A379155 -id:A379155 - 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

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

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)).

A378368 Positions (in A001597) of consecutive perfect powers with a unique prime between them.

Original entry on oeis.org

15, 20, 22, 295, 1257

Offset: 1

Gus Wiseman, Dec 17 2024

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

The perfect powers themselves are given by A001597(a(n)) = A378355(n).

			The 15th and 16th perfect powers are 125 and 128, and 127 is the only prime between them, so 15 is in the sequence.

These are the positions of 1 in A080769.
The next prime after A001597(a(n)) is A178700(n).
For no (instead of one) perfect powers we have A274605.
Swapping 'prime' and 'perfect power' gives A377434, unique case of A377283.
The next perfect power after A001597(a(n)) is A378374(n).
For prime powers instead of perfect powers we have A379155.
A000040 lists the primes, differences A001223.
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.
A081676 gives the greatest perfect power <= n.
A377432 counts perfect powers between primes, see A377436, A377466.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A046933, A052410, A053607, A067871, A076412, A080101, A375740, A377287.

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

We have A001597(a(n)) = A378355(n) < A178700(n) < A378374(n).

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).

A379158 Numbers m such that the consecutive prime powers A246655(m) and A246655(m+1) are both prime.

Original entry on oeis.org

1, 4, 8, 11, 12, 16, 19, 20, 21, 24, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 84, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Gus Wiseman, Dec 23 2024

nonn

Also positions of 2 in A366835.

			The 4th and 5th prime powers are 5 and 7, which are both prime, so 4 is in the sequence.
The 12th and 13th prime powers are 19 and 23, which are both prime, so 12 is in the sequence.

Positions of adjacent primes in A246655 (prime powers).
Positions of 2 in A366835.
For just one prime we have A379155, positions of prime powers in A379157.
For no primes we have A379156, positions of prime powers in A068315.
The primes powers themselves are A379541.
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.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A067871, A080769, A178700, A274605, A345531, A377281, A377287, A378368.

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

A246655(a(n)) = A379541(n).

A379541 Prime numbers such that the next greatest prime power is also prime.

Original entry on oeis.org

2, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 67, 71, 73, 83, 89, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 347, 349, 353

Offset: 1

Gus Wiseman, Dec 24 2024

nonn

			After 13 the next prime power is 16, which is not prime, so 13 is not in the sequence.
After 19 the next prime power is 23, which is prime, so 19 is in the sequence.

For no primes we have A068315, positions A379156.
Lesser of adjacent primes in A246655 (prime powers).
The indices of these primes are A377286.
For just one prime we have A379157, positions A379155.
Positions in the prime powers are A379158 = positions of 2 in A366835.
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.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A067871, A080769, A178700, A274605, A345531, A377281, A377287, A378368.

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

a(n) = A246655(A379158(n)).

cp's OEIS Frontend

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

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

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A378368 Positions (in A001597) of consecutive perfect powers with a unique prime between them.

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

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A379158 Numbers m such that the consecutive prime powers A246655(m) and A246655(m+1) are both prime.

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A379541 Prime numbers such that the next greatest prime power is also prime.

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