A336121 -id:A336121 - cp's OEIS frontend

A336120 a(n) = A292383(A122111(n)).

0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 5, 0, 5, 0, 8, 0, 16, 0, 10, 1, 32, 0, 16, 0, 10, 0, 11, 0, 64, 2, 8, 0, 128, 0, 20, 0, 20, 0, 32, 0, 256, 0, 22, 0, 8, 0, 64, 0, 11, 4, 40, 0, 512, 0, 16, 0, 1024, 0, 22, 8, 40, 0, 128, 0, 16, 0, 20, 0, 2048, 1, 256, 0, 80, 0, 44, 0, 4096, 0, 32, 16, 8192, 0, 80, 0, 22, 0, 512, 0, 16384, 32, 44, 0, 17, 0, 17, 0

Offset: 1

Antti Karttunen, Jul 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A122111, A252463, A253553, A253566, A292383, A336121, A336124, A336125, A336312.

Cf. A336119 (positions of zeros).

\\ Uses also code given in A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336120(n) = if(1==n,0,(3==A336124(n))+(2*A336120(A253553(n))));

a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + 2*a(A253553(n)).
a(n) = A292383(A122111(n)).
a(n) = A253566(n) - A336125(n).
A000120(a(n)) = A336121(n).

A336119 Numbers k such that A122111(k) [the conjugated prime factorization of k] is a square or twice a square.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 135, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163, 167, 169, 171, 173

Offset: 1

Antti Karttunen, Jul 14 2020

nonn

Sequence A122111(A028982(k)), k >= 1, sorted into ascending order.

Cf. A000040, A066207 (subsequences), A335909 (characteristic function).
Cf. A028982, A031215, A031368, A053866, A122111.
Positions of odd terms in A323173, positions of zeros in A336120 and A336121, positions of ones in A336312.

isA336119(n) = (A335909(n)); \\ See code in A335909.

A336312 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Restricted growth sequence transform of A278222(A336120(n)).
For all i, j:
  a(i) = a(j) => A336121(i) = A336121(j) => A335909(i) = A335909(j).

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

Cf. A336311, A336313.
Cf. A336119 (positions of ones).
Cf. also A292583, A332901, A335909, A336121.

up_to = 1024; \\ 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; };
\\ Needs also code from A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336120(n) = if(1==n,0,(3==A336124(n))+(2*A336120(A253553(n))));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
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));
v336312 = rgs_transform(vector(up_to,n,A278222(A336120(n))));
A336312(n) = v336312[n];

A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

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

Cf. A000120, A000244, A001222, A122111, A292375, A336121, A336124, A336125.

\\ Uses also code given in A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336123(n) = if(n<=2,n-1,(1==A336124(n))+A336123(A253553(n)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A336124(n) == 1] + a(A253553(n)).
a(n) = A000120(A336125(n)).
For n > 1, a(n) = A292375(A122111(n)).
a(n) = A001222(n) - A336121(n).
For all n >= 0, a(3^n) = n.

cp's OEIS Frontend

A336120 a(n) = A292383(A122111(n)).

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A336119 Numbers k such that A122111(k) [the conjugated prime factorization of k] is a square or twice a square.

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A336312 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) for all i, j >= 1.

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PARI

A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

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