cp's OEIS Frontend

A369646 Numbers k such that the difference A051903(k) - A328114(A003415(k)) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, A328114 is the maximal digit in the primorial base expansion of n, and A003415 is the arithmetic derivative.

%I A369646 #12 Feb 02 2024 16:11:47
%S A369646 1,8,16,832,1024,95232,131072,2097152,1006632960,1090519040
%N A369646 Numbers k such that the difference A051903(k) - A328114(A003415(k)) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, A328114 is the maximal digit in the primorial base expansion of n, and A003415 is the arithmetic derivative.
%H A369646 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A369646            k   factorization    max.exp.  k' in primorial  max digit  diff
%e A369646                                                   base
%e A369646            1                        0,              0,        0,       0
%e A369646            8 = 2^3,                 3,            200,        2,       1
%e A369646           16 = 2^4,                 4,           1010,        1,       3
%e A369646          832 = 2^6 * 13^1,          6,         111120,        2,       4
%e A369646         1024 = 2^10,               10,         222310,        3,       7
%e A369646        95232 = 2^10 * 3^1 * 31^1,  10,       10021220,        2,       8
%e A369646       131072 = 2^17,               17,       23132010,        3,      14
%e A369646      2097152 = 2^21,               21,      252354100,        5,      16
%e A369646   1006632960 = 2^26 * 3^1 * 5^1,   26,    23194866010,        9,      17
%e A369646   1090519040 = 2^24 * 5^1 * 13^1,  24,    22053155300,        5,      19.
%e A369646 Here k' stands for the arithmetic derivative of k, A003415(k). Primorial base expansion is obtained with A049345.
%o A369646 (PARI)
%o A369646 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369646 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A369646 A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
%o A369646 A351097(n) = (A328114(A003415(n))-A051903(n));
%o A369646 m=A351097(1); print1(1,", "); for(n=2,oo,x=A351097(n); if(x<m,print1(n,", "); m=x));
%Y A369646 Positions of records for -A351097(n).
%Y A369646 After the initial 1, a subsequence of A351098.
%Y A369646 Cf. A002110, A003415, A049345, A051903, A328114.
%Y A369646 Cf. also A369645, A369647.
%K A369646 nonn,more
%O A369646 1,2
%A A369646 _Antti Karttunen_, Feb 02 2024