A322867 -id:A322867 - cp's OEIS frontend

A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 3, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 20, 21, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 37, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 54, 55, 2, 15, 32, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 21, 2, 68, 69, 70, 2, 71, 2, 57

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Restricted growth sequence transform of function f, with f(1) = 0 and f(n) = [A322867(n), A323174(n)] for n > 1.
Equally, restricted growth sequence transform of function f, with f(1) = 0 and f(n) = A318310(A122111(n)) for n > 1.
For all i, j:
  a(i) = a(j) => A322867(i) = A322867(j),
  a(i) = a(j) => A323167(i) = A323167(j),
  a(i) = a(j) => A323174(i) = A323174(j).

Cf. A000203, A005187, A122111, A294898, A318310, A322867, A323167, A323173, A323174.

Cf. also A323240.

up_to = 4096;
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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318310aux(n) = [hammingweight(n), A294898(n)];
A323168aux(n) = if(1==n,0,A318310aux(A122111(n)));
v323168 = rgs_transform(vector(up_to, n, A323168aux(n)));
A323168(n) = v323168[n];

A322865 a(n) = A000265(A122111(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 9, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 27, 3, 25, 5, 1, 15, 1, 11, 9, 3, 27, 21, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 81, 45, 9, 5, 1, 35, 27, 7, 9, 3, 1, 21, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 33, 1, 3, 75, 5, 81, 15, 1, 11, 49, 3, 1, 21, 27, 3, 9, 7, 1, 35, 81, 5, 9, 3, 27, 13, 1, 135, 25, 63, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A122111, A322813, A322819, A322820, A322826, A322865, A322867.

Array[#/2^IntegerExponent[#, 2] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

A000265(n) = (n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322865(n) = A000265(A122111(n));

a(n) = A000265(A122111(n)).
a(n) = A122111(A322820(n)).
A000005(a(n)) = A322813(n).
A000203(a(n)) = A322819(n).
A122111(a(n)) = A322820(n).
A000120(a(n)) = A322867(n).

A322863 Permutation of natural numbers: a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 11, 6, 10, 25, 13, 15, 17, 49, 21, 8, 19, 16, 23, 35, 55, 121, 29, 27, 36, 169, 50, 77, 31, 81, 37, 12, 91, 289, 225, 30, 41, 361, 187, 125, 43, 625, 47, 143, 147, 529, 53, 45, 154, 90, 247, 221, 59, 28, 1225, 343, 391, 841, 61, 105, 67, 961, 605, 18, 5929, 2401, 71, 323, 551, 525, 73, 22, 79, 1369, 84, 437, 429, 14641, 83

Offset: 0

Antti Karttunen, Dec 30 2018

nonn

Note the indexing: the domain starts from 0, but the range excludes zero.

Inverse permutation: A322864.

Cf. A005940, A122111, A322866, A322867.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322863(n) = if(!n,1,A005940(1+A122111(n)));

a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).
For all n >= 1:
a(prime(n)) = prime(1+n).
A001222(a(n)) = A322867(n).

A323167 a(n) = A294898(A122111(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, -2, 0, 2, 3, -6, 0, 0, 0, -14, -5, 3, 0, 2, 0, -4, -21, -30, 0, 1, 10, -62, 16, -12, 0, -16, 0, 7, -53, -126, -16, 7, 0, -254, -117, -3, 0, -52, 0, -28, 4, -510, 0, 5, 38, 8, -245, -60, 0, 19, -68, -11, -501, -1022, 0, -15, 0, -2046, -20, 9, -172, -124, 0, -124, -1013, -58, 0, 16, 0, -4094, 22, -252, -42, -268, 0, 1, 38

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Cf. A000203, A005187, A294898, A122111, A322867, A323174, A323168.

Cf. also A295296, A323248.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A323167(n) = A294898(A122111(n));
\\ Or alternatively as:
A322867(n) = hammingweight(A122111(n));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }
A323167(n) = (A323174(n) - A322867(n));

a(n) = A294898(A122111(n)).

a(n) = A323174(n) - A322867(n).

A322866 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(A322863(i)) = A046523(A322863(j)) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Restricted growth sequence transform of A046523(A322863(n)).
Equally, restricted growth sequence transform of A278222(A322865(n)).
For all i, j: a(i) = a(j) => A322867(i) = A322867(j).

Cf. A005940, A046523, A122111, A278222, A286622, A322863, A322865, A322867.

up_to = 8192;
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; };
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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322863(n) = if(!n,1,A005940(1+A122111(n)));
v322866 = rgs_transform(vector(up_to,n,A046523(A322863(n))));
A322866(n) = v322866[n];

cp's OEIS Frontend

A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A322865 a(n) = A000265(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322863 Permutation of natural numbers: a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A323167 a(n) = A294898(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322866 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(A322863(i)) = A046523(A322863(j)) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI