A324186 -id:A324186 - cp's OEIS frontend

A366806 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324186(i) = A324186(j) for all i, j >= 0, where A324186 is the sum of odd divisors permuted by A163511.

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

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Restricted growth sequence transform of A324186.

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

Cf. A000265, A000593, A163511, A324186.

Cf. also A366804.

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; };
A000593(n) = sigma(n>>valuation(n, 2)); \\ From A000593
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));
A324186(n) = A000593(A163511(n));
v366806 = rgs_transform(vector(1+up_to,n,A324186(n-1)));
A366806(n) = v366806[1+n];

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A324184 a(n) = sigma(A163511(n)).

Original entry on oeis.org

1, 3, 7, 4, 15, 13, 12, 6, 31, 40, 39, 31, 28, 24, 18, 8, 63, 121, 120, 156, 91, 124, 93, 57, 60, 78, 72, 48, 42, 32, 24, 12, 127, 364, 363, 781, 280, 624, 468, 400, 195, 403, 372, 342, 217, 228, 171, 133, 124, 240, 234, 248, 168, 192, 144, 96, 90, 104, 96, 72, 56, 48, 36, 14, 255, 1093, 1092, 3906, 847, 3124, 2343, 2801, 600

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000203, A054429, A163511, A324054, A324183, A324185, A324186, A324187, A324188, A324189.

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(p+1)); n >>= 1); (t*p));
A324184(n) = sigma(A163511(n));

from sympy import nextprime
def A324184(n):
    if n:
        c, p = 1, 1
        while n:
            c *= ((p:=nextprime(p))**(s:=(~n&n-1).bit_length()+1)-1)//(p-1)
            n >>= s
        return c*(p**(s+1)-1)//(p**s-1)
    return 1 # Chai Wah Wu, Jul 25 2023

a(n) = A000203(A163511(n)).

For n >= 1, a(n) = A324054(A054429(n)).

A366881 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(A163511(i)) = A206787(A163511(j)) and A336651(A163511(n)) = A336651(A163511(j)) for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 04 2023

nonn

Restricted growth sequence transform of the ordered pair [A206787(A163511(n)), A336651(A163511(n))].
Restricted growth sequence transform of sequence b(n) = A351461(A163511(n)).
For all i, j >= 0:
  a(i) = a(j) => A324186(i) = A324186(j), (similarly for A366806)
  a(i) = a(j) => A366885(i) = A366885(j). (similarly for A366886).

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

Cf. A000265, A000593, A163511, A347385, A351461, A366887.
Cf. also A324186, A366806, A366885, A366886, A366888.
Differs from A366806 for the first time at n=105, where a(105) = 52, while A366806(105) = 19.

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));
A206787(n) = sumdiv(n, d, d*issquarefree(2*d));
A336651(n) = { my(f=factor(n>>valuation(n,2))); prod(i=1, #f~, f[i,1]^(f[i,2]-1)); };
A366881aux(n) = [A206787(A163511(n)), A336651(A163511(n))];
v366881 = rgs_transform(vector(1+up_to,n,A366881aux(n-1)));
A366881(n) = v366881[1+n];

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A366797 Number of odd divisors permuted by A163511: a(n) = A001227(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 4, 2, 2, 1, 5, 4, 4, 3, 6, 3, 3, 2, 6, 4, 4, 2, 4, 2, 2, 1, 6, 5, 5, 4, 8, 4, 4, 3, 9, 6, 6, 3, 6, 3, 3, 2, 8, 6, 6, 4, 8, 4, 4, 2, 6, 4, 4, 2, 4, 2, 2, 1, 7, 6, 6, 5, 10, 5, 5, 4, 12, 8, 8, 4, 8, 4, 4, 3, 12, 9, 9, 6, 12, 6, 6, 3, 9, 6, 6, 3, 6, 3, 3, 2, 10, 8, 8, 6, 12, 6, 6

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

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

Cf. A001227, A163511, A366798 (rgs-transform).

Cf. also A324186, A366873.

A001227(n) = numdiv(n>>valuation(n, 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));
A366797(n) = A001227(A163511(n));

a(n) = 2*A366873(n) - A324186(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)).

A366885 Dedekind psi function applied to the odd part of n, permuted by A163511: a(n) = A347385(A163511(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 12, 4, 6, 1, 36, 12, 30, 4, 24, 6, 8, 1, 108, 36, 150, 12, 120, 30, 56, 4, 72, 24, 48, 6, 32, 8, 12, 1, 324, 108, 750, 36, 600, 150, 392, 12, 360, 120, 336, 30, 224, 56, 132, 4, 216, 72, 240, 24, 192, 48, 96, 6, 96, 32, 72, 8, 48, 12, 14, 1, 972, 324, 3750, 108, 3000, 750, 2744, 36, 1800, 600, 2352

Offset: 0

Antti Karttunen, Nov 04 2023

nonn

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

Cf. A001615, A163511, A347385, A366886 (rgs-transform).

Cf. also A324186.

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));
A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A366885(n) = A347385(A163511(n));

A366892 a(n) = A336652(A163511(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 12, 3, 5, 1, 39, 12, 30, 3, 15, 5, 7, 1, 120, 39, 155, 12, 90, 30, 56, 3, 60, 15, 35, 5, 21, 7, 11, 1, 363, 120, 780, 39, 465, 155, 399, 12, 360, 90, 280, 30, 168, 56, 132, 3, 195, 60, 210, 15, 105, 35, 77, 5, 84, 21, 55, 7, 33, 11, 13, 1, 1092, 363, 3905, 120, 2340, 780, 2800, 39, 1860, 465, 1995

Offset: 0

Antti Karttunen, Nov 04 2023

nonn

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

Cf. A163511, A336652, A366893 (rgs-transform).

Cf. also A324186.

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));
A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));
A366892(n) = A336652(A163511(n));

A366894 a(n) = A336699(A163511(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1, 61, 3, 5, 7, 1, 1, 29, 1, 5, 1, 1, 1, 1, 1, 1, 1, 23, 61, 391, 3, 5, 5, 13, 7, 101, 1, 43, 1, 29, 29, 67, 1, 1, 5, 1, 1, 1, 1, 1, 1, 7, 1, 5, 1, 1, 1, 1, 1, 547, 23, 977, 61, 391, 391, 1401, 3, 127, 5, 19, 5, 13, 13, 23, 7, 39, 101, 221, 1, 43, 43, 67, 1, 371, 29, 25, 29

Offset: 0

Antti Karttunen, Jan 03 2024

nonn

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

Cf. A000265, A000593, A163511, A324186, A336699, A351565, A366895 (rgs-transform).

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));
A000265(n) = (n>>valuation(n,2));
A336699(n) = A000265(1+A000265(sigma(A000265(n))));
A366894(n) = A336699(A163511(n));

a(n) = A351565(A324186(n)).

A366895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366894(i) = A366894(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 4, 8, 3, 5, 5, 9, 2, 10, 1, 11, 1, 6, 6, 12, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 13, 7, 14, 4, 8, 8, 15, 3, 16, 5, 17, 5, 9, 9, 7, 2, 18, 10, 19, 1, 11, 11, 12, 1, 20, 6, 21, 6, 12, 12, 7, 1, 22, 1, 5, 5

Offset: 0

Antti Karttunen, Jan 03 2024

Restricted growth sequence transform of A366894.
For all i, j >= 0:
  A366881(i) = A366881(j) => A366806(i) = A366806(j) => a(i) = a(j).

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

Cf. A000265, A000593, A163511, A324186, A336699, A351565, A366894.

Cf. also A366806, A366881.

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; };
A000265(n) = (n>>valuation(n,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));
A336699(n) = A000265(1+A000265(sigma(A000265(n))));
A366894(n) = A336699(A163511(n));
v366895 = rgs_transform(vector(1+up_to,n,A366894(n-1)));
A366895(n) = v366895[1+n];

cp's OEIS Frontend

A366806 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324186(i) = A324186(j) for all i, j >= 0, where A324186 is the sum of odd divisors permuted by A163511.

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A324184 a(n) = sigma(A163511(n)).

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A366881 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(A163511(i)) = A206787(A163511(j)) and A336651(A163511(n)) = A336651(A163511(j)) for all i, j >= 0.

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A366797 Number of odd divisors permuted by A163511: a(n) = A001227(A163511(n)).

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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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A366885 Dedekind psi function applied to the odd part of n, permuted by A163511: a(n) = A347385(A163511(n)).

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A366892 a(n) = A336652(A163511(n)).

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A366894 a(n) = A336699(A163511(n)).

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A366895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366894(i) = A366894(j) for all i, j >= 0.

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