A394894 - OEIS

A394894

Number of integers x in range 2^(n-1) .. (2^n)-1 such that the u-th arithmetic derivative of A019565(x) is zero for some u.

1, 2, 3, 6, 13, 18, 36, 72, 123, 206, 423, 768, 1382, 2659, 4920, 9291, 17519, 34127, 64210, 122175, 234927, 449206, 860652, 1651460

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OFFSET

1,2

COMMENTS

Number of squarefree terms of A099308 whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n-1)-smooth].

Number of integers x in range A002110(n-1) .. A002110(n)-1 such that the u-th arithmetic derivative of A276086(x) is zero for some u, and A328114(x) = 1, where A002110(n) is the n-th primorial, and A328114 gives the largest digit in the primorial base.

Question: Are a(5) = 13 and a(11) = 423 the only cases where a(n) > 2*a(n-1)?

Ratio of a(n)/2^(n-1) shrinks like this:

1: 1/1 = 1.00

2: 2/2 = 1.00

3: 3/4 = 0.750

4: 6/8 = 0.750

5: 13/16 = 0.813

6: 18/32 = 0.563

7: 36/64 = 0.563

8: 72/128 = 0.563

9: 123/256 = 0.480

10: 206/512 = 0.402

11: 423/1024 = 0.413

12: 768/2048 = 0.375

13: 1382/4096 = 0.337

14: 2659/8192 = 0.325

15: 4920/16384 = 0.300

16: 9291/32768 = 0.284

17: 17519/65536 = 0.267

18: 34127/131072 = 0.260

19: 64210/262144 = 0.245

20: 122175/524288 = 0.233

21: 234927/1048576 = 0.224

22: 449206/2097152 = 0.214

23: 860652/4194304 = 0.205

24: 1651460/8388608 = 0.197

LINKS

Table of n, a(n) for n=1..24.

FORMULA

a(n) = A394890(n,1).

EXAMPLE

Of the four squarefree numbers in range A019565(2^(3-1)..(2^3)-1), i.e., A019565(4..7): 5, 10, 15, 30, only one (15) is in A099309, while the three others are in A099308, therefore a(3) = 3.

PROG

(PARI)

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));

is_A099308(n) = { while(n>1, n = A003415checked(n)); (n); };

A019565(n) = { my(m=1, p=2); while(n>0, if(n%2, m *= p); n >>= 1; p = nextprime(1+p)); (m); };

A394894(n) = sum(i=2^(n-1), (2^n)-1, is_A099308(A019565(i)));

CROSSREFS

The leftmost column of irregular triangle A394890.

Cf. A002110, A003415, A006530, A019565, A099308, A276086, A328114.

Sequence in context: A175281 A364606 A244790 * A111503 A239326 A075530

Adjacent sequences: A394891 A394892 A394893 * A394895 A394896 A394898

KEYWORD

nonn,more

AUTHOR

Antti Karttunen, Apr 07 2026

STATUS

approved