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.

Previous Showing 21-25 of 25 results.

A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

Original entry on oeis.org

4, 5, 8, 9, 12, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Gus Wiseman, Sep 11 2024

Keywords

Examples

			The nonprime numbers are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after term 4, term 5, term 8, etc.

Crossrefs

The complement appears to be A014689, except the first term.
Positions of 1's in A065310 (see also A054546, A073783).
First differences are A373403 (except first).
The version for non-prime-powers is A375713, differences A373672.
The version for prime-powers is A375734, differences A373671.
The version for non-perfect-powers is A375740.
The version for composite numbers is A375929.
A000040 lists the prime numbers, differences A001223.
A018252 lists the nonprimes, exclusive A002808.
A046933 counts composite numbers between primes.
Cf. A000961, A006549, A057820, A176246, A246655, A251092, A375708.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]
  • Python
    from sympy import primepi
def A375926(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 n+bisection(lambda y:primepi(x+1+y))-1
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

A375928 Positions of adjacent non-prime-powers (exclusive) differing by more than 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 12, 13, 14, 18, 21, 22, 25, 26, 29, 34, 35, 37, 39, 42, 43, 48, 49, 50, 55, 62, 65, 66, 69, 70, 73, 80, 83, 84, 86, 91, 92, 101, 102, 107, 112, 115, 116, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 175, 182
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2024

Keywords

Examples

			The non-prime-powers (exclusive) are 1, 6, 10, 12, 14, 15, 18, 20, ... which increase by more than 1 after positions 1, 2, 3, 4, 6, 7, ...

Crossrefs

For prime-powers inclusive (A000961) we have A376163, differences A373672.
For nonprime numbers (A002808) we have A014689, differences A046933.
First differences are A110969.
The complement is A375713.
For non-perfect-powers we have A375714, complement A375740.
The complement for prime-powers (exclusive) is A375734, differences A373671.
The complement for nonprime numbers is A375926, differences A373403.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A007916 lists non-perfect-powers, differences A375706.
A024619 lists non-prime-powers (inclusive), differences A375735.
A246655 lists prime-powers (exclusive), differences A174965.
A361102 lists non-prime-powers (exclusive), differences A375708.
Cf. A006549, A053289, A073783, A093555, A120430, A176246, A251092.

Programs

  • Mathematica
    ce=Select[Range[100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

Formula

The inclusive version is a(n+1) - 1.

A377043 The n-th perfect-power A001597(n) minus the n-th power of a prime A000961(n).

Original entry on oeis.org

0, 2, 5, 5, 11, 18, 19, 23, 25, 36, 48, 64, 81, 98, 100, 101, 115, 138, 164, 179, 184, 200, 209, 240, 271, 284, 300, 336, 374, 413, 439, 450, 495, 542, 587, 632, 683, 738, 793, 852, 887, 903, 964, 1029, 1097, 1165, 1194, 1230, 1295, 1370, 1443, 1518, 1561
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2024

Keywords

Comments

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

Crossrefs

Excluding 1 from the powers of primes gives A377044.
A000015 gives the least prime-power >= n.
A031218 gives the greatest prime-power <= n.
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, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102 (differences A375708).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A377051.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
per=Select[Range[1000],perpowQ];
per-NestList[NestWhile[#+1&,#+1,!PrimePowerQ[#]&]&,1,Length[per]-1]
  • Python
    from sympy import mobius, primepi, integer_nthroot
def A377043(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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def g(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024

Formula

a(n) = A001597(n) - A000961(n).

A377044 The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

Original entry on oeis.org

-1, 1, 4, 4, 9, 17, 18, 21, 23, 33, 47, 62, 77, 96, 98, 99, 113, 137, 159, 175, 182, 196, 207, 236, 265, 282, 297, 333, 370, 411, 433, 448, 493, 536, 579, 628, 681, 734, 791, 848, 879, 899, 962, 1028, 1094, 1159, 1192, 1220, 1293, 1364, 1437, 1514, 1559, 1591
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2024

Keywords

Comments

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

Crossrefs

Including 1 with the prime-powers gives A377043.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, A093555, A376596.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A031218 gives the greatest prime-power <= n.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102, A375708.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A376560, A376561, A377051.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
per=Select[Range[1000],perpowQ];
per-NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,2,Length[per]-1]
  • Python
    from sympy import mobius, primepi, integer_nthroot
def A377044(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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def g(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024

Formula

a(n) = A001597(n) - A246655(n).

A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

Original entry on oeis.org

4, 7, 8, 14, 15, 16, 18, 19, 22, 23, 26, 27, 29, 30, 31, 32, 35, 37, 39, 40, 43, 44, 45, 46, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 66, 67, 70, 71, 73, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2024

Keywords

Examples

			The non-prime-powers (inclusive) are 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ... which increase by 1 after positions 4, 7, 8, ...

Crossrefs

For prime-powers inclusive (A000961) we have A375734, differences A373671.
For nonprime numbers (A002808) we have A375926, differences A373403.
For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
First differences are A373672.
The exclusive version is a(n) - 1 = A375713.
Positions of 1's in A375735.
For non-perfect-powers we have A375740.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

Programs

  • Mathematica
    ce=Select[Range[2,100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],ce[[#+1]]==ce[[#]]+1&]
Previous Showing 21-25 of 25 results.

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