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

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

Author

Antti Karttunen, Feb 10 2022

Keywords

• nonn
• look

Comments

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).

Links

• 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

Crossrefs

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))].

Programs

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));
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, ", ")));

Formula

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