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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.
2

%I #12 Feb 02 2024 16:11:47

%S 1,8,16,832,1024,95232,131072,2097152,1006632960,1090519040

%N 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 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%e k factorization max.exp. k' in primorial max digit diff

%e base

%e 1 0, 0, 0, 0

%e 8 = 2^3, 3, 200, 2, 1

%e 16 = 2^4, 4, 1010, 1, 3

%e 832 = 2^6 * 13^1, 6, 111120, 2, 4

%e 1024 = 2^10, 10, 222310, 3, 7

%e 95232 = 2^10 * 3^1 * 31^1, 10, 10021220, 2, 8

%e 131072 = 2^17, 17, 23132010, 3, 14

%e 2097152 = 2^21, 21, 252354100, 5, 16

%e 1006632960 = 2^26 * 3^1 * 5^1, 26, 23194866010, 9, 17

%e 1090519040 = 2^24 * 5^1 * 13^1, 24, 22053155300, 5, 19.

%e Here k' stands for the arithmetic derivative of k, A003415(k). Primorial base expansion is obtained with A049345.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));

%o A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };

%o A351097(n) = (A328114(A003415(n))-A051903(n));

%o m=A351097(1); print1(1,", "); for(n=2,oo,x=A351097(n); if(x<m,print1(n,", "); m=x));

%Y Positions of records for -A351097(n).

%Y After the initial 1, a subsequence of A351098.

%Y Cf. A002110, A003415, A049345, A051903, A328114.

%Y Cf. also A369645, A369647.

%K nonn,more

%O 1,2

%A _Antti Karttunen_, Feb 02 2024