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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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).

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

Views

Author

Gus Wiseman, Dec 02 2024

Keywords

Comments

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, ...

Examples

			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}, ...

Crossrefs

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.

Programs

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

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}]

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)).
Previous Showing 11-16 of 16 results.

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