A322821 -id:A322821 - cp's OEIS frontend

A322869 a(n) = A000120(A048675(n)).

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

Offset: 1

Antti Karttunen, Dec 31 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. A000120, A001221, A001222, A048675, A097248, A322821, A322862, A322868.

Cf. A002110 (position of the first occurrence of n).

Array[If[# == 1, 0, DigitCount[Total@ Map[#2*2^(PrimePi@ #1 - 1) & @@ # &, FactorInteger[#]], 2, 1]] &, 105]  (* Michael De Vlieger, Dec 31 2018 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A322869(n) = hammingweight(A048675(n));

a(n) = A000120(A048675(n)) = A000120(A322821(n)).
a(n) = A001221(A097248(n)) = A001222(A097248(n)).
If n is squarefree, then a(n) = A001221(n) = A322862(n).
a(A002110(n)) = n.

A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

Original entry on oeis.org

0, 1, 3, 1, 7, 1, 5, 11, 15, 1, 9, 19, 11, 23, 25, 13, 31, 1, 17, 35, 19, 39, 41, 21, 23, 47, 49, 25, 53, 27, 7, 57, 63, 1, 33, 67, 35, 71, 73, 37, 39, 79, 81, 41, 85, 43, 11, 89, 47, 95, 97, 49, 101, 51, 13, 105, 109, 55, 7, 113, 29, 117, 119, 15, 127, 1, 65, 131, 67, 135, 137, 69, 71, 143, 145, 73, 149, 75, 19

Offset: 0

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000265, A005187, A283477, A322821, A324287, A324378, A324379.

A000265(n) = (n/2^valuation(n, 2));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324377(n) = if(!n,n,A000265(A005187(n)));

a(0) = 0; for n > 0, a(n) = A000265(A005187(n)) = A005187(n) / 2^A324379(n).

a(n) = A322821(A283477(n)).

A322868 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A048675(i)) = A278222(A048675(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 3, 2, 2, 4, 2, 3, 4, 4, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 4, 3, 3, 2, 4, 4, 5, 2, 6, 2, 4, 2, 4, 2, 3, 2, 4, 4, 4, 2, 5, 4, 6, 4, 4, 2, 2, 2, 4, 3, 3, 4, 6, 2, 4, 4, 6, 2, 5, 2, 4, 4, 4, 3, 6, 2, 2, 2, 4, 2, 3, 4, 4, 4, 6, 2, 4, 4, 4, 4, 4, 4, 5, 2, 4, 4, 4, 2, 6, 2, 6, 5

Offset: 1

Antti Karttunen, Dec 31 2018

nonn

Restricted growth sequence transform of A278222(A048675(n)), or equivalently, of A278222(A322821(n)).

For all i, j: a(i) = a(j) => A322869(i) = A322869(j).

Cf. A005940, A046523, A048675, A278222, A322821, A322861, A322869.

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
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A278222(n) = A046523(A005940(1+n));
v322868 = rgs_transform(vector(up_to,n,A278222(A048675(n))));
A322868(n) = v322868[n];

A324286 a(n) = A002487(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 3, 5, 1, 3, 1, 6, 2, 3, 1, 3, 1, 3, 4, 7, 2, 2, 1, 8, 5, 3, 1, 5, 1, 4, 1, 9, 1, 2, 1, 4, 6, 5, 1, 3, 3, 5, 7, 10, 1, 1, 1, 11, 2, 2, 4, 7, 1, 6, 8, 5, 1, 3, 1, 12, 3, 7, 2, 9, 1, 1, 1, 13, 1, 2, 5, 14, 9, 7, 1, 4, 3, 8, 10, 15, 6, 3, 1, 5, 3, 3, 1, 11, 1, 9, 3

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Like A323902 and A323903, this also has quite a moderate growth rate, even though some terms of A048675 soon grow quite big.

Cf. A002487, A048675, A283477, A322821, A323902, A323903, A324287.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A324286(n) = A002487(A048675(n));

a(n) = A002487(A048675(n)) = A002487(A322821(n)).

a(A283477(n)) = A324287(n).

cp's OEIS Frontend

A322869 a(n) = A000120(A048675(n)).

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A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

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A322868 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A048675(i)) = A278222(A048675(j)) for all i, j.

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A324286 a(n) = A002487(A048675(n)).

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PARI

Formula