A297112 -id:A297112 - cp's OEIS frontend

A324180 SumXOR variant of A297168.

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 3, 0, 9, 6, 7, 0, 5, 0, 3, 10, 17, 0, 3, 4, 33, 6, 3, 0, 5, 0, 15, 18, 65, 12, 7, 0, 129, 34, 15, 0, 9, 0, 3, 6, 257, 0, 3, 8, 9, 66, 3, 0, 9, 20, 23, 130, 513, 0, 15, 0, 1025, 6, 31, 36, 17, 0, 3, 258, 9, 0, 3, 0, 2049, 10, 3, 24, 33, 0, 23, 14, 4097, 0, 23, 68, 8193, 514, 39, 0, 9, 40, 3

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Cf. A061395, A156552, A297106, A297112, A297167, A297168, A324181 (rgs-transform), A324120 (number of 1-bits).

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324180(n) = { my(v=0); fordiv(n, d, if(dA297112(d)))); (v); };

a(n) = Cumulative XOR of A297112(d), where d ranges over the proper divisors d of n.

A324196 Lexicographically earliest sequence such that a(i) = a(j) => A324195(i) = A324195(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 7, 8, 9, 10, 7, 11, 12, 8, 13, 14, 7, 15, 13, 16, 17, 18, 13, 19, 20, 21, 22, 23, 7, 24, 25, 26, 27, 19, 13, 28, 29, 30, 25, 31, 32, 33, 34, 21, 35, 36, 25, 37, 38, 39, 40, 41, 13, 42, 43, 44, 45, 46, 13, 47, 48, 49, 43, 50, 51, 52, 53, 54, 38, 55, 25, 56, 57, 21, 58, 37, 59, 60, 43, 49, 61, 62, 25, 63, 64, 65, 66, 67, 13, 68, 69, 70, 71, 72, 43

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Restricted growth sequence transform of A324195.

For all i, j: a(i) = a(j) => A324197(i) = A324197(j) => A324190(i) = A324190(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324197.

Cf. also A324181.

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; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
v324196 = rgs_transform(vector(up_to, n, A324195(n)));
A324196(n) = v324196[n];

A324197 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324195(n) for all other numbers except f(2) = -1 and f(n) = -2 when n is an odd prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

For all i, j: a(i) = a(j) => A324190(i) = A324190(j).

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

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324196.

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; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
Aux324197(n) = if(isprime(n),-(n%2)-1,A324195(n));
v324197 = rgs_transform(vector(up_to, n, Aux324197(n)));
A324197(n) = v324197[n];

A329372 Dirichlet convolution of the identity function with A156552.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 8, 17, 12, 22, 16, 44, 32, 40, 32, 49, 64, 61, 128, 78, 56, 76, 256, 132, 32, 142, 50, 136, 512, 152, 1024, 129, 104, 274, 88, 209, 2048, 532, 188, 230, 4096, 256, 8192, 252, 148, 1048, 16384, 356, 80, 159, 356, 454, 32768, 240, 160, 392, 680, 2078, 65536, 504, 131072, 4128, 248, 321, 280, 464, 262144, 858, 1328, 400

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Equally, Dirichlet convolution of sigma (A000203) with A297112 (Möbius transform of A156552).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A061395, A156552, A297112, A297167, A329374.

Cf. also A329371, A329373.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A329372(n) = sumdiv(n,d,(n/d)*A156552(d));

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));

a(n) = Sum_{d|n} d * A156552(n/d).
a(n) = Sum_{d|n} A000203(n/d) * A297112(d).
A000265(a(n)) = A329374(n).

A364568 a(n) = A290077(n) - A364567(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, -2, 0, 4, 0, 12, 2, 10, 0, -6, -2, 4, 0, 16, 4, 16, 0, 26, 12, 32, 4, 84, 10, 38, 0, -20, -6, 4, -4, 24, 4, 20, 0, 44, 16, 40, 8, 104, 16, 56, 0, 78, 26, 68, 24, 152, 32, 104, 8, 262, 84, 184, 20, 468, 38, 130, 0, -48, -20, -8, -12, 16, 4, 28, -8, 40, 24, 64, 8, 168, 20, 76, 0, 88, 44, 104, 32

Offset: 0

Antti Karttunen, Aug 05 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000010, A005940, A033265, A290077, A297112, A364558, A364567.

Cf. also A364499.

A290077(n) = { my(p=2,z=1); while(n, if(!(n%2), p=nextprime(1+p), z *= (p-(1==(n%4)))); n>>=1); (z); };
A364567(n) = if(!n,n, my(i=1); while(n>1, if((n%4)!=1, i<<=1); n >>= 1); (i));
A364568(n) = (A290077(n) - A364567(n));

For n > 0, a(n) = -A364558(A005940(1+n)) = A000010(A005940(1+n)) - 2^A033265(n).

A297169 Restricted growth sequence transform of a(1) = -1, a(n) = A297168(n) for n > 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 27 2018

nonn

For all i, j: A300827(i) = A300827(j) => a(i) = a(j). - Antti Karttunen, Mar 13 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A156552, A297112, A297167, A297168, A300827.

allocatemem(2^30);
up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A297168v1(n) = if(1==n,-1,sumdiv(n,d,(dA297112(d)));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297168v1(n))),"b297169.txt");
\\ (More efficient PARI program) - Antti Karttunen, Mar 13 2018

A323914 Lexicographically earliest sequence such that a(i) = a(j) => A322994(i) = A322994(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 12 2019

nonn

Restricted growth sequence transform of A322994.

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

Cf. A156552, A297112, A322993, A322994.

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/2^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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A322993(n) = if(1==n,0,A000265(A156552(n)));
A322994(n) = sumdiv(n,d,moebius(n/d)*A322993(d));
v323914 = rgs_transform(vector(up_to,n,A322994(n)));
A323914(n) = v323914[n];

A329374 a(1) = 0; for n > 1, a(n) = A000265(A329372(n)), where A329372 is Dirichlet convolution of the identity function with A156552.

Original entry on oeis.org

0, 1, 1, 5, 1, 3, 1, 17, 3, 11, 1, 11, 1, 5, 1, 49, 1, 61, 1, 39, 7, 19, 1, 33, 1, 71, 25, 17, 1, 19, 1, 129, 13, 137, 11, 209, 1, 133, 47, 115, 1, 1, 1, 63, 37, 131, 1, 89, 5, 159, 89, 227, 1, 15, 5, 49, 85, 1039, 1, 63, 1, 129, 31, 321, 35, 29, 1, 429, 83, 25, 1, 605, 1, 4115, 111, 409, 15, 101, 1, 307, 45, 8213, 1, 13, 65, 8203, 655, 179, 1, 335, 25

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..32768
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A061395, A156552, A297112, A297167, A329372, A329373.

A000265(n) = (n/2^valuation(n, 2));
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));
A329374(n) = if(1==n, 0, A000265(A329372(n)));

a(1) = 0; and for n > 1, a(n) = A000265(A329372(n)).

A329034 Möbius transform of A122111.

Original entry on oeis.org

1, 1, 3, 1, 7, 1, 15, 2, 5, 3, 31, 3, 63, 7, 7, 2, 127, 4, 255, 7, 17, 15, 511, 2, 19, 31, 16, 15, 1023, 7, 2047, 4, 37, 63, 31, 2, 4095, 127, 77, 6, 8191, 15, 16383, 31, 27, 255, 32767, 6, 65, 14, 157, 63, 65535, 4, 69, 14, 317, 511, 131071, 1, 262143, 1023, 59, 2, 145, 31, 524287, 127, 637, 25, 1048575, 8, 2097151, 2047, 38, 255, 115, 63

Offset: 1

Antti Karttunen, Nov 08 2019

sign

a(144) = -2 is the first negative term.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A122111, A329035.

Cf. also A297112.

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)));
A329034(n) = sumdiv(n,d,moebius(n/d)*A122111(d));

a(n) = Sum_{d|n} A008683(n/d)*A122111(d).

A324285 a(n) = A002487(A297168(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 3, 0, 4, 2, 3, 0, 4, 0, 5, 3, 5, 0, 4, 1, 6, 2, 7, 0, 5, 0, 4, 4, 7, 2, 7, 0, 8, 5, 7, 0, 7, 0, 9, 3, 9, 0, 5, 1, 5, 6, 11, 0, 8, 3, 10, 7, 10, 0, 9, 0, 11, 5, 5, 4, 13, 0, 13, 8, 6, 0, 10, 0, 12, 4, 15, 2, 19, 0, 9, 3, 13, 0, 11, 5, 14, 9, 13, 0, 11, 3, 17, 10, 15, 6, 6, 0, 6, 7, 9, 0, 25, 0, 16, 5

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Cf. A002487, A156552, A297168.

Cf. also A323902, A324115, A324116, A324049.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A297168(n) = sumdiv(n,d,(dA297112(d)); \\ Antti Karttunen, Feb 22 2019
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); };
A324285(n) = A002487(A297168(n));

a(n) = A002487(A297168(n)).

cp's OEIS Frontend

A324180 SumXOR variant of A297168.

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A324196 Lexicographically earliest sequence such that a(i) = a(j) => A324195(i) = A324195(j) for all i, j >= 1.

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A324197 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324195(n) for all other numbers except f(2) = -1 and f(n) = -2 when n is an odd prime.

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A329372 Dirichlet convolution of the identity function with A156552.

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A364568 a(n) = A290077(n) - A364567(n).

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A297169 Restricted growth sequence transform of a(1) = -1, a(n) = A297168(n) for n > 1.

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A323914 Lexicographically earliest sequence such that a(i) = a(j) => A322994(i) = A322994(j), for all i, j >= 1.

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A329374 a(1) = 0; for n > 1, a(n) = A000265(A329372(n)), where A329372 is Dirichlet convolution of the identity function with A156552.

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A329034 Möbius transform of A122111.

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