A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(j), for all i, j >= 1.

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

Offset: 1

Author

Antti Karttunen, Jan 31 2022

Keywords

• nonn
• easy

Comments

Restricted growth sequence transform of the ordered pair [A351091(n), A351092(n)], or equally, of the ordered pair [A351093(n), A351094(n)].

For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A000593(i) = A000593(j).

Examples

			Consider two odd semiprimes, 689 and 697. The divisors of 689 are 1, 13, 53, 689, and the divisors of 697 are 1, 17, 41, 697. Applying A019565(A289813(x)) to the former gives [2, 30, 7, 105], while with the latter it gives [2, 5, 105, 42], and the product of both sequences is 44100. Applying A019565(A289814(x)) to the former gives [1, 1, 30, 286], while with the latter it gives [1, 6, 2, 715]. Product of both sequences is 8580. Therefore, because A351091(689) = A351091(697) and A351092(689) = A351092(697), also a(689) = a(697).
		

Links

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

Crossrefs

Cf. A000593, A351037, A351091, A351092, A351093, A351094.

Differs from A003602 for the first time at n=697, where a(697) = 345 while A003602(697) = 349.

Cf. also A293226, A351030.

Programs

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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); }; \\ From A289813
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); }; \\ From A289814
A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };
Aux351090(n) = [A351091(n),A351092(n)];
v351090 = rgs_transform(vector(up_to, n, Aux351090(n)));
A351090(n) = v351090[n];