A351072 -id:A351072 - cp's OEIS frontend

A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130

Offset: 1

T. D. Noe, Oct 12 2004

nonn

The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.

For all n, A003415(a(n)) is also a term of the sequence. A351255 gives the nonzero terms as ordered by their position in A276086. - Antti Karttunen, Feb 14 2022

			18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths).
Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1 - charfun).
Cf. A276086, A328116, A351255 (permutation of nonzero terms), A351257, A351259, A351261, A351072 (number of prime(k)-smooth terms > 1).
Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
Union of A359544 and A359545.

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]

\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Åhlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
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));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022

For all n >= 0, A328309(a(n)) = n. - Antti Karttunen, Feb 14 2022

A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 30, 25, 150, 375, 750, 5625, 7, 14, 21, 42, 126, 70, 105, 315, 350, 1575, 3150, 1750, 2625, 49, 98, 882, 490, 735, 4410, 2450, 3675, 11025, 12250, 30625, 61250, 183750, 686, 3430, 5145, 25725, 77175, 385875, 1929375, 3858750, 4802, 72030, 120050, 180075, 33614, 100842, 117649, 705894, 26471025

Offset: 1

Antti Karttunen, Feb 10 2022

Equal to nonzero terms of A099308 when sorted into ascending order. In this order, which is dictated by the primorial base expansion of n (A049345) and mapped to products of prime powers by A276086, all terms of A099308 that are prime(k)-smooth appear before the terms that are not prime(k)-smooth.
Number of terms whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n-1)-smooth] is given by A351071(n): 1, 4, 8, 44, 216, 1474, 11130, ...
For all n > 1, A003415(a(n)) is also a term of the sequence.
Note that only 451 of the first 105367 terms (all 19-smooth terms) are such that there occurs a 19-smooth number (A080682) larger than 1 on the path before 1 is encountered, when starting from x = a(n) and iterating with map x -> A003415(x).

Antti Karttunen, Table of n, a(n) for n = 1..12878 (all 17-smooth terms of this sequence)
Antti Karttunen, 105368 initial terms, without indices (all 19-smooth terms of this sequence, and also A276086(9699690) = 23, the first 23-smooth term)
Index entries for sequences related to primorial base

Cf. A003415, A049345, A099307, A099308, A276086, A328116, A351071, A351072 (number of prime(n)-smooth terms).

Cf. A351256 [= A051903(a(n))], A351257 [= A099307(a(n))], A351258, A351259 [= A351078(a(n))], A351261 [= A351079(a(n))].

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));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(u, ", ")));

a(n) = A276086(A328116(n)).

A351071 Number of integers x in range A002110(n) .. A002110(1+n)-1 such that the k-th arithmetic derivative of A276086(x) is zero for some k, where A002110(n) is the n-th primorial.

Original entry on oeis.org

1, 4, 8, 44, 216, 1474, 11130, 92489

Offset: 0

Antti Karttunen, Feb 02 2022

a(n) is the number of terms of A328116 in range A002110(n) .. A002110(1+n)-1.
a(n) is the number of terms in A351255 (and in A099308) whose largest prime factor (A006530) is A000040(1+n).
Ratio a(n) / A061720(n) develops as:
  0:     1 / 1       = 1.0
  1:     4 / 4       = 1.0
  2:     8 / 24      = 0.333...
  3:    44 / 180     = 0.244...
  4:   216 / 2100    = 0.1029...
  5:  1474 / 27720   = 0.05317...
  6: 11130 / 480480  = 0.02316...
  7: 92489 / 9189180 = 0.01006...
Computing term a(8) would need processing over 213393180 integers whose greatest prime factor is 23, from single A351255(105368) = 23 at start to product (2^1)*(3^2)*(5*4)*(7^6)*(11^10)*(13^12)*(17^16)*(19^18)*(23^22) at the end of the batch [number whose size in binary is 346 bits], and would required factoring integers of comparable size and more (see A351261), that might not all be easily factorable.

			There are eight terms [6, 7, 9, 12, 15, 20, 21, 28] that are >= A002110(2) and < A002110(3) in A328116 for which the corresponding terms [5, 10, 30, 25, 150, 375, 750, 5625] in A276086 (and A351255) are all in A099308, therefore a(2) = 8.

Cf. A002110, A003415, A061720, A099308, A328306, A328307, A328308, A328116, A351067, A351069, A351255, A351261, A351072 (partial sums).

Cf. also A327969.

\\ Memoization would work quite badly here. (See comments in A351255. In practice sequence A328306 was computed first, up to its term a(9699690). Same data is available in A328116.)
A002110(n) = prod(i=1,n,prime(i));
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));
A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328306(n) = A328308(A276086(n));
A351071(n) = sum(k=A002110(n),A002110(1+n)-1,A328306(k));

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} A328306(k).

a(n) = A328307(A002110(1+n)) - A328307(A002110(n)).

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A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

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A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

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A351071 Number of integers x in range A002110(n) .. A002110(1+n)-1 such that the k-th arithmetic derivative of A276086(x) is zero for some k, where A002110(n) is the n-th primorial.

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