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 1-8 of 8 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&]

A378035 Greatest perfect power < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 9, 16, 16, 16, 27, 27, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 125, 128, 128, 128, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 216, 225, 225, 225, 225, 225, 243, 256, 256, 256, 256
Offset: 1

Views

Author

Gus Wiseman, Nov 23 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

Restriction of A081676 to the primes.
Positions of last appearances are also A377283.
A version for squarefree numbers is A378032.
The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.
The union is A378253.
Terms appearing exactly once are A378355.
Run lengths are A378356, first differences of A377283, complement A377436.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the nonperfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A080769 counts primes between perfect powers, prime powers A067871.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]
  • PARI
    a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024
  • Python
    from sympy import mobius, integer_nthroot, prime
def A378035(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(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = (p:=prime(n)-1)-f(p)
    return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024

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}]]

A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

Original entry on oeis.org

125, 216, 243, 64000, 1295029, 2535525316, 542939080312
Offset: 1

Views

Author

Gus Wiseman, Nov 26 2024

Keywords

Comments

These are perfect-powers p such that the interval from p to the next perfect power contains a unique prime.
Is this sequence infinite? See A178700.

Examples

			We have 125 because 127 is the only prime between 125 and 128.

Crossrefs

The next prime is A178700.
Singletons in A378035 (union A378253), restriction of A081676.
The next perfect power is A378374.
Swapping primes and perfect powers gives A379154, unique case of A377283.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the not 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 least perfect power > prime(n) (run-lengths A378251), restrict of A377468.
Cf. A000961, A031218, A045542, A052410, A067871, A076412, A080769, A131605, A188951, A216765, A378250, 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

A151800(a(n)) = A178700(n).

A378253 Perfect powers p such that there are no other perfect powers between p and the least prime > p.

Original entry on oeis.org

1, 4, 9, 16, 27, 36, 49, 64, 81, 100, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849
Offset: 1

Views

Author

Gus Wiseman, Nov 26 2024

Keywords

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.
Each term is the greatest perfect power < prime(k) for some k.

Examples

			The first number line below shows the perfect powers. The second shows each prime. To get a(n), we take the last perfect power in each interval between consecutive primes, omitting 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==

Crossrefs

Union of A378035, restriction of A081676 to the primes.
The opposite is A378250, union of A378249 (run-lengths A378251).
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.
A377283 ranks perfect powers between primes, differences A378356.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
Cf. A000015, A000961, A052410, A067871, A076412, A131605, A188951, A216765, A345531, A377057, A377468, A378355.

Programs

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

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).

A378365 Next prime index after each perfect power, duplicates removed.

Original entry on oeis.org

1, 3, 5, 7, 10, 12, 16, 19, 23, 26, 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, 328, 330, 343
Offset: 1

Views

Author

Gus Wiseman, Nov 26 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 n at position prime(n). To get a(n), we take the first prime between each pair of consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

Crossrefs

The opposite version is A377283.
Positions of first appearances in A378035.
First differences are A378251.
Union of A378356.
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.
A080769 counts primes between perfect powers.
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, A378250, A378253, A378355.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Union[1+Table[PrimePi[n],{n,Select[Range[100],perpowQ]}]]

Formula

These are the distinct elements of the set {1 + A000720(A151800(n)), n>0}.

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 1-8 of 8 results.

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