A369645 - OEIS

%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