A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

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

Offset: 1

Author

Antti Karttunen, Jul 20 2022

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

Examples

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A000265, A009194, A246278, A348717, A355931.

Cf. also A300230, A305800, A305897, A355833.

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; };
A000265(n) = (n>>valuation(n,2));
A009194(n) = gcd(n, sigma(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];