A351578 -id:A351578 - cp's OEIS frontend

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

1, 1, 1, 2, 1, 3, 3, 1, 2, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 5, 3, 5, 3, 6, 3, 5, 2, 7, 1, 5, 6, 3, 5, 5, 3, 5, 6, 3, 8, 2, 9, 3, 5, 4, 5, 5, 3, 8, 5, 1, 8, 5, 3, 7, 5, 3, 10, 6, 3, 6, 6, 5, 6, 5, 6, 5, 5, 4, 8, 3, 9, 5, 5, 7, 2, 11, 3, 10, 3, 10, 6, 6, 9, 5, 8, 6, 5, 8, 5, 10, 6, 12, 6, 10, 6, 5, 10, 4, 6, 8, 5, 13

Offset: 1

Antti Karttunen, Apr 06 2022

Restricted growth sequence transform of A278222(A109812(n)), or equally of, A278222(A351965(n)).

For all i, j: A351578(i) = A351578(j) => a(i) = a(j) => A352884(i) = A352884(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A005940, A046523, A109812, A278222, A286622, A351965, A352884.

Cf. also A351578, A352888.

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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351963 = rgs_transform(vector(up_to, n, A046523(A005940(1+A109812(n)))));
A351963(n) = v351963[n];

A351964 The 2-adic valuation of A109812(n); The exponent of the highest power of 2 dividing A109812(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 07 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A007814, A109812, A351578, A351965.

v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A007814(n) = valuation(n,2);
A351964(n) = A007814(A109812(n));

from itertools import count, islice
def A351964_gen(): # generator of terms
    yield 0
    l1, s, b = 1, 2, set()
    while True:
        for i in count(s):
            if not (i in b or i & l1):
                yield (~i&i-1).bit_length()
                l1 = i
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
A351964_list = list(islice(A351964_gen(),30)) # Chai Wah Wu, Jul 16 2022

a(n) = A007814(A109812(n)).

A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 4, 2, 5, 6, 7, 5, 6, 5, 2, 1, 5, 6, 4, 3, 7, 6, 4, 8, 7, 6, 5, 1, 2, 9, 10, 7, 11, 12, 6, 9, 13, 7, 3, 5, 6, 7, 6, 5, 6, 9, 4, 6, 9, 2, 6, 12, 6, 5, 11, 7, 13, 14, 7, 15, 13, 12, 16, 9, 14, 11, 12, 5, 6, 7, 11, 12, 6, 5, 5, 1, 4, 13, 7, 8, 15, 16, 12, 11, 9, 10, 12, 11, 12, 10, 14, 17, 14, 16, 18

Offset: 1

Antti Karttunen, Apr 07 2022

Restricted growth sequence transform of A278219(A109812(n)).

For all i, j: a(i) = a(j) => A352889(i) = A352889(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A109812, A278219, A352889.

Cf. also A286619, A351578, A351963.

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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
A109812(n) = v109812[n];
v352888 = rgs_transform(vector(up_to, n, A278219(A109812(n))));
A352888(n) = v352888[n];

cp's OEIS Frontend

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

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A351964 The 2-adic valuation of A109812(n); The exponent of the highest power of 2 dividing A109812(n).

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A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

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