A093555 -id:A093555 - cp's OEIS frontend

A057820 First differences of sequence of consecutive prime powers (A000961).

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

Offset: 1

Labos Elemer, Nov 08 2000

nonn

a(n) = 1 iff A000961(n) = A006549(k) for some k. - Reinhard Zumkeller, Aug 25 2002
Also run lengths of distinct terms in A070198. - Reinhard Zumkeller, Mar 01 2012
Does this sequence contain all positive integers? - Gus Wiseman, Oct 09 2024

			Odd differences arise in pairs in neighborhoods of powers of 2, like {..,2039,2048,2053,..} gives {..,11,5,..}

Michael B. Porter, Table of n, a(n) for n = 1..10000

Cf. A000961, A036616, A001223.
For perfect-powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
Positions of ones are A375734.
Run-compression is A376308.
Run-lengths are A376309.
Sorted positions of first appearances are A376340.
The second (instead of first) differences are A376596, zeros A376597.
Prime-powers:
- terms: A000961 or A246655, complement A024619
- differences: A057820 (this), first appearances A376341
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708 (ones A375713)
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670)
- anti-runs: A373679, A373575, A255346, A373672
Cf. A000015, A014210, A014963, A025475, A025528, A027833, A037201, A046933, A076259, A078147, A093555, A345531, A376310.

a057820_list = zipWith (-) (tail a000961_list) a000961_list
-- Reinhard Zumkeller, Mar 01 2012

A057820 := proc(n)
        A000961(n+1)-A000961(n) ;
end proc: # R. J. Mathar, Sep 23 2016

Map[Length, Split[Table[Apply[LCM, Range[n]], {n, 1, 150}]]] (* Geoffrey Critzer, May 29 2015 *)
Join[{1},Differences[Select[Range[500],PrimePowerQ]]] (* Harvey P. Dale, Apr 21 2022 *)

isA000961(n) = (omega(n) == 1 || n == 1)
n_prev=1;for(n=2,500,if(isA000961(n),print(n-n_prev);n_prev=n)) \\ Michael B. Porter, Oct 30 2009

from sympy import primepi, integer_nthroot
def A057820(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r, k = m, f(m)+1
    while r != k: r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Sep 12 2024

a(n) = A000961(n+1) - A000961(n).

Offset corrected and b-file adjusted by Reinhard Zumkeller, Mar 03 2012

A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

Original entry on oeis.org

5, 4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-prime-powers (exclusive) are listed by A361102.

Warning: For this sequence, 1 is not a prime-power but is a non-prime-power.

			The 6th non-prime-power (exclusive) is 15, and the 7th is 18, so a(6) = 3.

For prime-powers (A000961, A246655) we have A057820, gaps A093555.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A361102, inclusive A024619.
Positions of 1's are A375713.
If 1 is considered a prime power we have A375735.
Runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
A013929 lists the nonsquarefree numbers, differences A078147.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A046933, A061399, A176246, A251092, A375709.

Differences[Select[Range[100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375708(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 09 2024

A110969 Length of the runs of ones in A014963.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 3, 1, 3, 1, 3, 5, 1, 2, 2, 3, 1, 5, 1, 1, 5, 7, 3, 1, 3, 1, 3, 7, 3, 1, 2, 5, 1, 9, 1, 5, 5, 3, 1, 3, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 1, 7, 4, 5, 5, 1, 5, 3, 1, 5, 3, 13, 3, 1, 3, 13, 5, 5, 3, 1, 3, 5, 1, 5, 5, 5, 3, 5, 7, 3, 7

Offset: 1

Franz Vrabec, Sep 27 2005

nonn

Unbounded sequence.
From A373669 we see that 10 first appears at a(28195574) = 10.
Also run-lengths of non-prime-powers (assuming 1 is not a prime-power), where a run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one. Also nonzero differences of consecutive prime-powers minus one. - Gus Wiseman, Jun 18 2024

			a(5)=2 because the fifth run of ones in A014963 is of length 2.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A014963.
Positions of first appearances are A373670, sorted A373669.
For runs of prime-powers:
- length A174965, antiruns A373671
- min A373673, antiruns A120430
- max A373674, antiruns A006549
- sum A373675, antiruns A373576
For runs of non-prime-powers:
- length A110969 (this sequence), antiruns A373672
- min A373676, antiruns A373575
- max A373677, antiruns A255346
- sum A373678, antiruns A373679
A000961 lists all powers of primes. A246655 lists just prime-powers.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Various run-lengths: A053797, A120992, A175632, A176246.
Various antirun-lengths: A027833, A373127, A373403, A373409.
Cf. A001359, A008864, A067774, A251092.

Length /@ SplitBy[Table[Exp[MangoldtLambda[n]], {n, 400}], # != 1 &][[ ;; -1 ;; 2]] (* Michael De Vlieger, Mar 21 2024 *)
DeleteCases[Differences[Select[Range[100],PrimePowerQ]]-1,0] (* Gus Wiseman, Jun 18 2024 *)

\\ b(n) returns boolean of A014963(n) == 1.
b(n)={my(t); !isprime(if(ispower(n, ,&t), t, n))}
seq(n)={my(k=1, i=0, L=List()); while(#Lk, listput(L, i-k)); k = i+1)); Vec(L)} \\ Andrew Howroyd, Jan 02 2020

Terms a(41) and beyond from Andrew Howroyd, Jan 02 2020

A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 4, 1, 0, 0, 5, 1, 0, 0, 0, 7, 2, 1, 1, 1, 1, 8, 1, -1, -2, -3, -4, -5, 9, 1, 0, 1, 3, 6, 10, 15, 11, 2, 1, 1, 0, -3, -9, -19, -34, 13, 2, 0, -1, -2, -2, 1, 10, 29, 63, 16, 3, 1, 1, 2, 4, 6, 5, -5, -34, -97, 17, 1, -2, -3, -4, -6, -10, -16, -21, -16, 18, 115

Offset: 0

Gus Wiseman, Oct 20 2024

Row k of the array is the k-th differences of A000961.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     4     5     7     8     9    11
  k=1:   1     1     1     1     2     1     1     2     2
  k=2:   0     0     0     1    -1     0     1     0     1
  k=3:   0     0     1    -2     1     1    -1     1    -3
  k=4:   0     1    -3     3     0    -2     2    -4     6
  k=5:   1    -4     6    -3    -2     4    -6    10    -8
  k=6:  -5    10    -9     1     6   -10    16   -18     5
  k=7:  15   -19    10     5   -16    26   -34    23     9
  k=8: -34    29    -5   -21    42   -60    57   -14   -42
  k=9:  63   -34   -16    63  -102   117   -71   -28   104
Triangle form:
    1
    2    1
    3    1    0
    4    1    0    0
    5    1    0    0    0
    7    2    1    1    1    1
    8    1   -1   -2   -3   -4   -5
    9    1    0    1    3    6   10   15
   11    2    1    1    0   -3   -9  -19  -34
   13    2    0   -1   -2   -2    1   10   29   63
   16    3    1    1    2    4    6    5   -5  -34  -97

Row k=0 is A000961, exclusive A246655.
Row k=1 is A057820.
Row k=2 is A376596.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, nonsquarefree A377046.
Triangle row-sums are A377052, absolute version A377053.
Column n = 1 is A377054, for primes A007442 or A030016.
First position of 0 in each row is A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.

nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).

A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, 1, 0, 1, -2, 4, -4, 0, 4, 2, -4, -2, 2, -2, 2, 4, -4, -2, -1, 2, 3, -4, 8, -8, 4, 0, -2, -2, 2, 2, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 0, 6, -3, -4, 5, 0, -4, 4, -2, -2

Offset: 1

Gus Wiseman, Oct 02 2024

sign

For the exclusive version, shift left once.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
Positions of zeros are A376597, complement A376598.
Sorted positions of first appearances are A376653, exclusive A376654.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A045542, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2]

from sympy import primepi, integer_nthroot
def A376596(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Original entry on oeis.org

1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 01 2024

sign

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

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros are A376588, complement A376589.
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
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A064113, A069623, A093555, A174965, A182853, A336416, A336417, A361102.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ],2]

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A376562(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)
    return next(i for i in count(1-k) if not perfect_power(r+i)) # Chai Wah Wu, Oct 02 2024

A375735 First differences of non-prime-powers (inclusive).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 04 2024

nonn

Inclusive means 1 is a prime-power but not a non-prime-power.

Non-prime-powers (inclusive) are listed by A024619.

			The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.

For perfect powers (A001597) we have the latter terms of A053289.
For nonprime numbers (A002808) we have the latter terms of A073783.
For squarefree numbers (A005117) we have the latter terms of A076259.
First differences of A024619.
For prime-powers (A246655) we have the latter terms of A057820.
Essentially the same as the exclusive version, A375708.
Positions of 1's are A375713(n) - 1.
For runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, first differences A001223.
A000961 lists prime-powers (inclusive).
A007916 lists non-perfect-powers, first differences A375706.
A013929 lists the nonsquarefree numbers, first differences A078147.
A246655 lists prime-powers (exclusive).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power anti-runs: A373576, min A120430, max A006549, length A373671.
Non-prime-power anti-runs: A373679, min A373575, max A255346, len A373672.
Cf. A046933, A061399, A093555, A176246, A251092, A323054, A375709, A375736.

Differences[Select[Range[2,100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375735(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

Original entry on oeis.org

-2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375735, ones A375713(n) - 1.
Positions of zeros are A376600, complement A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.
A321346/A321378 count integer partitions without prime-powers, factorizations A322452.
For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]

from sympy import primepi, integer_nthroot
def A376599(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A373671 Length of the n-th maximal antirun of prime-powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 26, 27, 1007, 5558, 5734, 31209

Offset: 1

Gus Wiseman, Jun 14 2024

An antirun of a sequence (in this case A000961 without 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of prime-powers begin:
   2
   3
   4
   5   7
   8
   9  11  13  16
  17  19  23  25  27  29  31

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

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671 (this sequence)
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
A000961 lists the powers of primes (including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers (not including 1 A024619).
Cf. A000040, A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

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

Partial sums are A025528(A006549(n)).

A373672 Length of the n-th maximal antirun of non-prime-powers.

Original entry on oeis.org

5, 3, 1, 6, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Jun 14 2024

nonn

An antirun of a sequence (in this case A361102 or A024619 with 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns 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

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

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672 (this sequence), firsts (3,7,2,25,1,4)
- min A373575
- max A255346
A000961 lists all powers of primes. A246655 lists just prime-powers.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

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

Partial sums are A356068(A255346(n)).

cp's OEIS Frontend

A057820 First differences of sequence of consecutive prime powers (A000961).

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A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

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A110969 Length of the runs of ones in A014963.

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A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

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A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

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A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

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A375735 First differences of non-prime-powers (inclusive).

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A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.