A369645 -id:A369645 - cp's OEIS frontend

A351038 Numbers whose maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization.

2, 3, 4, 6, 7, 8, 9, 12, 16, 30, 31, 32, 33, 36, 37, 38, 39, 40, 44, 45, 48, 60, 63, 64, 68, 72, 75, 76, 80, 81, 96, 104, 108, 112, 128, 144, 160, 192, 210, 211, 212, 213, 216, 217, 218, 219, 220, 224, 225, 232, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 252, 256, 270, 272, 275, 276, 279, 280, 284, 288, 304

Offset: 1

Antti Karttunen, Feb 01 2022

Numbers k for which A328114(k) <= A051903(k).

Numbers such that when the map x -> A276086(x) is applied to them, the maximal exponent in the prime factorization (A051903) doesn't increase.

Antti Karttunen, Table of n, a(n) for n = 1..28472; terms <= 9699690
Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A351039 (characteristic function), A351069 and A351070 (counts).
Positions of nonpositive terms in A350074.
Subsequences: A350075, A351075, also A369645 (after its initial 1).
Cf. also A341518.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351038(n) = (A051903(A276086(n)) <= A051903(n));

A369646 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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 02 2024

			           k   factorization    max.exp.  k' in primorial  max digit  diff
                                                  base
           1                        0,              0,        0,       0
           8 = 2^3,                 3,            200,        2,       1
          16 = 2^4,                 4,           1010,        1,       3
         832 = 2^6 * 13^1,          6,         111120,        2,       4
        1024 = 2^10,               10,         222310,        3,       7
       95232 = 2^10 * 3^1 * 31^1,  10,       10021220,        2,       8
      131072 = 2^17,               17,       23132010,        3,      14
     2097152 = 2^21,               21,      252354100,        5,      16
  1006632960 = 2^26 * 3^1 * 5^1,   26,    23194866010,        9,      17
  1090519040 = 2^24 * 5^1 * 13^1,  24,    22053155300,        5,      19.
Here k' stands for the arithmetic derivative of k, A003415(k). Primorial base expansion is obtained with A049345.

Index entries for sequences related to primorial base

Positions of records for -A351097(n).
After the initial 1, a subsequence of A351098.
Cf. A002110, A003415, A049345, A051903, A328114.
Cf. also A369645, A369647.

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]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A351097(n) = (A328114(A003415(n))-A051903(n));
m=A351097(1); print1(1,", "); for(n=2,oo,x=A351097(n); if(x

cp's OEIS Frontend

A351038 Numbers whose maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization.

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