A369650 -id:A369650 - cp's OEIS frontend

A373146 a(n) = A276085(n) - A003415(n), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

0, 0, 1, -2, 5, -2, 29, -9, -2, 0, 209, -12, 2309, 22, 0, -28, 30029, -16, 510509, -16, 22, 198, 9699689, -39, 2, 2296, -21, 0, 223092869, -22, 6469693229, -75, 198, 30012, 24, -54, 200560490129, 510490, 2296, -59, 7420738134809, -8, 304250263527209, 164, -29, 9699666, 13082761331670029, -106, 46, -32, 30012, 2256

Offset: 1

Antti Karttunen, May 26 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A003415, A276085, A369650 (positions of 0's), A373145.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373146(n) = (A276085(n) - A003415(n));

A373490 Numbers k for which A003415(k) is a multiple of A276085(k), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 15, 16, 20, 28, 32, 36, 64, 100, 108, 128, 144, 162, 180, 189, 192, 216, 225, 256, 400, 480, 500, 512, 686, 784, 972, 1000, 1024, 1250, 1280, 1296, 1350, 1728, 1890, 2048, 2160, 2304, 2700, 2835, 2880, 3375, 3500, 3600, 3840, 4032, 4096, 4500, 4800, 4860, 5005, 5096, 5120, 6300, 6400, 6480, 6750

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

Conjecture: if term present here is squarefree, then it must be a term of A369650: 1, 2, 10, 15, 5005, and possibly no others.

Antti Karttunen, Table of n, a(n) for n = 1..233; terms <= 2^20

Cf. A003415, A276085, A373489 (characteristic function).
Subsequences: A000079, A369650.
After the initial 1, gives the positions of 0's in A373147.
Cf. also A373487, A373488.

PARI
```
isA373490 = A373489;
```

cp's OEIS Frontend

A373146 a(n) = A276085(n) - A003415(n), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

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