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

Author

Antti Karttunen, Sep 30 2019

Keywords

• nonn

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]

Examples

			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).
		

Links

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

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.

Programs

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];

Formula

a(p) = 2 for all primes p.