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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A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 0, 3, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0, 5, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 6, 0, 2, 0, 2, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 16 2024

Keywords

Comments

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Crossrefs

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

Programs

  • Mathematica
    Differences[Table[NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]]

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

A378367 Greatest non prime power <= n, allowing 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 6, 6, 6, 10, 10, 12, 12, 14, 15, 15, 15, 18, 18, 20, 21, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 48, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 63, 65, 66, 66
Offset: 1

Views

Author

Gus Wiseman, Nov 29 2024

Keywords

Comments

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

Examples

			The greatest non prime power <= 7 is 6, so a(7) = 6.

Crossrefs

Sequences obtained by subtracting each term from n are placed in parentheses below.
For prime we have A007917 (A064722).
For nonprime we have A179278 (A010051 almost).
For perfect power we have A081676 (A069584).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For non perfect power we have A378363.
The opposite is A378372, subtracting n A378371.
For prime power we have A031218 (A276781 - 1).
Subtracting from n gives (A378366).
A000015 gives the least prime power >= n (A378370).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n (A013632), weak version A007918 (A007920).
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A065514, A113646, A345531 (A377281), A377051, A377054, A377282, A377468 (A074984), A378358, A378457.
Cf. A356068.

Programs

  • Mathematica
    Table[NestWhile[#-1&,n,PrimePowerQ[#]&],{n,100}]

Formula

a(n) = n - A378366(n).
a(n) = A361102(A356068(n)). - Ridouane Oudra, Aug 22 2025

A379157 Prime powers p such that the interval from p to the next prime power contains a unique prime number.

Original entry on oeis.org

3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, 79, 81, 113, 125, 127, 128, 167, 169, 241, 243, 251, 256, 283, 289, 337, 343, 359, 361, 509, 512, 523, 529, 619, 625, 727, 729, 839, 841, 953, 961, 1021, 1024, 1327, 1331, 1367, 1369, 1669, 1681, 1847, 1849
Offset: 1

Views

Author

Gus Wiseman, Dec 22 2024

Keywords

Examples

			The next prime power after 32 is 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 32 is in the sequence.

Crossrefs

For no primes we have A068315/A379156, for perfect powers A116086/A274605.
The previous instead of next prime power we have A175106.
For perfect powers instead of prime powers we have A378355.
The positions of these prime powers (in A246655) are A379155.
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.
A246655 lists the prime powers.
A366833 counts prime powers between primes, see A053607, A304521.
A366835 counts primes between prime powers, for perfect powers A080769.
Cf. A046933, A067871, A080101, A178700, A345531, A377281, A377287, A377434, A378374.

Programs

  • Mathematica
    v=Select[Range[100],PrimePowerQ]
nextpripow[n_]:=NestWhile[#+1&,n+1,!PrimePowerQ[#]&]
Select[v,Length[Select[Range[#,nextpripow[#]],PrimeQ]]==1&]

Formula

a(n) = A246655(A379155(n)).

A378457 Difference between n and the greatest prime power <= n, allowing 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 0, 1, 2, 3, 4
Offset: 1

Views

Author

Gus Wiseman, Nov 29 2024

Keywords

Comments

Prime powers allowing 1 are listed by A000961.

Examples

			The greatest prime power <= 6 is 5, so a(6) = 1.

Crossrefs

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we have A010051 (almost) (A179278).
Subtracting from n gives (A031218).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
Adding one gives A276781.
For nonsquarefree we have (A378033).
For non perfect power we have (A378363).
For non prime power we have A378366 (A378367).
The opposite is A378370 = A377282-1.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007920, A013632, A065514, A074984, A377051, A377054, A377281, A377289, A377468, A378357, A378371.

Programs

  • Mathematica
    Table[n-NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]

Formula

a(n) = n - A031218(n).
a(n) = A276781(n) - 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).

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

A377780 First differences of A000015 (smallest prime-power >= n).

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 3, 0, 0, 1, 2, 0, 4, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 5, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 2, 0, 2, 0, 6, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 13 2024

Keywords

Crossrefs

First differences of A000015, restriction to primes A345531.
The opposite is A377782, restriction to primes A377781, differences of A065514.
For squarefree instead of prime-power see A067535, A112925, A112926, A120327.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A080101 counts prime-powers between primes (exclusive).
A361102 lists the non-powers of primes, differences A375708.
A366833 counts prime-powers between primes.
Cf. A001597, A007916, A007949, A031218, A053289, A053607, A343249, A343250, A377282, A377289, A377468.

Programs

  • Mathematica
    Differences[Table[NestWhile[#+1&,n,!PrimePowerQ[#]&],{n,100}]]

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.
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