A340382 -id:A340382 - cp's OEIS frontend

A340377 Numbers k such that there are no 2-digits in the ternary expansion of A048673(k).

1, 3, 5, 9, 13, 17, 19, 21, 35, 47, 53, 59, 67, 71, 73, 91, 93, 95, 121, 123, 129, 143, 145, 157, 163, 173, 175, 179, 207, 211, 229, 233, 239, 255, 267, 291, 297, 299, 321, 327, 351, 355, 371, 381, 405, 413, 437, 451, 477, 479, 485, 487, 499, 503, 505, 523, 527, 541, 547, 549, 557, 595, 643, 645, 647, 661, 691, 701

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

All terms are odd, because A048673(2n) = 3*A048673(n) - 1, which forces the least significant digit in the ternary expansion of A048673(2n) to be "2".

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros in A291759 and in A340379. Positions of ones in A340382.

Cf. A048673, A340376.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
isA340377(n) = (0==A291759(n));

A340381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A304759(i)) = A278222(A304759(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

For all i, j: A304758(i) = A304758(j) => a(i) = a(j) => A340378(i) = A340378(j).

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

Cf. A278222, A304758, A304759, A340378, A340382, A340383.
Cf. A340376 (positions of 2's).
Cf. also A305301.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(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
A278222(n) = A046523(A005940(1+n));
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; };
v340381 = rgs_transform(vector(up_to,n,A278222(A304759(n))));
A340381(n) = v340381[n];

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A340381(n), A340382(n)], or equally, of the function f(n) = A290093(A048673(n)).

For all i, j: a(i) = a(j) => A286586(i) = A286586(j) => A286585(i) = A286585(j).

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

Cf. A048673, A278222, A290093, A291759, A304759, A340381, A340382, A340684.

Cf. also A290094, A305303.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(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
A278222(n) = A046523(A005940(1+n));
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; };
Aux340383(n) = [A278222(A291759(n)),A278222(A304759(n))];
v340383 = rgs_transform(vector(up_to,n,Aux340383(n)));
A340383(n) = v340383[n];

cp's OEIS Frontend

A340377 Numbers k such that there are no 2-digits in the ternary expansion of A048673(k).

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A340381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A304759(i)) = A278222(A304759(j)), for all i, j >= 1.

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A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

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PARI