A351097 -id:A351097 - cp's OEIS frontend

A351098 Numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is less than the maximal exponent in the prime factorization of k.

8, 9, 16, 24, 28, 32, 40, 45, 48, 81, 96, 108, 112, 120, 125, 128, 136, 160, 184, 189, 192, 198, 208, 212, 225, 236, 244, 250, 256, 270, 288, 296, 352, 361, 459, 507, 625, 640, 768, 800, 832, 864, 896, 928, 960, 972, 1008, 1024, 1056, 1088, 1104, 1120, 1152, 1168, 1184, 1232, 1272, 1280, 1320, 1344, 1350, 1408, 1440

Offset: 1

Antti Karttunen, Feb 03 2022

nonn

Numbers k such that A328390(k) < A051903(k).
Numbers k for which A051903(A327859(n)) < A051903(k).
These seem to be rarer than A351075. All terms are nonsquarefree.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base

Positions of negative terms in A351097. Subsequence of A013929 and of A351099.
Cf. A003415, A051903, A276086, A327859, A328390, A369637 (characteristic function).
Cf. also A351075.

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); };
isA351098(n) = (A328114(A003415(n)) < A051903(n));

PARI
```
\\ Or see A369637.
```

A351099 Composite numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is not larger than the maximal exponent in the prime factorization of k.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 24, 25, 28, 30, 32, 36, 40, 45, 48, 49, 50, 54, 56, 58, 62, 64, 68, 74, 81, 87, 96, 98, 99, 108, 112, 120, 125, 128, 136, 155, 156, 160, 161, 162, 184, 189, 192, 196, 198, 203, 204, 208, 209, 210, 212, 217, 220, 221, 224, 225, 236, 244, 246, 247, 250, 252, 256, 268, 270, 272, 280, 282, 288

Offset: 1

Antti Karttunen, Feb 03 2022

nonn

Composite k such that A328390(k) <= A051903(k).

Composite k for which A051903(A327859(n)) <= A051903(k).

Cf. A003415, A051903, A276086, A327859, A328390, A351097, A351098 (subsequence).

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); };
isA351099(n) = (n>1&&!isprime(n)&&(A328114(A003415(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

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A351098 Numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is less than the maximal exponent in the prime factorization of k.

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A351099 Composite numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is not larger than the maximal exponent in the prime factorization of k.

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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.

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