A373594 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007814(n), A065339(n), A083025(n), A373591(n), A373592(n)].

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

Offset: 1

Author

Antti Karttunen, Jun 13 2024

Keywords

• nonn

Comments

Restricted growth sequence transform of the function f given in the definition.

Note that for composite n, f(n) can be defined in general as a quintuple vector [v(n), w(n), x(n), y(n), z(n)], where v, w, x, y and z are any five of these six sequences: A007814, A007949, A065339, A083025, A373591, A373592. This follows because A007814(n) + A065339(n) + A083025(n) = A007949(n) + A373591(n) + A373592(n) = A001222(n), so the omitted sixth element can be always worked out from the remaining five.

For all i, j > 1:

A305900(i) = A305900(j) => a(i) = a(j) => A373595(i) = A373595(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A007814, A007949, A065339, A083025, A373591, A373592.

Cf. also A305900, A373595.

Programs

PARI

up_to = 100000;
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; };
A007814(n) = valuation(n,2);
A065339(n) = sum(i=1, #n=factor(n)~, (3==n[1, i]%4)*n[2, i]);
A083025(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%4)*n[2, i]);
A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]);
A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]);
Aux373594(n) = if(n<=3, n, if(isprime(n), 0, [A007814(n), A083025(n), A065339(n), A373591(n), A373592(n)]));
v373594 = rgs_transform(vector(up_to, n, Aux373594(n)));
A373594(n) = v373594[n];