A375708 -id:A375708 - cp's OEIS frontend

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

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).

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

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

Gus Wiseman, Sep 02 2024

nonn

			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.

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.

Join@@Position[Differences[Select[Range[100],!PrimePowerQ[#]&]],1]

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

A376308 Run-compression of the sequence of first differences of prime-powers.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 4, 2, 4, 6, 2, 8, 5, 1, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 4, 2, 4, 6, 2, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Sep 20 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
The run-compression is A376308 (this sequence).

For primes instead of prime-powers we have A037201, halved A373947.
For squarefree numbers instead of prime-powers we have A376305.
For run-lengths instead of compression we have A376309.
For run-sums instead of compression we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A024619 and A361102 list non-prime-powers, differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A034296, A053289, A076259, A110969, A120430, A124767, A174965, A374251.

First/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A375705 Sum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

5, 18, 75, 164, 26, 118, 102, 510, 791, 1160, 1629, 2210, 369, 253, 2040, 3756, 4745, 3914, 1764, 3978, 2994, 8720, 10421, 6003, 5984, 14459, 16820, 19425, 13446, 8328, 25415, 28824, 32525, 36530, 40851, 45500, 50489, 55830, 37259, 23276, 67616, 74085, 80954

Offset: 1

Gus Wiseman, Aug 29 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 begins with A375703(n), ends with A375704(n), adds up to a(n), and has length A375702(n).

For nonprime numbers we have A054265, anti-runs A373404.
For nonsquarefree numbers we have A373414, anti-runs A373412.
For squarefree numbers we have A373413, anti-runs A373411.
For prime-powers we have A373675, anti-runs A373576.
For non-prime-powers we have A373678, anti-runs A373679.
The anti-run version is A375737, sums of A375736.
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
For runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705 (this)
Cf. A006093, A006549, A251092, A255346, A371201, A373197, A375708.

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

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

Gus Wiseman, Sep 04 2024

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

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

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.

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

Numbers k such that A246655(k+1) - A246655(k) = 1.

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

a(14) from Amiram Eldar, Sep 24 2024

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

Gus Wiseman, Sep 11 2024

nonn

			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.

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.

Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]

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

cp's OEIS Frontend

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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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.

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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.

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A376308 Run-compression of the sequence of first differences of prime-powers.

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A375705 Sum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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

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