A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 11, 12, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 16, 2, 17, 2, 18, 19, 4, 2, 20, 3, 21, 4, 22, 2, 23, 4, 24, 4, 4, 2, 25, 2, 4, 26, 27, 4, 28, 2, 29, 4, 30, 2, 31, 2, 4, 32, 33, 4, 34, 2, 35, 36, 4, 2, 37, 4, 4, 4, 38, 2, 39, 4, 40, 4, 4, 4, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47

Offset

1,2

Comments

Restricted growth sequence transform of A319356.

The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.

For all i, j:

a(i) = a(j) => A000005(i) = A000005(j),

a(i) = a(j) => A319683(i) = A319683(j),

a(i) = a(j) => A319686(i) = A319686(j),

a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

Prog

(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319356(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(1+A003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to, n, A319356(n)));
A319357(n) = v319357[n];

Crossrefs

Cf. A003415, A319356.

Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),

Cf. also A300245, A300249, A319353, A319693.

Sequence in context: A324203 A290110 A300250 * A237433 A236515 A318469

Adjacent sequences: A319354 A319355 A319356 * A319358 A319359 A319360

Keyword

nonn

Author

Antti Karttunen, Oct 02 2018

Status

approved