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.

Showing 1-5 of 5 results.

A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 31, 34, 39, 44, 47, 48, 53, 54, 61, 66, 68, 72, 78, 85, 92, 97, 99, 105, 114, 122, 129, 137, 146, 154, 162, 168, 172, 181, 191, 200, 210, 217, 219, 228, 240, 251, 263, 269, 274, 283, 295, 306, 309, 319, 327, 329, 342, 357
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2024

Keywords

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Examples

			The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

Crossrefs

A version for prime powers is A377057, exclusive A377287.
A version for squarefree numbers is A377431.
Positions of positive terms in A377432 (counts perfect powers between primes).
The case of a unique choice is A377434 (a subset).
The complement (no choices) is A377436.
The case of at least two choices is A377466 (a subset).
Positions of last appearances in A378249.
First-differences are A378251.
This is A378365 - 1, union of A378356 - 1.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
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.
A131605 lists perfect powers that are not prime powers.
Cf. A000015, A045542, A065514, A076412, A081676, A188951, A216765, A345531, A377468, A378035, A378250.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]

A378251 Number of primes between consecutive perfect powers, zeros omitted.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 1, 3, 5, 5, 3, 1, 5, 1, 7, 5, 2, 4, 6, 7, 7, 5, 2, 6, 9, 8, 7, 8, 9, 8, 8, 6, 4, 9, 10, 9, 10, 7, 2, 9, 12, 11, 12, 6, 5, 9, 12, 11, 3, 10, 8, 2, 13, 15, 10, 11, 15, 7, 9, 12, 13, 11, 12, 17, 2, 11, 16, 16, 13, 17, 15, 14, 16, 15
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2024

Keywords

Comments

First differences of A377283 and A378365. Run-lengths of A378035 and A378249.
Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Examples

			The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==

Links

Crossrefs

Same as A080769 with 0's removed (which were at positions A274605).
First differences of A377283 and A378365 (union of A378356).
Run-lengths of A378035 (union A378253) and A378249 (union A378250).
The version for nonprime prime powers is A378373, with zeros A067871.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, run-lengths of A377468.
A007916 lists the non-perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
Cf. A007918, A023055, A045542, A052410, A076412, A081676, A188951.
Cf. A216765, A377057, A377431, A378355.

Programs

  • Maple
    N:= 10^6: # to use perfect powers up to N
PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
PP:= sort(convert(PP,list)):
M:= map(numtheory:-pi, PP):
subs(0=NULL, M[2..-1]-M[1..-2]): # Robert Israel, Jan 23 2025
  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A378356 Prime index of the next prime after the n-th perfect power.

Original entry on oeis.org

1, 3, 5, 5, 7, 10, 10, 12, 12, 16, 19, 23, 26, 31, 31, 32, 35, 40, 45, 48, 49, 54, 55, 62, 67, 69, 73, 79, 86, 93, 98, 100, 106, 115, 123, 130, 138, 147, 155, 163, 169, 173, 182, 192, 201, 211, 218, 220, 229, 241, 252, 264, 270, 275, 284, 296, 307, 310, 320
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2024

Keywords

Links

Crossrefs

First differences are A080769.
Union is A378365.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
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, see A377434, A377436, A377466.
A378249 gives the least perfect power > prime(n), restriction of A377468.
Cf. A045542, A052410, A065514, A068361, A076412, A081676, A216765, A345531, A377283, A378035, A378250, A378251, A378253.

Programs

  • Mathematica
    Table[PrimePi[NextPrime[n]],{n,Select[Range[1000],perpowQ]}]

Formula

a(n) = A000720(A001597(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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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