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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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

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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)).

A378614 Number of composite numbers (A002808) between consecutive perfect powers (A001597), exclusive.

Original entry on oeis.org

0, 1, 0, 4, 5, 1, 2, 3, 8, 11, 12, 15, 15, 3, 1, 12, 19, 21, 16, 7, 12, 11, 25, 29, 16, 13, 32, 33, 35, 22, 14, 40, 39, 42, 45, 46, 47, 50, 52, 32, 19, 55, 56, 59, 60, 27, 35, 65, 64, 67, 68, 40, 30, 75, 74, 77, 19, 57, 62, 9, 9, 81, 81, 88, 89, 87, 32, 55, 94
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2024

Keywords

Comments

The inclusive version is a(n) + 2.

Examples

			The composite numbers counted by a(n) cover A106543 with the following disjoint sets:
  .
  6
  .
  10 12 14 15
  18 20 21 22 24
  26
  28 30
  33 34 35
  38 39 40 42 44 45 46 48
  50 51 52 54 55 56 57 58 60 62 63

Links

Crossrefs

For prime instead of perfect power we have A046933.
For prime instead of composite we have A080769.
For nonsquarefree instead of perfect power we have A378373, for primes A236575.
For nonprime prime power instead of perfect power we have A378456.
A001597 lists the perfect powers, differences A053289.
A002808 lists the composite numbers.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A106543 lists the composite non perfect powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378365 gives the least prime > each perfect power, opposite A377283.
Cf. A045542, A052410, A065890, A076412, A081676, A216765, A276781, A377057, A377468, A378035, A378251.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
v=Select[Range[100],perpowQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]
  • Python
    from sympy import mobius, integer_nthroot, primepi
def A378614(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return -(a:=bisection(f,n,n))+(b:=bisection(lambda x:f(x)+1,a+1,a+1))-primepi(b)+primepi(a)-1 # Chai Wah Wu, Dec 03 2024

A378617 First differences of A378249 (next perfect power after prime(n)).

Original entry on oeis.org

0, 4, 0, 8, 0, 9, 0, 0, 7, 0, 17, 0, 0, 0, 15, 0, 0, 17, 0, 0, 0, 19, 0, 0, 21, 0, 0, 0, 0, 7, 16, 0, 0, 25, 0, 0, 0, 0, 27, 0, 0, 0, 0, 20, 0, 0, 9, 18, 0, 0, 0, 0, 13, 33, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 19, 0, 18, 0, 0, 0, 39, 0, 0, 0, 0, 0, 41, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2024

Keywords

Comments

This is the next perfect power after prime(n+1), minus the next perfect power after prime(n).
Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Crossrefs

Positions of positives are A377283.
Positions of zeros are A377436.
The restriction to primes has first differences A377468.
A version for nonsquarefree numbers is A377784, differences of A377783.
The opposite is differences of A378035 (restriction of A081676).
First differences of A378249, run-lengths A378251.
Without zeros we have differences of A378250.
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.
A377432 counts perfect powers between primes.
A378356 - 1 gives next prime after perfect powers, union A378365 - 1.
Cf. A000015, A007918, A023055, A045542, A052410, A065514, A076412, A188951, A216765, A377431, A377434.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,Prime[n],Not@*perpowQ],{n,100}]//Differences
Previous Showing 31-33 of 33 results.

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