A297167 -id:A297167 - cp's OEIS frontend

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

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

A324195 Cumulative bitwise-OR of A297112(d), where d ranges over the divisors d of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 8, 7, 6, 5, 16, 7, 32, 9, 6, 15, 64, 7, 128, 15, 10, 17, 256, 15, 12, 33, 14, 27, 512, 7, 1024, 31, 18, 65, 12, 15, 2048, 129, 34, 31, 4096, 11, 8192, 51, 14, 257, 16384, 31, 24, 13, 66, 99, 32768, 15, 20, 63, 130, 513, 65536, 15, 131072, 1025, 30, 63, 36, 19, 262144, 195, 258, 13, 524288, 31, 1048576, 2049, 14, 387, 24, 35, 2097152, 63, 30

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

A324180 differs from this one in that it uses XOR instead of OR, and uses only the proper divisors of n.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000120, A003986, A297112, A297167, A297168, A061395, A324190, A324196, A324197.

Cf. also A324180.

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); };

A000120(a(n)) = A324190(n).

A324538 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324537(n) for all other numbers, except f(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 07 2019

nonn

For all i, j:
  a(i) = a(j) => A069513(i) = A069513(j),
  a(i) = a(j) => A324191(i) = A324191(j).

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

Cf. A000961 (positions of terms <= 2), A069513, A297167, A324191, A324537.

Cf. also A300827, A324181, A324203, A324196, A324197.

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));
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A324537(n) = { my(m=1); fordiv(n, d, if(d>2, m *= prime(A297167(d)))); A003557(m); };
Aux324538(n) = if(1==n,0,A324537(n));
v324538 = rgs_transform(vector(up_to,n,Aux324538(n)));
A324538(n) = v324538[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)).

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

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

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A324195 Cumulative bitwise-OR of A297112(d), where d ranges over the divisors d of n.

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A324538 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324537(n) for all other numbers, except f(1) = 0.

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

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