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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A373676 First element of each maximal run of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 20, 24, 26, 28, 30, 33, 38, 42, 44, 48, 50, 54, 60, 62, 65, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 129, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234
Offset: 1

Views

Author

Gus Wiseman, Jun 16 2024

Keywords

Comments

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The last element of the same run is A373677.
Consists of 1 and all non-prime-powers k such that k-1 is a power of a prime.

Examples

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Links

Crossrefs

See link for prime, composite, squarefree, and nonsquarefree runs/antiruns.
For runs of powers of primes:
- length A174965
- min A373673
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676 (this sequence)
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes. A246655 is just prime-powers so lacks 1.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A007674, A014963, A038664, A053806, A054265, A068781, A072284, A073051, A373400.

Programs

  • Mathematica
    Select[Range[100],#==1||!PrimePowerQ[#]&&PrimePowerQ[#-1]&]

A373677 Last element of each maximal run of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 15, 18, 22, 24, 26, 28, 30, 36, 40, 42, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238
Offset: 1

Views

Author

Gus Wiseman, Jun 16 2024

Keywords

Comments

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373676.
Consists of all non-prime-powers k such that k+1 is a prime-power.

Examples

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Links

Crossrefs

See link for prime, composite, squarefree, and nonsquarefree runs/antiruns.
For runs of powers of primes:
- length A174965
- min A373673
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677 (this sequence)
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes. A246655 is just prime-powers so lacks 1.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A038664, A068780, A068781, A073051, A373400, A373401.

Programs

  • Mathematica
    Select[Range[100],!PrimePowerQ[#]&&PrimePowerQ[#+1]&]

A373678 Sums of maximal runs of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 29, 18, 63, 24, 26, 28, 30, 138, 117, 42, 135, 48, 153, 280, 60, 125, 131, 207, 72, 380, 80, 82, 430, 651, 297, 102, 315, 108, 333, 819, 369, 126, 259, 670, 138, 1296, 150, 770, 800, 495, 168, 513, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675
Offset: 1

Views

Author

Gus Wiseman, Jun 16 2024

Keywords

Comments

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.

Examples

			The maximal runs of non-powers of primes begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Links

Crossrefs

A000040 lists the primes, differences A001223.
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A038664, A046933, A054265, A065855, A071148, A293697, A371201, A373400.

Programs

  • Mathematica
    Total/@Split[Select[Range[100],!PrimePowerQ[#]&],#1+1==#2&]//Most

A373674 Last element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Gus Wiseman, Jun 16 2024

Keywords

Comments

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373673.
Consists of all powers of primes k such that k+1 is not a power of primes.

Examples

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Links

Crossrefs

For prime antiruns we have A001359, min A006512, length A027833.
For composite runs we have A006093, min A008864, length A176246.
For prime runs we have A067774, min A025584, length A251092 or A175632.
For squarefree runs we have A373415, min A072284, length A120992.
For nonsquarefree runs we have min A053806, length A053797.
For runs of prime-powers:
- length A174965
- min A373673
- max A373674 (this sequence)
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A068780, A068781, A073051, A373400.

Programs

  • Mathematica
    pripow[n_]:=n==1||PrimePowerQ[n];
Max/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

A375714 Positions of non-successions of consecutive non-perfect-powers. Numbers k such that the k-th non-perfect-power is at least two fewer than the next.

Original entry on oeis.org

2, 5, 11, 19, 20, 24, 27, 39, 53, 69, 87, 107, 110, 112, 127, 151, 177, 196, 204, 221, 233, 265, 299, 317, 334, 372, 412, 454, 481, 497, 543, 591, 641, 693, 747, 803, 861, 921, 959, 982, 1046, 1112, 1180, 1250, 1284, 1321, 1395, 1471, 1549, 1629, 1675, 1710
Offset: 1

Views

Author

Gus Wiseman, Sep 10 2024

Keywords

Comments

Non-perfect-powers (A007916) are numbers with no proper integer roots.

Examples

			The initial non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, which increase by more than one after term 2, term 5, term 11, etc.

Crossrefs

First differences are A375702.
Positions of terms > 1 in A375706 (differences of A007916).
The complement for non-prime-powers is A375713, differences A373672.
The complement is A375740.
The version for non-prime-powers is A375928, differences A110969.
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
Cf. A000040, A001223, A046933, A053289, A073783, A246655, A251092, A375734.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
ce=Select[Range[100],radQ];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]
  • Python
    from itertools import count, islice
from sympy import perfect_power
def A375714_gen(): # generator of terms
    a, b = -1, 0
    for n in count(1):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&(c^1):
            yield a
        b = c
A375714_list = list(islice(A375714_gen(),52)) # Chai Wah Wu, Sep 11 2024

Formula

A007916(a(n)+1) - A007916(a(n)) > 1.

A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Gus Wiseman, Sep 10 2024

Keywords

Comments

Positions in A007916 of numbers k such that k+1 is also a member.
Positions of 1's in A375706 (first differences of A007916).
Non-perfect-powers (A007916) are numbers with no proper integer roots.

Examples

			The non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, ... which increase by one after positions 1, 3, 4, 6, ...

Crossrefs

The version for non-prime-powers is A375713, differences A373672.
The complement is A375714, differences A375702.
The version for prime-powers is A375734, differences A373671.
The complement for non-prime-powers is A375928, differences A110969.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A001597 lists perfect-powers, differences A053289.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprime numbers, differences A065310.
Non-perfect-powers:
- terms: A007916
- differences: A375706
- runs: sum A375705, A375703, A375704, A375702
- anti-runs: A375737, A375738, A375739, A375736.
Non-prime-powers (exclusive):
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- anti-runs: A373679, A373575, A255346, A373672
Cf. A006549, A046933, A093555, A174965, A246655, A251092.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]
  • Python
    from itertools import count, islice
from sympy import perfect_power
def A375740_gen(): # generator of terms
    a, b = -1, 0
    for n in count(2):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&c:
            yield a
    b = c
A375740_list = list(islice(A375740_gen(), 52)) # Chai Wah Wu, Sep 11 2024

A375713 Indices of consecutive non-prime-powers (A361102) differing by 1. Numbers k such that the k-th and (k+1)-th non-prime-powers differ by just one.

Original entry on oeis.org

5, 8, 9, 15, 16, 17, 19, 20, 23, 24, 27, 28, 30, 31, 32, 33, 36, 38, 40, 41, 44, 45, 46, 47, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 67, 68, 71, 72, 74, 75, 76, 77, 78, 79, 81, 82, 85, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99, 100, 103, 104, 105, 106
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2024

Keywords

Examples

			The initial non-prime-powers are 1, 6, 10, 12, 14, 15, 18, 20, 21, which first increase by one after the fifth and eighth terms.

Crossrefs

The inclusive version is a(n) - 1.
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.
Positions of 1's in A375708.
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
    Join@@Position[Differences[Select[Range[100],!PrimePowerQ[#]&]],1]

Formula

A361102(k+1) - A361102(k) = 1.

A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

Original entry on oeis.org

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53
Offset: 1

Views

Author

Gus Wiseman, Jun 14 2024

Keywords

Comments

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
Are there only 9 terms?
From David A. Corneth, Jun 14 2024: (Start)
No. a(10) exists.
Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

Examples

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Links

Crossrefs

For composite runs we have A073051, sorted A373400, firsts of A176246.
For squarefree runs we have firsts of A120992.
For prime-powers runs we have firsts of A174965.
For prime runs we have firsts of A251092 or A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199, firsts of A053797.
The sorted version is A373670.
For antiruns we have firsts of A373672.
For runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
A000961 lists the powers of primes (including 1).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

Programs

  • Mathematica
    q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]

A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2024

Keywords

Comments

Positions of 1's in A073783 (see also A054546, A065310).

Examples

			The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...

Crossrefs

Positions in A002808 of each element of A068780.
The complement is A065890 shifted.
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 nonprime numbers is A375926.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprime numbers, differences A065310.
A046933 counts composite numbers between primes.
Cf. A014689, A054546, A176246, A246655, A251092.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],CompositeQ]],1]
  • Python
    from sympy import primepi
def A375929(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+2+y))-2
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024
  • Python
    # faster for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    pic, prevc = 0, -1
    for i in count(4):
        if not isprime(i):
            if i == prevc + 1:
                yield pic
            pic, prevc = pic+1, i
print(list(islice(agen(), 10000))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = A375926(n) - 1.

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700
Offset: 1

Views

Author

Gus Wiseman, Sep 04 2024

Keywords

Comments

The corresponding prime-powers A246655(a(n)) are given by A006549.
From A006549, it is not known whether this sequence is infinite.

Examples

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.

Crossrefs

For nonprime numbers (A002808) we have A375926, differences A373403.
Positions of 1's in A057820.
First differences are A373671.
For nonsquarefree numbers we have A375709, differences A373409.
For non-prime-powers we have A375713.
For non-perfect-powers we have A375740.
For squarefree numbers we have A375927, differences A373127.
Prime-powers:
- terms: A000961, complement A024619.
- differences: A057820.
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
- anti-runs: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A025528 counts prime-powers up to n.
Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Formula

Numbers k such that A246655(k+1) - A246655(k) = 1.
The inclusive version is a(n) + 1 shifted.

Extensions

a(14) from Amiram Eldar, Sep 24 2024
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