A369642 -id:A369642 - cp's OEIS frontend

A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

1, 1, 2, 2, 9, 2, 18, 2, 25, 5, 10, 2, 225, 2, 30, 15, 21, 2, 750, 2, 625, 45, 50, 2, 525, 45, 150, 3750, 21, 2, 14, 2, 18375, 75, 250, 25, 49, 2, 750, 225, 735, 2, 630, 2, 875, 210, 1250, 2, 385875, 75, 1050, 375, 13125, 2, 36750, 225, 1029, 1125, 14, 2, 1029, 2, 42, 5250, 2941225, 125, 98, 2, 1225, 1875, 78750

Offset: 0

Antti Karttunen, Sep 30 2019

Sequence contains only terms of A048103.
Are there fixed points other than 1, 2, 10, 15, 5005? (There are none in the range 5006 .. 402653184.) See A369650.
Records occur at n = 0, 2, 4, 6, 8, 12, 18, 27, 32, 48, 64, 80, 144, 224, 256, 336, 448, 480, 512, 1728, ... (see also A131117).
a(n) and n are never multiples of 9 at the same time, thus the fixed points certainly exclude any terms of A008591. For a proof, consider my comment in A047257 and that A003415(9*n) is always a multiple of 3. - Antti Karttunen, Feb 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A008591, A048103, A131117, A276086, A327858, A327860, A341517 [= mu(a(n))], A341518 (k where a(k) is squarefree), A369641 (composite k where a(k) is squarefree), A369642.

Cf. A370114 (where a(k) is a multiple of k), A370115 (where k is a multiple of a(k)), A369650.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327859(n) = A276086(A003415(n));

a(n) = A276086(A003415(n)).

a(p) = 2 for all primes p.

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

Original entry on oeis.org

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,", "))));

A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

9, 10, 14, 15, 16, 28, 30, 45, 58, 62, 74, 87, 108, 112, 136, 155, 161, 189, 198, 203, 209, 210, 212, 217, 221, 225, 236, 244, 246, 247, 282, 290, 299, 323, 361, 374, 399, 422, 435, 478, 482, 507, 717, 1055, 1205, 1477, 1480, 1631, 1673, 1687, 1940, 2132, 2189, 2212, 2308, 2356, 2519, 2524, 2561, 2587, 2655, 2766

Offset: 1

Antti Karttunen, Jan 31 2024

nonn

Composite terms of A341518, i.e., composite numbers k such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.

Antti Karttunen, Table of n, a(n) for n = 1..20054 (terms less than 2^30)

Setwise difference A341518 \ A158611.
Cf. A003415, A276086, A276156, A327859, A341517, A369640 (characteristic function).
Cf. A327978, A328243, A369642 (subsequences).

PARI
```
\\ See A369640
```

A369639 Numbers k, not squarefree, such that the maximal digit in the primorial base representation of k' is <= 3, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 24, 25, 28, 32, 36, 40, 44, 45, 48, 49, 50, 54, 56, 60, 63, 68, 76, 81, 92, 96, 98, 99, 108, 112, 120, 121, 125, 136, 147, 153, 156, 160, 175, 184, 189, 192, 196, 198, 204, 208, 212, 220, 225, 228, 234, 236, 244, 250, 252, 268, 270, 280, 284, 289, 296, 300, 315, 316, 328, 333, 338, 340, 344, 361

Offset: 1

Antti Karttunen, Feb 01 2024

nonn

Nonsquarefree numbers k (A013929) such that A327859(k) = A276086(A003415(k)) is biquadratefree number (A046100), or equally that A328114(A003415(k)) <= 3.

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

Cf. A003415, A013929, A046100, A327859, A328114.

Nonsquarefree terms of A369642 form a subsequence.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
isA369639(n) = (n>0 && !issquarefree(n) && ismaxprimobasedigit_at_most(A003415(n),3));

cp's OEIS Frontend

A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369639 Numbers k, not squarefree, such that the maximal digit in the primorial base representation of k' is <= 3, where k' stands for the arithmetic derivative of k, A003415.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI