A322806 -id:A322806 - cp's OEIS frontend

A322807 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = A285330(n) for all other numbers.

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

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

Cf. A048675, A285328, A285330, A285332, A322806.

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; };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
A322807aux(n) = if((n%2)&&isprime(n),-1,A285330(n));
v322807 = rgs_transform(vector(up_to,n,A322807aux(n)));
A322807(n) = v322807[n];

A322811 a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

For all i, j:
  A322806(i) = A322806(j) => a(i) = a(j),
  A322807(i) = A322807(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001221, A048675, A285330, A322806, A322807, A322812.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
A322811(n) = if(1==n,0,omega(A285330(n)));

a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

If n > 1 is squarefree, a(n) = A322812(n) = A001221(A048675(n)), otherwise a(n) = A001221(A285328(n)) = A001221(n).

A322861 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A285330(i)) = A278222(A285330(j)) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 31 2018

nonn

Restricted growth sequence transform of A278222(A285330(n)).
For all i, j:
  A322806(i) = A322806(j) => a(i) = a(j),
  A322807(i) = A322807(j) => a(i) = a(j),
  a(i) = a(j) => A322862(i) = A322862(j).

Cf. A048675, A278222, A285330, A286622, A322806, A322807, A322862, A322868.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(moebius(n)<>0,A048675(n),A285328(n));
A278222(n) = A046523(A005940(1+n));
v322861 = rgs_transform(vector(up_to,n,A278222(A285330(n))));
A322861(n) = v322861[n];

cp's OEIS Frontend

A322807 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = A285330(n) for all other numbers.

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A322811 a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

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A322861 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A285330(i)) = A278222(A285330(j)) for all i, j.

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