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Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.
2

%I #20 Feb 02 2024 16:10:19

%S 1,2,8,32,256,2560,30720,32768,4194304,20971520,58720256,234881024,

%T 536870912,1342177280

%N Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.

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

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

%e base

%e 1 0, 1, 1, -1

%e 2 = 2^1, 1, 10, 1, 0

%e 8 = 2^3, 3, 110, 1, 2

%e 32 = 2^5, 5, 1010, 1, 4

%e 256 = 2^8, 8, 11220, 2, 6

%e 2560 = 2^9 * 5^1, 9, 111120, 2, 7

%e 30720 = 2^11 * 3^1 * 5^1, 11, 1032000, 3, 8

%e 32768 = 2^15, 15, 1120110, 2, 13

%e 4194304 = 2^22, 22, 83876020, 8, 14

%e 20971520 = 2^22 * 5^1, 22, 231462310, 6, 16

%e 58720256 = 2^23 * 7^1, 23, 610501410, 6, 17

%e 234881024 = 2^25 * 7^1, 25, 1141710210, 7, 18

%e 536870912 = 2^29, 29, 296AA71010, 10, 19

%e 1342177280 = 2^28 * 5^1, 28, 6071712310, 7, 21.

%e On the penultimate row, letter "A" in the primorial base expansion stands for ten (10 in decimal), as 2^29 = 0*prime(0)# + 1*prime(1)# + 0*prime(2)# + 1*prime(3)# + 7*prime(4)# + 10*prime(5)# + 10*prime(6)# + 6*prime(7)# + 9*prime(8)# + 2*prime(9)#, where prime(n)# = A002110(n).

%o (PARI)

%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 A350074(n) = (A328114(n) - A051903(n));

%o m=A350074(1); print1(1,", "); for(n=2,oo,x=A350074(n); if(x<m,print1(n,", "); m=x));

%Y Positions of records for -A350074(n).

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

%Y Cf. also A369646, A369647.

%Y After the initial 1, subsequence of A351038, after the two initial terms, subsequence of A350075.

%K nonn,more

%O 1,2

%A _Antti Karttunen_, Feb 01 2024