A351035 Lexicographically earliest infinite sequence such that a(i) = a(j) => A347385(i) = A347385(j) and A336158(i) = A336158(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, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42

Offset

1,3

Comments

Restricted growth sequence transform of the ordered pair [A347385(n), A336158(n)], where A347385(n) is the Dedekind psi function applied to the odd part of n, i.e., A001615(A000265(n)), and A336158(n) is the least representative of the prime signature of the odd part of n.

For all i, j >= 1: A003602(i) = A003602(j) => a(i) = a(j).

Links

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

Example

a(33) = a(35) as both 33 = 3*11 and 35 = 5*7 are odd nonsquare semiprimes, thus A336158 gives equal values for them, and also A347385(33) = A001615(33) = A347385(35) = A001615(35) = 48.

Prog

(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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A347385(n) = if(1==n, n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
Aux351035(n) = [A347385(n), A336158(n)];
v351035 = rgs_transform(vector(up_to, n, Aux351035(n)));
A351035(n) = v351035[n];

Crossrefs

Cf. A000265, A001615, A003602, A046523, A336158, A351035.

Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.

Differs from A351036 for the first time at n=175, where a(175) = 78, while A351036(175) = 80.

Sequence in context: A336392 A336935 A336162 * A351036 A351040 A347374

Adjacent sequences: A351032 A351033 A351034 * A351036 A351037 A351038

Keyword

nonn,easy

Author

Antti Karttunen, Jan 30 2022

Status

approved