A342666 -id:A342666 - cp's OEIS frontend

A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

1, 2, 2, 3, 2, 3, 2, 4, 3, 5, 2, 6, 2, 3, 3, 7, 2, 4, 2, 8, 5, 9, 2, 10, 3, 11, 4, 11, 2, 11, 2, 12, 3, 13, 3, 14, 2, 3, 9, 11, 2, 8, 2, 15, 6, 16, 2, 17, 3, 5, 11, 18, 2, 19, 5, 20, 13, 21, 2, 22, 2, 23, 8, 24, 3, 25, 2, 26, 3, 6, 2, 9, 2, 27, 4, 28, 3, 26, 2, 29, 7, 30, 2, 31, 9, 32, 16, 33, 2, 31, 5, 34, 21, 35

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A342666(n), A350063(n)], when assuming that A342666(1) = 0.
Restricted growth sequence transform of the function f(1) = 0, f(n) = A336470(A156552(n)) for n > 1.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A350065(i) = A350065(j).
For all i, j >= 2: a(i) = a(j) => A342651(i) = A342651(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A156552, A322993, A336466, A342666, A350063, A336470.

Cf. also A305897, A350065, A342651.

up_to = 3003;
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));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
Aux350067(n) = if(1==n,1,my(u=A000265(A156552(n))); [A046523(u),A336466(u)]);
v350067 = rgs_transform(vector(up_to, n, Aux350067(n)));
A350067(n) = v350067[n];

A342651 a(n) = A329697(A156552(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 3, 1, 3, 2, 3, 0, 3, 0, 3, 1, 5, 1, 3, 0, 1, 3, 3, 0, 3, 0, 4, 2, 6, 0, 4, 1, 2, 3, 4, 0, 3, 2, 4, 5, 4, 0, 4, 0, 7, 3, 4, 1, 4, 0, 5, 1, 2, 0, 3, 0, 4, 2, 4, 1, 5, 0, 4, 2, 8, 0, 3, 3, 7, 6, 4, 0, 3, 2, 6, 4, 9, 3, 4, 0, 4, 3, 2, 0, 5, 0, 5, 3

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..6381 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A156552, A329697, A322993, A342652, A350067.

Cf. A000040 (positions of 0's), A350069 (of 1's).

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = A329697(A156552(n));

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = if(isprime(n),0,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); sum(i=2-(ps[1]%2),#ps,es[i]*(1+A329697(ps[i]-1)))); \\ Antti Karttunen, Jan 29 2022

a(n) = A329697(A156552(n)) = A329697(A322993(n)).
a(n) = A329697(A342666(n)) + A342656(n).
a(p) = 0 for all primes p.
a(A003961(n)) = a(n).

cp's OEIS Frontend

A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A342651 a(n) = A329697(A156552(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula