Range[100],PrimePowerQ];
Select[Range[Length[v]-1],FreeQ[Range[v[[#]],v[[#+1]]],_?PrimeQ]&]

A246655(a(n)) = A068315(n).

cp's OEIS Frontend

A366835 In the pair (A246655(n), A246655(n+1)), how many primes are there?

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A378370 Distance between n and the least prime power >= n, allowing 1.

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A379155 Numbers k such that there is a unique prime between the k-th and (k+1)-th prime powers (A246655).

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A068315 For numbers k such that A025474(k) > 1 and A025474(k+1) > 1, sequence gives A000961(k).

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

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A378367 Greatest non prime power <= n, allowing 1.

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A379157 Prime powers p such that the interval from p to the next prime power contains a unique prime number.

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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