A110969 -id:A110969 - 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[[#]]]&]

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

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]]]

A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

Original entry on oeis.org

2, 8, 10, 15, 17, 18, 24, 28, 30, 37, 38, 43, 45, 47, 48, 52, 56, 59, 65, 67, 69, 73, 80, 85, 92, 93, 94, 100, 106, 108, 111, 115, 122, 125, 128, 133, 134, 137, 138, 141, 143, 145, 148, 153, 158, 165, 166, 171, 178, 183, 184, 192, 196, 198, 203, 205, 207, 210

Offset: 1

Gus Wiseman, Sep 01 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1) (this)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by one after the 2nd and 8th terms.

Positions of 1's in A078147.
For prime-powers (A246655) we have A375734.
First differences are A373409.
For prime numbers we have A375926.
For squarefree instead of nonsquarefree we have A375927.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373410, A373573.

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

Complement of A375710 U A375711 U A375712.

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

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Original entry on oeis.org

6, 1, 18, 8, 4, 2, 10, 52, 678

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The sorted version is A373574.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			The maximal antiruns of nonsquarefree numbers begin:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The a(n)-th rows are:
    49
     4    8
   148  150  152
    64   68   72   75
    28   32   36   40   44
     9   12   16   18   20   24
    81   84   88   90   92   96   98
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite runs we have A073051, firsts of A176246, sorted A373400.
For squarefree runs we have the triple (5,3,1), firsts of A120992.
For prime runs we have the triple (1,3,2), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127, sorted A373200.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373401, firsts of A027833, sorted A373402.
For composite antiruns we have the triple (2,7,1), firsts of A373403.
Positions of first appearances in A373409.
The sorted version is A373574.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A110969, A251092, A294242, A373410, A373412.

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

A376309 Run-lengths of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 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, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			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, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000

For runs of prime-powers increasing by one we have A174965.
For primes instead of prime-powers we have A333254.
For squarefree numbers instead of prime-powers we have A376306.
For compression instead of run-lengths we have A376308.
For run-sums instead of run-lengths 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, 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.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

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

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

More terms from Antti Karttunen, Jan 18 2025

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			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, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
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.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A376309 Run-lengths of the sequence of first differences of prime-powers.

Views

Author