cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Gus Wiseman, Dec 22 2024

Keywords

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Dec 17 2024

Keywords

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root.
The perfect powers themselves are given by A001597(a(n)) = A378355(n).

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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&]

Formula

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

A378374 Perfect powers p such that the interval from the previous perfect power to p contains a unique prime.

Original entry on oeis.org

128, 225, 256, 64009, 1295044
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2024

Keywords

Comments

Also numbers appearing exactly once in A378249.

Examples

			The consecutive perfect powers 125 and 128 have interval (125, 126, 127, 128) with unique prime 127, so 128 is in the sequence.

Crossrefs

The previous prime is A178700.
For prime powers instead of perfect powers we have A345531, difference A377281.
Opposite singletons in A378035 (union A378253), restriction of A081676.
For squarefree numbers we have A378082, see A377430, A061398, A377431, A068360.
Singletons in A378249 (run-lengths A378251), restriction of A377468 to the primes.
If the same interval contains at least one prime we get A378250.
For next instead of previous perfect power we have A378355.
Swapping "prime" with "perfect power" gives A378364.
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.
A080769 counts primes between perfect powers.
Cf. A023055, A045542, A052410, A216765, A376560, A377287, A378356.

Programs

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

Formula

We have a(n) < A178700(n) < A378355(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

Views

Author

Gus Wiseman, Dec 22 2024

Keywords

Comments

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

Crossrefs

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.

Programs

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

Formula

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

A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

Original entry on oeis.org

1, 0, 4, 5, 1, 2, 12, 11, 12, 31, 3, 1, 32, 59, 11, 25, 46, 13, 125, 14, 80, 88, 94, 103, 52, 261, 35, 267, 147, 172, 120, 9, 9, 163, 355, 279, 313, 207, 329, 347, 376, 108, 257, 805, 283, 262, 25, 917, 242, 1081, 702, 365, 752, 389, 251, 535, 1679, 877, 447
Offset: 1

Views

Author

Gus Wiseman, Nov 30 2024

Keywords

Comments

The inclusive version is a(n) + 2.
Nonprime prime powers (A246547) begin: 4, 8, 9, 16, 25, 27, 32, 49, ...

Examples

			The initial terms count the following composite numbers:
  {6}, {}, {10,12,14,15}, {18,20,21,22,24}, {26}, {28,30}, ...
The composite numbers for a(77) = 6 together with their prime indices are the following. We have also shown the nonprime prime powers before and after:
  32761: {42,42}
  32762: {1,1900}
  32763: {2,19,38}
  32764: {1,1,1028}
  32765: {3,847}
  32766: {1,2,14,31}
  32767: {4,11,36}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Crossrefs

For prime instead of composite we have A067871.
For nonsquarefree numbers we have A378373, for primes A236575.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A002808 lists the composite numbers.
A031218 gives the greatest prime-power <= n.
A046933 counts composite numbers between primes.
A053707 gives first differences of nonprime prime powers.
A080101 = A366833 - 1 counts prime powers between primes.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the nearest prime power after prime(n) + 1, difference A377281.
Cf. A377286, A377287, A377288 (primes A053706).
Cf. A024619, A053607, A065890, A076259, A078147, A243348, A276781, A377057, A377282.

Programs

  • Mathematica
    nn=1000;
v=Select[Range[nn],PrimePowerQ[#]&&!PrimeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A378364 Prime numbers such that the interval from the previous prime number contains a unique perfect power.

Original entry on oeis.org

2, 5, 17, 53, 67, 83, 101, 131, 149, 173, 197, 223, 227, 251, 257, 293, 331, 347, 367, 401, 443, 487, 521, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1009, 1031, 1091, 1163, 1229, 1297, 1361, 1373, 1447, 1523, 1601, 1693, 1733, 1777, 1861, 1949, 2027, 2053
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2024

Keywords

Comments

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

Examples

			The prime before 17 is 13, and the interval (13,14,15,16,17) contains only one perfect power 16, so 17 is in the sequence.
The prime before 29 is 23, and the interval (23,24,25,26,27,28,29) contains two perfect powers 25 and 27, so 29 is not in the sequence.

Crossrefs

For non prime powers we have A006512.
For zero instead of one perfect power we have the prime terms of A345531.
The indices of these primes are the positions of 1 in A377432.
The indices of these primes are 1 + A377434(n-1).
For more than one perfect power see A377466.
Swapping "prime" with "perfect power" gives A378374.
For next instead of previous prime we have A379154.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A081676 gives the greatest perfect power <= n.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A052410, A053607, A069623, A376560, A377287, A377436.

Programs

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

A379154 Prime numbers p such that the interval from p to the next prime number contains a unique perfect power.

Original entry on oeis.org

3, 13, 47, 61, 79, 97, 127, 139, 167, 193, 211, 223, 241, 251, 283, 317, 337, 359, 397, 439, 479, 509, 523, 571, 619, 673, 727, 773, 839, 887, 953, 997, 1021, 1087, 1153, 1223, 1291, 1327, 1367, 1439, 1511, 1597, 1669, 1723, 1759, 1847, 1933, 2017, 2039, 2113
Offset: 1

Views

Author

Gus Wiseman, Dec 18 2024

Keywords

Comments

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

Examples

			The prime after 13 is 17, and the interval (13,14,15,16,17) contains only one perfect power 16, so 13 is in the sequence.

Links

Crossrefs

The indices of these primes are one plus the positions of 1 in A377432.
For zero instead of one perfect power we have the primes indexed by A377436.
The indices of these primes are A377434.
Swapping "prime" with "perfect power" gives A378355, indices A378368.
For previous instead of next prime we have A378364.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A081676 gives the greatest perfect power <= n.
A116086 gives perfect powers with no primes between them and the next perfect power.
A366833 counts prime powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A052410, A069623, A376560, A377283, A377287, A377466, A378374.

Programs

  • Maple
    N:= 10^4: # to get all entries <= N
S:={seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}:
S:= sort(convert(S,list)):
J:= select(i -> nextprime(S[i]) < S[i+1] and prevprime(S[i]) > S[i-1], [$2..nops(S)-1]):
J:= [1,op(J)]:
map(prevprime, S[J]); # Robert Israel, Jan 19 2025
  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[1000],PrimeQ[#]&&Length[Select[Range[#,NextPrime[#]],perpowQ]]==1&]
  • PARI
    is_a379154(n) = isprime(n) && #select(x->ispower(x), [n+1..nextprime(n+1)-1])==1 \\ Hugo Pfoertner, Dec 19 2024

Formula

a(n) = A151799(A378364(n+1)).

A378616 Greatest non prime power <= prime(n).

Original entry on oeis.org

1, 1, 1, 6, 10, 12, 15, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 255, 262, 268, 270
Offset: 1

Views

Author

Gus Wiseman, Dec 06 2024

Keywords

Comments

Conjecture: Equal to A006093(n) = prime(n) - 1 except at terms of A159611.

Examples

			The first number line below shows the non prime powers. The second shows the primes:
--1-------------6----------10----12----14-15-------18----20-21-22----24--
=====2==3====5=====7==========11====13==========17====19==========23=====

Crossrefs

For nonprime instead of non prime power we have A156037.
Restriction of A378367.
Lengths are A378615.
For nonsquarefree: A378032 (diffs A378034), restriction of A378033 (diffs A378036).
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Prime powers between primes:
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A006093, A053607, A143731, A159611, A304521, A343249, A345531, A356068, A368748, A377281, A377289, A377703, A377781.

Programs

  • Mathematica
    Table[Max[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

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

Views

Author

Gus Wiseman, Dec 23 2024

Keywords

Comments

Also positions of 2 in A366835.

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Dec 24 2024

Keywords

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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