A068780 -id:A068780 - cp's OEIS frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (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 maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
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.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A373403 Length of the n-th maximal antirun of composite numbers differing by more than one.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

This antirun ranges from A005381 (with 4 prepended) to A068780, with sum A373404.

An antirun of a sequence (in this case A002808) is an interval of positions such that consecutive terms differ by more than one.

			Row-lengths of:
   4   6   8
   9
  10  12  14
  15
  16  18  20
  21
  22  24
  25
  26
  27
  28  30  32
  33
  34
  35
  36  38
  39
  40  42  44

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

Functional neighbors: A005381, A027833 (partial sums A029707), A068780, A176246 (rest of A046933, firsts A073051), A373127, A373404, A373409.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A038664, A053767, A067970, A174965, A196274, A373400, A373573.

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

a(2n) = 1.

a(2n - 1) = A196274(n) for n > 1.

A005381 Numbers k such that k and k-1 are composite.

Original entry on oeis.org

9, 10, 15, 16, 21, 22, 25, 26, 27, 28, 33, 34, 35, 36, 39, 40, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 63, 64, 65, 66, 69, 70, 75, 76, 77, 78, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

N. J. A. Sloane

Position where the composites first outnumber the primes by n, among the first natural numbers. - Lekraj Beedassy, Jul 11 2006

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Stauduhar, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
B. W. J. Irwin, Recursive Modular Conjecture for pi(n).

Equals A068780 + 1. Cf. A007921.

Cf. A093515 (complement, apart from 1 which is in neither sequence), A323162 (characteristic function).

isA005381 := proc(n)
    not isprime(n) and not isprime(n-1) ;
end proc:
A005381 := proc(n)
    local a;
    option remember;
    if n = 1 then
        9;
    else
        for a from procname(n-1)+1 do
            if isA005381(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 14 2015
# second Maple program:
q:= n-> ormap(isprime, [n, n-1]):
remove(q, [$2..130])[];  # Alois P. Heinz, Dec 26 2021

Select[Range[2, 200], ! PrimeQ[# - 1] && ! PrimeQ[#] &]

is(n)=!isprime(n)&&!isprime(n-1) \\ M. F. Hasler, Jan 07 2019

from sympy import isprime
def ok(n): return n > 3 and not isprime(n) and not isprime(n-1)
print([k for k in range(122) if ok(k)]) # Michael S. Branicky, Dec 26 2021

Conjecture: pi(n)=Sum_{k=1..n} k mod a(m) mod a(m-1) ... mod a(1) mod 2, for all values 1Benedict W. J. Irwin, May 04 2016
As a check, take n=9, m=2, a(m)=10. Then we must take the numbers 1 through 9 and reduce them mod 10 then mod 9 then mod 2. The results are 1,0,1,0,1,0,1,0,0, whose sum is 4 = pi(9), as predicted. - N. J. A. Sloane, May 05 2016
For an attempt at a proof for the conjecture above, see the link. If it is true, then for n>2, isprime(n)=(n mod x) mod 2, where x is the largest a(n)<=n. - Benedict W. J. Irwin, May 06 2016

A007921 Numbers that are not the difference of two primes.

Original entry on oeis.org

7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, 103, 109, 113, 115, 117, 119, 121, 123, 127, 131, 133, 139, 141, 143, 145, 151, 153, 157, 159, 163, 167, 169, 173, 175, 181, 183, 185, 187, 193

Offset: 1

R. Muller

Conjecturally, odd numbers k such that k+2 is composite.
Is this the same as A068780(2n-1) - 1? - J. Stauduhar, Aug 23 2012
A092953(a(n)) = 0. - Reinhard Zumkeller, Nov 10 2012
It seems that the sequence contains the squares of all primes except for 2 and 3. - Ivan N. Ianakiev, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - Jon E. Schoenfield, May 03 2024]
Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - Bernard Schott, Jan 08 2023
If it exists, the least even term is > 10^12 (see 1st comment in A020483). - Bernard Schott, Jan 09 2023

F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)

T. D. Noe, Table of n, a(n) for n = 1..10000
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
C. Dumitrescu and V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 94.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.
Index entries for primes, gaps between.

Cf. A048859.
Complement of A030173. Cf. A001223.
Cf. also A005408, A010051.
Cf. A123556, A342309.
Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).

a007921 n = a007921_list !! (n-1)
a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
-- Reinhard Zumkeller, Jul 03 2015

filter :=  d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # Bernard Schott, Jan 08 2023

Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* James C. McMahon, May 04 2024 *)

is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; Charles R Greathouse IV, Jun 28 2013

from sympy import isprime
print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # Indranil Ghosh, Jun 15 2017, after Charles R Greathouse IV

A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

Original entry on oeis.org

1, 3, 8, 23, 33, 45, 29, 281, 98, 153, 188, 262, 366, 428, 589, 737, 216, 1182, 3301, 2190, 1878, 1830, 7969, 3076, 3426, 2224, 3792, 8027, 4611, 4521, 3643, 8687, 14861, 12541, 15782, 3384, 34201, 19025, 17005, 44772, 23282, 38589, 14356

Offset: 1

Robert G. Wilson v, Aug 15 2002

nonn

Also, least k such that 2n = A001223(k-1) = prime(k+1) - prime(k), where prime(k) = A001223(n). - Alexander Adamchuk, Jul 30 2006

Also the least number k>0 such that the k-th maximal run of composite numbers has length 2n-1. For example, the 8th such run (24,25,26,27,28) is the first of length 2(3)-1, so a(3) = 8. Also positions of first appearances in A176246 (A046933 without first term). - Gus Wiseman, Jun 12 2024

			a(3) = 8 because 1+0+2-2+2-2+2+2 = 5 and (5+1)/2 = 3.

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

Cf. A000230, A001223.
Position of first appearance of 2n+1 in A176246.
For nonsquarefree runs we have a bisection of A373199.
A000040 lists the primes, first differences A001223.
A002808 lists the composite numbers, differences A073783, sums A053767.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005381, A027833, A038664, A045881, A068780, A174965, A371201, A373403.

NextPrim[n_Integer] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {50}]; s = 0; k = 1; p = 0; q = 2; r = 3; While[k < 10^6, p = q; q = r; r = NextPrim[q]; s = s + p + r - 2q; If[s < 101 && a[[(s + 1)/2]] == 0, a[[(s + 1)/2]] = k]; k++ ]; a

a001223(n) = prime(n+1) - prime(n);
a(n) = {my(k = 1); while(2*n != A001223(k+1), k++); k;} \\ Michel Marcus, Nov 20 2016

a(n) = A038664(n) - 1. - Filip Zaludek, Nov 19 2016

A373673 First element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

1, 7, 11, 13, 16, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

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 A373674.
Consists of all powers of primes k such that k-1 is not a power of primes.

			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

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

For composite antiruns we have A005381, max A068780, length A373403.
For prime antiruns we have A006512, max A001359, length A027833.
For composite runs we have A008864, max A006093, length A176246.
For prime runs we have A025584, max A067774, length A251092 or A175632.
For runs of prime-powers:
- length A174965
- min A373673 (this sequence)
- max A373674
- 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, A007674, A014963, A053806, A054265, A068781, A072284, A073051, A120992, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Min/@Split[Select[Range[100],pripow],#1+1==#2&]//Most

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

Gus Wiseman, Jun 16 2024

nonn

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.

			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

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

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.

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

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

Gus Wiseman, Jun 16 2024

nonn

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.

			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

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

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.

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

A373404 Sum of the n-th maximal antirun of composite numbers differing by more than one.

Original entry on oeis.org

18, 9, 36, 15, 54, 21, 46, 25, 26, 27, 90, 33, 34, 35, 74, 39, 126, 45, 94, 49, 50, 51, 106, 55, 56, 57, 180, 63, 64, 65, 134, 69, 216, 75, 76, 77, 158, 81, 166, 85, 86, 87, 178, 91, 92, 93, 94, 95, 194, 99, 306, 105, 324, 111, 226, 115, 116, 117, 118, 119

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this antirun is given by A373403.

An antirun of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by more than one.

			Row sums of:
   4   6   8
   9
  10  12  14
  15
  16  18  20
  21
  22  24
  25
  26
  27
  28  30  32
  33
  34
  35
  36  38
  39
  40  42  44

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

Partial sums are a subset of A053767 (partial sums of composite numbers).
Functional neighbors: A005381, A054265, A068780, A373403, A373405, A373411, A373412.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A003114, A027833, A038664, A293697, A350842 (strict A350844), A371201.

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

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).

cp's OEIS Frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

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A373403 Length of the n-th maximal antirun of composite numbers differing by more than one.

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A005381 Numbers k such that k and k-1 are composite.

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A007921 Numbers that are not the difference of two primes.

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A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

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A373673 First element of each maximal run of powers of primes (including 1).

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

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