A366875 -id:A366875 - cp's OEIS frontend

A366878 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366875(i) = A366875(j) for all i, j >= 0.

1, 2, 3, 4, 3, 1, 5, 6, 3, 2, 2, 2, 3, 7, 8, 9, 3, 10, 1, 11, 3, 12, 1, 9, 3, 10, 13, 14, 3, 15, 12, 16, 3, 6, 17, 18, 3, 19, 20, 21, 3, 1, 9, 22, 3, 23, 12, 24, 3, 25, 17, 26, 3, 27, 28, 29, 3, 10, 30, 23, 3, 31, 32, 11, 3, 11, 8, 33, 3, 34, 35, 36, 3, 11, 37, 38, 3, 39, 40, 41, 3, 5, 2, 42, 3, 43, 44, 45, 3, 1, 46, 47, 3

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Restricted growth sequence transform of A366875.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A113415, A163511, A349915, A366875.

Cf. also A366383, A366874.

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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A366875(n) = A349915(A163511(n));
v366878 = rgs_transform(vector(1+up_to,n,A366875(n-1)));
A366878(n) = v366878[1+n];

A366873 a(n) = A113415(A163511(n)), where A113415(n) is the average of number of and sum of odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 8, 3, 4, 1, 22, 8, 17, 3, 14, 4, 5, 1, 63, 22, 80, 8, 65, 17, 30, 3, 42, 14, 26, 4, 18, 5, 7, 1, 185, 63, 393, 22, 316, 80, 202, 8, 206, 65, 174, 17, 117, 30, 68, 3, 124, 42, 127, 14, 100, 26, 50, 4, 55, 18, 38, 5, 26, 7, 8, 1, 550, 185, 1956, 63, 1567, 393, 1403, 22, 1020, 316, 1204, 80, 804, 202

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A113415, A163511, A366874 (rgs-transform).

Cf. also A324186, A366797, A366875.

A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366873(n) = A113415(A163511(n));

a(n) = (1/2) * (A324186(n)+A366797(n)).

A366879 a(n) = A326938(A163511(n)), where A326938 is the Dirichlet inverse of the sum of divisors d of n such that n/d is odd.

Original entry on oeis.org

1, -2, 0, -4, 0, 3, 8, -6, 0, 0, -6, 5, 0, 24, 12, -8, 0, 0, 0, 0, 0, -20, -10, 7, 0, -18, -48, 48, 0, 32, 16, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, -42, 0, -28, -14, 11, 0, 0, 36, -40, 0, -192, -96, 96, 0, -24, -64, 72, 0, 48, 24, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -30, 35, 0, 168, 84

Offset: 0

Antti Karttunen, Oct 27 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A002131, A163511, A326938, A366880 (rgs-transform).

Cf. also A366875.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A326938(n) = sumdiv(n, d, if(n/d%2, moebius(n/d)*moebius(d)*d));
A366879(n) = A326938(A163511(n));

cp's OEIS Frontend

A366878 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366875(i) = A366875(j) for all i, j >= 0.

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A366873 a(n) = A113415(A163511(n)), where A113415(n) is the average of number of and sum of odd divisors of n.

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Formula

A366879 a(n) = A326938(A163511(n)), where A326938 is the Dirichlet inverse of the sum of divisors d of n such that n/d is odd.

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PARI