A327931 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 75

Offset

1,2

Comments

Restricted growth sequence transform of A327930, or equally, of the ordered pair [A003415(n), A319356(n)].

It seems that the sequence takes duplicated values only on primes (A000040) and some subset of squarefree semiprimes (A006881). If this holds, 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) => A319684(i) = A319684(j),

a(i) = a(j) => A319685(i) = A319685(j),

a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above and in A319357]

Links

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

Formula

a(p) = 2 for all primes p.

Example

Divisors of 39 are [1, 3, 13, 39], while the divisors of 55 are [1, 5, 11, 55]. Taking their arithmetic derivatives (A003415) yields in both cases [0, 1, 1, 16], thus a(39) = a(55) (= 28, as allotted by restricted growth sequence transform).

Prog

(PARI)
up_to = 8192;
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
v003415 = vector(up_to, n, A003415(n));
A327930(n) = { my(m=1); fordiv(n, d, if((d>1), m *= prime(v003415[d]))); (m); };
v327931 = rgs_transform(vector(up_to, n, A327930(n)));
A327931(n) = v327931[n];

Crossrefs

Cf. A000005, A000040 (positions of 2's), A003415, A006881, A101296, A319356, A319357, A319684, A319685, A327930.

Differs from A300249 for the first time at n=105, where a(105)=75, while A300249(105)=56.

Differs from A300235 for the first time at n=153, where a(153)=110, while A300235(153)=106.

Differs from A305895 for the first time at n=3283, where a(3283)=2502, while A305895(3283)=1845.

Sequence in context: A373379 A351260 A305895 * A369050 A300833 A373150

Adjacent sequences: A327928 A327929 A327930 * A327932 A327933 A327934

Keyword

nonn

Author

Antti Karttunen, Sep 30 2019

Status

approved