A053289 -id:A053289 - cp's OEIS frontend

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least 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.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 8, 1, 4, 3, 12, 14, 16, 18, 20, 3, 2, 15, 24, 26, 19, 8, 17, 12, 32, 34, 18, 17, 38, 40, 42, 27, 16, 46, 48, 50, 52, 54, 56, 58, 60, 38, 23, 64, 66, 68, 70, 34, 37, 74, 76, 78, 80, 46, 35, 84, 86, 88, 22, 67, 70, 9, 11, 94, 96, 98, 100, 102, 39, 64

Offset: 1

Gus Wiseman, Aug 27 2024

nonn

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

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n has length a(n), first A375703, last A375704, sum A375705.

For nonsquarefree numbers we have A053797, anti-runs A373409.
For squarefree numbers we have A120992, anti-runs A373127.
For nonprime numbers we have A176246, anti-runs A373403.
For prime-powers we have A373675, anti-runs A373576.
For non-prime-powers we have A373678, anti-runs A373679.
The anti-run version is A375736, sum A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 (this).
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
Cf. A007674, A045542, A061399, A216765, A251092, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Select[Range[100],radQ],#1+1==#2&]//Most

For n > 2 we have a(n) = A053289(n+1) - 1.

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

Gus Wiseman, Sep 10 2024

nonn

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

			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.

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.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
ce=Select[Range[100],radQ];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

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

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

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

Original entry on oeis.org

1, 4, 9, 12, 18, 24, 34, 47, 60, 79, 117, 178, 198, 206, 215, 244, 311, 402, 465, 614, 782, 1078, 1109, 1234, 1890, 1939, 1961, 2256, 2290, 3149, 3377, 3460, 3502, 3722, 3871, 4604, 4694, 6634, 8073, 8131, 8793, 12370, 12661, 14482, 14990, 15912, 17140, 19166

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    34: {1,7}
    47: {15}
    60: {1,1,2,3}
    79: {22}
   117: {2,2,6}
   178: {1,24}
   198: {1,2,2,5}
   206: {1,27}
   215: {3,14}
   244: {1,1,18}

For compression instead of sorted firsts we have A376308.
For run-lengths instead of sorted firsts we have A376309.
For run-sums instead of sorted firsts we have A376310.
The version for squarefree numbers is the unsorted version of A376311.
The unsorted version is A376341.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A078147, A110969, A120430, A174965, A373948, A375706.

q=Differences[Select[Range[100],PrimePowerQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

Original entry on oeis.org

1, 3, 1, 5, 3, 3, 4, 2, 6, 1, 9, 2, 4, 2, 10, 2, 3, 7, 2, 6, 2, 8, 8, 4, 2, 4, 2, 4, 8, 7, 9, 2, 10, 2, 6, 6, 4, 2, 10, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 13, 7, 6, 2, 6, 4, 2, 6, 18, 4, 2, 4, 14, 6, 6, 6, 4, 6, 2, 12, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Gus Wiseman, Nov 07 2024

nonn

What is the union of this sequence? In particular, does it contain 17?

First differences of A345531.
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.
A080101 counts prime-powers between primes (exclusive).
A246655 lists the prime-powers, differences A057820 without first term.
A361102 lists the non-powers of primes, differences A375708.
A366833 counts prime-powers between primes, see A053607, A304521, A377057 (positive), A377286 (zero), A377287 (one), A377288 (two).
A377432 counts perfect-powers between primes, see A377434 (one), A377436 (zero), A377466 (multiple).
Cf. A000040, A008864, A031218, A377281, A377282, A377286, A377289, A377468.

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

from sympy import factorint, prime, nextprime
def A377703(n): return -next(filter(lambda m:len(factorint(m))<=1, count((p:=prime(n))+1)))+next(filter(lambda m:len(factorint(m))<=1, count(nextprime(p)+1))) # Chai Wah Wu, Nov 14 2024

A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

2, 5, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729

Offset: 1

Gus Wiseman, Aug 28 2024

nonn

Non-perfect-powers A007916 are numbers without a proper integer root.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n has length A375702, first a(n), last A375704, sum A375705.

For prime numbers we have A045344.
For nonsquarefree numbers we have A053806, anti-runs A373410.
For nonprime numbers we have A055670, anti-runs A005381.
For squarefree numbers we have A072284, anti-runs A373408.
The anti-run version is A216765 (same as A375703 with 2 exceptions).
For non-prime-powers we have A373673, anti-runs A120430.
For prime-powers we have A373676, anti-runs A373575.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1.
- first: A375703 (this)
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
Cf. A007674, A045542, A375708, A375713, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&!radQ[#-1]&]

Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.

A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

3, 7, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727, 1763

Offset: 1

Gus Wiseman, Aug 29 2024

nonn

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

Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).

For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.
For prime numbers we have A045344.
Inserting 8 after 7 gives A045542.
For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.
For squarefree numbers we have A373415, anti-runs A007674.
For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).
Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).
The anti-run version is A375739.
A001597 lists perfect-powers, differences A053289.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (this) (same as A045542 with 8 removed)
- sum: A375705
Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&!radQ[#+1]&]

For n > 2 we have a(n) = A045542(n+1).

A375736 Length of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

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

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose lengths are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For squarefree numbers we have A373127, runs A120992.
For nonprime numbers we have A373403, runs A176246.
For nonsquarefree numbers we have A373409, runs A053797.
For prime-powers we have A373576, runs A373675.
For non-prime-powers (exclusive) we have A373672, runs A110969.
For runs instead of anti-runs we have A375702.
For anti-runs of non-perfect-powers:
- length: A375736 (this)
- first: A375738
- last: A375739
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A216765, A251092, A373679, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

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

Gus Wiseman, Sep 10 2024

nonn

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.

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

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.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]

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

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

Gus Wiseman, Nov 21 2024

nonn

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

			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==

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.

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&]

cp's OEIS Frontend

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

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A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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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.

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A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

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A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

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A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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A375736 Length of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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