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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A378363 Greatest number <= n that is 1 or not a perfect-power.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 24, 24, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67
Offset: 1

Views

Author

Gus Wiseman, Nov 24 2024

Keywords

Comments

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

Examples

			In the non-perfect-powers ... 5, 6, 7, 10, 11 ... the greatest term <= 8 is 7, so a(8) = 7.

Crossrefs

The union is A007916, complement A001597.
The version for prime numbers is A007917 or A151799, opposite A159477.
The version for prime-powers is A031218, opposite A000015.
The version for squarefree numbers is A067535, opposite A070321.
The version for perfect-powers is A081676, opposite A377468.
The version for composite numbers is A179278, opposite A113646.
Terms appearing multiple times are A375704, opposite A375703.
The run-lengths are A375706.
Terms appearing only once are A375739, opposite A375738.
The version for nonsquarefree numbers is A378033, opposite A120327.
The opposite version is A378358.
Subtracting n gives A378364, opposite A378357.
The version for non-prime-powers is A378367 (subtracted A378371), opposite A378372.
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.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A014234, A045542, A052410, A065514, A076412, A162966, A188951, A345531.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#-1&,n,#>1&&perpowQ[#]&],{n,100}]
  • Python
    from sympy import mobius, integer_nthroot
def A378363(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = n-f(n)
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2024

Keywords

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of first appearances (A376268):
  1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...

Crossrefs

These are the sorted positions of first appearances in A053289 (union A023055).
The complement is A376519.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2024

Keywords

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

Crossrefs

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

Programs

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

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}]]

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).

A377702 Perfect-powers except for powers of 2.

Original entry on oeis.org

9, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197
Offset: 1

Views

Author

Gus Wiseman, Nov 05 2024

Keywords

Comments

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

Examples

			The terms together with their prime indices begin:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   169: {6,6}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   324: {1,1,2,2,2,2}

Crossrefs

Including the powers of 2 gives A001597, counted by A377435.
For prime-powers we have A061345.
These terms are counted by A377467, for non-perfect-powers A377701.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A188951 counts perfect-powers less than 2^n.
A377468 gives the least perfect-power > n.
Cf. A014210, A014234, A023055, A045542, A052410, A065514, A069623, A246655, A304521, A336416, A345531, A366833.

Programs

  • Mathematica
    Select[Range[1000],GCD@@FactorInteger[#][[All,2]]>1&&!IntegerQ[Log[2,#]]&]
  • Python
    from sympy import mobius, integer_nthroot
def A377702(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-2+x+(l:=x.bit_length())+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,l)))
    return bisection(f,n+1,n+1) # Chai Wah Wu, Nov 06 2024

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}.

A377433 Number of non-perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 5, 1, 4, 3, 1, 5, 2, 5, 7, 2, 1, 3, 1, 3, 11, 2, 5, 1, 8, 1, 5, 5, 3, 4, 5, 1, 9, 1, 2, 1, 11, 10, 2, 1, 3, 5, 1, 8, 4, 5, 5, 1, 5, 3, 1, 8, 13, 3, 1, 3, 12, 5, 8, 1, 3, 5, 6, 5, 5, 3, 5, 7, 2, 7, 9, 1, 9, 1, 5, 2
Offset: 1

Views

Author

Gus Wiseman, Nov 02 2024

Keywords

Comments

Non-perfect-powers (A007916) are numbers without a proper integer root.
Positions of terms > 1 appear to be A049579.

Examples

			Between prime(4) = 7 and prime(5) = 11 the only non-perfect-power is 10, so a(4) = 1.

Crossrefs

Positions of 1 are latter terms of A029707.
Positions of terms > 1 appear to be A049579.
For prime-powers instead of non-perfect-powers we have A080101.
For non-prime-powers instead of non-perfect-powers we have A368748.
Perfect-powers in the same range are counted by A377432.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706.
A065514 gives the greatest prime-power < prime(n), difference A377289.
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A064113, A065890, A244508, A377282, A377434, A377436, A377466, A377467.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],radQ]],{n,100}]

Formula

a(n) + A377432(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 7, 8, 11, 16, 24, 32, 42, 61, 82, 118, 166, 231, 322, 453, 635, 892, 1253, 1767, 2487, 3505, 4936, 6959, 9816, 13850, 19538, 27578, 38933, 54972, 77641, 109668, 154922, 218879, 309277, 437047, 617658, 872968, 1233896, 1744153, 2465547, 3485478
Offset: 0

Views

Author

Gus Wiseman, Nov 04 2024

Keywords

Comments

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
Also the number of perfect-powers with n bits.

Examples

			The perfect-powers in each prescribed range (rows):
    1
    .
    4
    8    9
   16   25   27
   32   36   49
   64   81  100  121  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
Their binary expansions (columns):
  1  .  100  1000  10000  100000  1000000  10000000  100000000
             1001  11001  100100  1010001  10010000  100100001
                   11011  110001  1100100  10101001  101000100
                                  1111001  11000100  101010111
                                  1111101  11011000  101101001
                                           11100001  110010000
                                           11110011  110111001
                                                     111100100

Crossrefs

The union of all numbers counted is A001597, without powers of two A377702.
The version for squarefree numbers is A077643.
These are the first differences of A188951.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Not counting powers of 2 gives A377467.
The version for non-perfect-powers is A377701.
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.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A345531, A377434.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n,2^(n+1)-1],perpowQ]],{n,0,15}]
  • Python
    from sympy import mobius, integer_nthroot
def A377435(n):
    if n==0: return 1
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

Formula

For n != 1, a(n) = A377467(n) + 1.

Extensions

a(26)-a(46) from Chai Wah Wu, Nov 05 2024
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