A341355 -id:A341355 - cp's OEIS frontend

A341353 Greatest k such that 3^k divides A156552(n); the 3-adic valuation of A156552(n).

0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 3, 3, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0

Offset: 2

Antti Karttunen, Feb 14 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..65539
Index entries for sequences computed from indices in prime factorization

Cf. A007949, A156552, A329609 (positions of nonzeros), A329903, A341354 (even bisection), A341355.

Cf. also A055396 (the 2-adic valuation + 1), A292251.

A007949(n) = valuation(n,3);
A156552(n) = { my(f = factor(n), 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 };
A341353(n) = A007949(A156552(n));

a(n) = A007949(A156552(n)).
a(p) = 0 for all primes p.
a(n^2) > 0 for all n >= 2.
a(n) > 0 iff A329903(n) = 0.

A341354 Greatest k such that 3^k divides A156552(2*n); number of trailing 1-digits in the ternary expansion of A156552(n).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 4, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 4, 0, 0

Offset: 1

Antti Karttunen, Feb 14 2021

nonn

The 3-adic valuation of A156552(2*n).

Antti Karttunen, Table of n, a(n) for n = 1..65539
Index entries for sequences computed from indices in prime factorization

Even bisection of A341353.
Cf. A000040, A007949, A156552, A341355.
Cf. A329604 (positions of nonzero terms).

A007949(n) = valuation(n,3);
A156552(n) = { my(f = factor(n), 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 };
A341354(n) = A007949(A156552(2*n));

a(n) = A341353(2*n) = A007949(A156552(2*n)) = A007949(1+(2*A156552(n))).

For all n >= 1, a(A000040(2*n)) = a(n^2) = 0.

A353277 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A020639(n), A341353(n)], with f(1) = 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 10 2022

nonn

Restricted growth sequence transform of function f(1) = 1, and for n > 1, f(n) = [A007814(u), A007949(u)], where u = A156552(n).

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

Cf. A007814, A007949, A020639, A156552, A341353, A353278 (ordinal transform).

Cf. also A322026, A340680, A341355.

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; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
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 };
Aux353277(n) = if(1==n,1,my(u=A156552(n)); [A007814(u), A007949(u)]);
v353277 = rgs_transform(vector(up_to, n, Aux353277(n)));
A353277(n) = v353277[n];

cp's OEIS Frontend

A341353 Greatest k such that 3^k divides A156552(n); the 3-adic valuation of A156552(n).

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A341354 Greatest k such that 3^k divides A156552(2*n); number of trailing 1-digits in the ternary expansion of A156552(n).

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A353277 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A020639(n), A341353(n)], with f(1) = 1.

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