A322994 -id:A322994 - cp's OEIS frontend

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

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];

A322993 a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 3, 9, 1, 11, 1, 17, 5, 15, 1, 13, 1, 19, 9, 33, 1, 23, 3, 65, 7, 35, 1, 21, 1, 31, 17, 129, 5, 27, 1, 257, 33, 39, 1, 37, 1, 67, 11, 513, 1, 47, 3, 25, 65, 131, 1, 29, 9, 71, 129, 1025, 1, 43, 1, 2049, 19, 63, 17, 69, 1, 259, 257, 41, 1, 55, 1, 4097, 13, 515, 5, 133, 1, 79, 15, 8193, 1, 75, 33, 16385, 513, 135, 1, 45, 9

Offset: 1

Antti Karttunen, Jan 02 2019

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000265, A156552, A246277, A305897 (restricted growth sequence transform), A322994 (Möbius transform).

Cf. also A322995.

Array[#/2^IntegerExponent[#, 2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 91] (* Michael De Vlieger, Jan 03 2019 *)

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)));

a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).
For n > 1, a(n) = A156552(2*A246277(n)).
A000120(a(n)) = A001222(n) for all n >= 1.

A324542 Möbius transform of A324118, where A324118(n) = A000593(A156552(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 4, 3, 11, 1, 3, 1, 16, 4, 16, 1, 5, 1, 4, 11, 46, 1, 8, 3, 82, 4, 27, 1, 10, 1, 8, 16, 174, 4, 20, 1, 256, 46, 32, 1, 4, 1, 17, 3, 798, 1, 8, 3, 15, 82, 45, 1, 12, 11, 20, 174, 1300, 1, 2, 1, 2734, 4, 72, 16, 27, 1, 125, 256, 8, 1, 20, 1, 4354, 5, 363, 4, 25, 1, 8, 16, 10926, 1, 53, 46, 20518, 798, 168, 1, 35, 11, 317

Offset: 1

Antti Karttunen, Mar 07 2019

sign

The first three zeros after a(1) occur at n = 192, 288, 3645.

The first negative term is a(150) = -1. There are 184 negative terms among the first 4473 terms.

Cf. A000593, A008683, A156552, A324118, A324543.

Cf. also A322993, A322994.

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))));
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A322993(n) = A156552(2*A246277(n));
memoA324118 = Map();
A324118(n) = if(1==n, 0, my(v); if(mapisdefined(memoA324118,n,&v), v, v=sigma(A322993(n)); mapput(memoA324118,n,v); (v)));
A324542(n) = sumdiv(n,d,moebius(n/d)*A324118(d));

a(n) = Sum_{d|n} A008683(n/d) * A324118(d).

A329373 Dirichlet convolution of the identity function with A322993.

Original entry on oeis.org

0, 1, 1, 5, 1, 10, 1, 17, 6, 16, 1, 40, 1, 26, 13, 49, 1, 49, 1, 66, 19, 46, 1, 124, 8, 80, 25, 108, 1, 114, 1, 129, 31, 148, 17, 185, 1, 278, 49, 206, 1, 182, 1, 192, 65, 538, 1, 340, 10, 111, 85, 330, 1, 190, 25, 336, 151, 1056, 1, 428, 1, 2082, 97, 321, 35, 318, 1, 606, 283, 258, 1, 557, 1, 4136, 87, 1128, 23, 530, 1, 566, 90, 8236, 1, 684, 55, 16430

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Equally, Dirichlet convolution of sigma (A000203) with A322994 (Möbius transform of A322993).

Antti Karttunen, Table of n, a(n) for n = 1..2001
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..32768
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A000265, A156552, A322993, A322994, A324542, A329372, A329374.

A000265(n) = (n/2^valuation(n, 2));
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
A322993(n) = if(1==n,0,A000265(A156552(n)));
A329373(n) = sumdiv(n,d,(n/d)*A322993(d));

a(n) = Sum_{d|n} d * A322993(n/d).

a(n) = Sum_{d|n} A000203(n/d) * A322994(d).

A354185 Möbius transform of A348717.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 7, 1, 4, 1, 11, 3, 8, 1, 10, 1, 8, 7, 19, 1, 8, 2, 23, 4, 12, 1, 13, 1, 16, 11, 31, 3, 12, 1, 35, 19, 16, 1, 17, 1, 20, 4, 43, 1, 16, 2, 38, 23, 24, 1, 32, 7, 24, 31, 55, 1, 16, 1, 59, 8, 32, 11, 29, 1, 32, 35, 45, 1, 24, 1, 71, 10, 36, 3, 29, 1, 32, 8, 79, 1, 24, 19, 83, 43, 40, 1, 44, 7

Offset: 1

Antti Karttunen, May 19 2022

nonn

Question: Are all terms positive?

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A348717.

Cf. also A322994.

A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A354185(n) = sumdiv(n,d,moebius(n/d)*A348717(d));

a(n) = Sum_{d|n} A008683(n/d) * A348717(d).

cp's OEIS Frontend

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

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A322993 a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).

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A324542 Möbius transform of A324118, where A324118(n) = A000593(A156552(n)).

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A329373 Dirichlet convolution of the identity function with A322993.

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A354185 Möbius transform of A348717.

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