A046933 -id:A046933 - cp's OEIS frontend

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 20, 24, 27, 29, 31, 33, 37, 38, 42, 45, 46, 50, 52, 56, 61, 62, 64, 67, 68, 71, 78, 81, 84, 86, 92, 93, 96, 100, 103, 105, 109, 110, 117, 118, 121, 122, 130, 139, 141, 142, 145, 149, 150, 154, 158, 162, 166, 167, 170, 172, 174, 180

Offset: 1

Labos Elemer, May 24 2002

nonn

Also the number of squarefree numbers <= prime(n). - Gus Wiseman, Dec 08 2024

			a(25)=61 because A005117(61) = prime(25) = 97.
From _Gus Wiseman_, Dec 08 2024: (Start)
The squarefree numbers up to prime(n) begin:
n = 1  2  3  4   5   6   7   8   9  10
    ----------------------------------
    2  3  5  7  11  13  17  19  23  29
    1  2  3  6  10  11  15  17  22  26
       1  2  5   7  10  14  15  21  23
          1  3   6   7  13  14  19  22
             2   5   6  11  13  17  21
             1   3   5  10  11  15  19
                 2   3   7  10  14  17
                 1   2   6   7  13  15
                     1   5   6  11  14
                         3   5  10  13
                         2   3   7  11
                         1   2   6  10
                             1   5   7
                                 3   6
                                 2   5
                                 1   3
                                     2
                                     1
The column-lengths are a(n).
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A013928.
The strict version is A112929.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A070321 gives the greatest squarefree number up to n.
Other families: A014689, A027883, A378615, A065890.
Squarefree numbers between primes: A061398, A068360, A373197, A373198, A377430, A112925, A112926.
Nonsquarefree numbers: A057627, A378086, A061399, A068361, A120327, A377783, A378032, A378033.
Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531, A008864.

Position[Select[Range[300], SquareFreeQ], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Aug 17 2023 *)

lista(nn)=sqfs = select(n->issquarefree(n), vector(nn, i, i)); for (i = 1, #sqfs, if (isprime(sqfs[i]), print1(i, ", "));); \\ Michel Marcus, Sep 11 2013

a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ Charles R Greathouse IV, Sep 13 2013

a(n,p=prime(n))=my(s); forfactored(k=1, sqrtint(p), s+=p\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Nov 27 2017

first(n)=my(v=vector(n),pr,k); forsquarefree(m=1,n*logint(n,2)+3, k++; if(m[2][,2]==[1]~, v[pr++]=k; if(pr==n, return(v)))) \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import prime, mobius
def A071403(n): return (p:=prime(n))+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # Chai Wah Wu, Jul 20 2024

A005117(a(n)) = A000040(n) = prime(n).
a(n) ~ (6/Pi^2) * n log n. - Charles R Greathouse IV, Nov 27 2017
a(n) = A013928(A008864(n)). - Ridouane Oudra, Oct 15 2019
From Gus Wiseman, Dec 08 2024: (Start)
a(n) = A112929(n) + 1.
a(n+1) - a(n) = A373198(n) = A061398(n) - 1.
(End)

A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

2, 6, 15, 18, 22, 25, 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, 329, 342, 357, 367, 378, 393, 400

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

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

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains two perfect-powers (8,9), so 4 is not in the sequence.
Primes 5 and 6 are 11 and 13, and the interval (12) contains no perfect-powers, so 5 is not in the sequence.
Primes 6 and 7 are 13 and 17, and the interval (14,15,16) contains just one perfect-power (16), so 6 is in the sequence.

For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A377467.
For prime-powers we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360.
These are the positions of 1 in A377432.
For no perfect-powers we have A377436.
For more than one perfect-power we have A377466.
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.
A031218 gives the greatest prime-power <= n.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A345531 gives the least prime-power > prime(n), difference A377281.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect-power > n.
Cf. A006549, A023055, A045542, A052410, A069623, A080101, A216765, A224363, A246655, A336416, A375740, A376560, A376561, A377057.

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

A373675 Sums of maximal runs of powers of primes A000961.

Original entry on oeis.org

15, 24, 11, 13, 33, 19, 23, 25, 27, 29, 63, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 255, 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 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.

A000040 lists the primes, differences A001223.
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).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A029707, A038664, A046933, A054265, A065855, A071148, A293697, A371201.

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

A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

3, 4, 9, 10, 13, 14, 15, 22, 26, 33, 39, 48, 59, 60, 65, 85, 88, 89, 93, 104, 113, 116, 122, 142, 143, 147, 148, 155, 181, 188, 198, 201, 209, 212, 213, 224, 226, 234, 235, 244, 254, 264, 265, 268, 287, 288, 313, 320, 328, 332, 333, 341, 343, 353, 361, 366

Offset: 1

Gus Wiseman, Oct 29 2024

nonn

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains only squarefree 10, so 4 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

For composite instead of squarefree we have A029707.
These are the positions of 1 in A061398, or 2 in A373198.
For no squarefree numbers we have A068360.
For prime-power instead of squarefree we have A377287.
For at least one squarefree number we have A377431.
For perfect-power instead of squarefree we have A377434.
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composites, complement A008578.
A005117 lists the squarefree numbers, complement A013929.
A377038 gives k-differences of squarefree numbers.
Cf. A007674, A013603, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

R:= NULL: count:= 0: q:= 2:
for k from 1 while count < 100 do
  p:= q; q:= nextprime(q);
  if nops(select(numtheory:-issqrfree,[$p+1 .. q-1]))=1 then
    R:= R,k; count:= count+1;
 fi
od:
R; # Robert Israel, Nov 29 2024

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]==1&]

is(n,p=prime(n))=my(q=nextprime(p+1),s); for(k=p+1,q-1, if(issquarefree(k) && s++>1, return(0))); s==1 \\ Charles R Greathouse IV, Nov 29 2024

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

A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

Original entry on oeis.org

2, 3, 11, 17, 11, 42, 17, 62, 49, 59, 133, 114, 83, 89, 98, 223, 59, 254, 206, 71, 302, 161, 341, 462, 97, 203, 314, 107, 330, 824, 386, 398, 275, 856, 149, 460, 635, 494, 337, 702, 179, 1294, 191, 582, 197, 1635, 1950, 449, 227, 690, 943, 239, 983, 1013, 1036

Offset: 1

Gus Wiseman, May 29 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A373198:
   2
   3
   5   6
   7  10
  11
  13  14  15
  17
  19  21  22
  23  26
  29  30
  31  33  34  35
  37  38  39
  41  42
  43  46
  47  51
  53  55  57  58

Counting all numbers (not just squarefree) gives A371201.
For the sectioning of A005117 (squarefree between prime):
- sum is A373197 (this sequence)
- length is A373198 = A061398 - 1
- min is A000040
- max is A112925, opposite A112926
For squarefree numbers between powers of two:
- sum is A373123
- length is A077643, partial sums A143658
- min is A372683, delta A373125, indices A372540, firsts of A372475
- max is A372889, delta A373126
For primes between powers of two:
- sum is A293697 (except initial terms)
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, delta A013603
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A046933, A049093, A049094, A076259, A077641.

Table[Total[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,15}]

A373678 Sums of maximal runs of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 29, 18, 63, 24, 26, 28, 30, 138, 117, 42, 135, 48, 153, 280, 60, 125, 131, 207, 72, 380, 80, 82, 430, 651, 297, 102, 315, 108, 333, 819, 369, 126, 259, 670, 138, 1296, 150, 770, 800, 495, 168, 513, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675

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 A361102) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of non-powers of primes 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.

A000040 lists the primes, differences A001223.
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).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A038664, A046933, A054265, A065855, A071148, A293697, A371201, A373400.

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

A375707 First differences minus 1 of nonsquarefree numbers.

Original entry on oeis.org

3, 0, 2, 3, 1, 1, 3, 0, 1, 0, 3, 3, 3, 3, 0, 2, 0, 0, 1, 1, 1, 3, 2, 0, 3, 3, 2, 0, 3, 0, 2, 3, 1, 1, 3, 1, 0, 0, 3, 3, 3, 3, 0, 2, 0, 2, 0, 0, 1, 3, 2, 0, 3, 3, 2, 0, 1, 1, 0, 2, 3, 1, 1, 3, 0, 1, 0, 2, 0, 3, 3, 3, 0, 2, 3, 1, 1, 3, 2, 0, 3, 3, 3, 3, 0, 2, 3

Offset: 1

Gus Wiseman, Sep 16 2024

Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).
Delete all 0's to get A120992.
The image is {0,1,2,3}.
Add 1 to all terms for A078147.

			The runs of squarefree numbers begin:
  (5,6,7)
  ()
  (10,11)
  (13,14,15)
  (17)
  (19)
  (21,22,23)
  ()
  (26)
  ()
  (29,30,31)
  (33,34,35)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.
For runs of squarefree numbers:
- length: A120992, anti A373127
- min: A072284, anti A373408
- max: A373415, anti A007674
- sum: A373413, anti A373411
For runs of nonsquarefree numbers:
- length: A053797, anti A373409
- min: A053806, anti A373410
- max: A376164, anti A068781
- sum: A373414, anti A373412
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A046933 counts composite numbers between consecutive primes.
A073784 counts primes between consecutive composite numbers.
A093555 counts non-prime-powers between consecutive prime-powers.
Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.

Differences[Select[Range[100],!SquareFreeQ[#]&]]-1

lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ Amiram Eldar, Sep 17 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - Amiram Eldar, Sep 17 2024

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

cp's OEIS Frontend

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

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A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

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A373675 Sums of maximal runs of powers of primes A000961.

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Mathematica

A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.

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Maple

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PARI

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

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Mathematica

PARI

Formula

A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

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Mathematica

A373678 Sums of maximal runs of non-prime-powers.

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