A369645 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.

1, 2, 8, 32, 256, 2560, 30720, 32768, 4194304, 20971520, 58720256, 234881024, 536870912, 1342177280

Offset: 1

Author

Antti Karttunen, Feb 01 2024

Keywords

• nonn
• more

Examples

			           k   factorization   max.exp.  in primorial  max digit  diff
                                             base
           1                       0,            1,       1,      -1
           2 = 2^1,                1,           10,       1,       0
           8 = 2^3,                3,          110,       1,       2
          32 = 2^5,                5,         1010,       1,       4
         256 = 2^8,                8,        11220,       2,       6
        2560 = 2^9 * 5^1,          9,       111120,       2,       7
       30720 = 2^11 * 3^1 * 5^1,  11,      1032000,       3,       8
       32768 = 2^15,              15,      1120110,       2,      13
     4194304 = 2^22,              22,     83876020,       8,      14
    20971520 = 2^22 * 5^1,        22,    231462310,       6,      16
    58720256 = 2^23 * 7^1,        23,    610501410,       6,      17
   234881024 = 2^25 * 7^1,        25,   1141710210,       7,      18
   536870912 = 2^29,              29,   296AA71010,      10,      19
  1342177280 = 2^28 * 5^1,        28,   6071712310,       7,      21.
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).
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Positions of records for -A350074(n).

Cf. A002110, A049345, A051903, A328114.

Cf. also A369646, A369647.

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

Programs

PARI

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A350074(n) = (A328114(n) - A051903(n));
m=A350074(1); print1(1,", "); for(n=2,oo,x=A350074(n); if(x