A369646 -id:A369646 - cp's OEIS frontend

A369647 Nonzero numbers k in A341518 for which A051903(k) attains novel values.

1, 2, 9, 16, 108, 9024, 2990880, 995336192, 1805726080

Offset: 1

Antti Karttunen, Feb 02 2024

Indices of record values of A051903 attained for the terms of A341518 is a subsequence: 1, 2, 9, 16, 9024, 995336192.

			           k   factorization           max.exp         k'    A049345(k')
           1                              0,           0,            0
           2 = 2^1,                       1,           1,            1
           9 = 3^2,                       2,           6,          100
          16 = 2^4,                       4,          32,         1010
         108 = 2^2 * 3^3,                 3,         216,        10100
        9024 = 2^6 * 3 * 47,              6,       30272,      1011010
     2990880 = 2^5 * 3^2 * 5 * 31 * 67,   5,    10210416,    110010100
   995336192 = 2^13 * 121501,            13,  6469693440,  10000010000
  1805726080 = 2^7 * 5 * 157 * 17971,     7,  6692788416,  11000100100.
See also the examples at A351073 and A369649.

Cf. A003415, A049345, A051903, A276086, A327859, A328114, A341518.

Cf. also A351073, A369642, A369646, A369649.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
isA341518(n) = ismaxprimobasedigit_at_most(A003415(n),1);
m=Map(); for(n=1,2990880,if(isA341518(n),e=A051903(n);if(!mapisdefined(m,e),mapput(m,e,n);print1(n,", "))));

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 01 2024

			           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).

Index entries for sequences related to primorial base

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.

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

cp's OEIS Frontend

A369647 Nonzero numbers k in A341518 for which A051903(k) attains novel values.

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