A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

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

Offset: 1

Author

Antti Karttunen, Jan 10 2019

Keywords

• nonn

Comments

Restricted growth sequence transform of function f, with f(1) = 0 and f(n) = [A322867(n), A323174(n)] for n > 1.

Equally, restricted growth sequence transform of function f, with f(1) = 0 and f(n) = A318310(A122111(n)) for n > 1.

For all i, j:

a(i) = a(j) => A322867(i) = A322867(j),

a(i) = a(j) => A323167(i) = A323167(j),

a(i) = a(j) => A323174(i) = A323174(j).

Links

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

Crossrefs

Cf. A000203, A005187, A122111, A294898, A318310, A322867, A323167, A323173, A323174.

Cf. also A323240.

Programs

PARI

up_to = 4096;
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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318310aux(n) = [hammingweight(n), A294898(n)];
A323168aux(n) = if(1==n,0,A318310aux(A122111(n)));
v323168 = rgs_transform(vector(up_to, n, A323168aux(n)));
A323168(n) = v323168[n];