A064664 -id:A064664 - cp's OEIS frontend

A349614 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation, A064413) with the Dirichlet inverse of A064413.

1, 0, 1, -3, 7, -7, 2, 6, -8, -10, 5, 9, 14, 2, -41, -1, 17, 27, 15, -6, -38, -18, 13, 10, -32, -29, 18, 33, 18, 62, 29, -13, -31, -53, -107, 25, 48, -51, -86, 13, 30, 116, 58, 23, 88, -34, 37, -47, -30, 56, -113, 3, 45, -39, -137, -154, -73, -67, 41, 160, 84, -91, 174, 56, -154, 152, 91, 6, -113, 246, 58, -185, 56

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Obviously, convolving this with A064413 gives its inverse permutation A064664.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A349400, A349613 (Dirichlet inverse), A349615 (sum with it), A349617.

Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
memoA349400 = Map();
A349400(n) = if(1==n,1,my(v); if(mapisdefined(memoA349400,n,&v), v, v = -sumdiv(n,d,if(dA064413(n/d)*A349400(d),0)); mapput(memoA349400,n,v); (v)));
A349614(n) = sumdiv(n,d,A064664(d)*A349400(n/d));

a(n) = Sum_{d|n} A064664(d) * A349400(n/d).

A304526 Möbius transform of A064664, the inverse of EKG-sequence.

Original entry on oeis.org

1, 1, 4, 1, 9, -2, 13, 5, 1, -2, 19, 2, 27, -2, -3, 9, 32, 7, 36, 8, -3, -2, 42, 4, 14, -2, 16, 12, 56, 15, 60, 14, -3, -2, 2, 15, 66, -2, -3, 17, 73, 23, 80, 21, 27, -2, 88, 20, 36, 23, -3, 19, 99, 20, 24, 18, -3, -2, 106, 18, 114, -2, 35, 33, 17, 41, 127, 32, -3, 33, 133, 24, 137, -2, 44, 34, 45, 44, 150, 23, 54, -2, 159, 27, 42, -2, -3, 33, 166, 25, 52

Offset: 1

Antti Karttunen, May 18 2018

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Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A000010, A008683, A064423, A064664, A304527, A327867 (even bisection), A323411, A349617.

A304526(n) = sumdiv(n, d, moebius(n/d)*A064664(d));

a(n) = Sum_{d|n} A008683(n/d)*A064664(d).
a(n) = A064664(n) - A304527(n).
For all n >= 1, a(A000040(n)) = A064423(n).
For n >= 2, a(2*A000040(n)) = -2.
For n >= 3, a(3*A000040(n)) = -3.
a(n) = Sum_{d|n} A000010(n/d)*A349617(d). - Antti Karttunen, Jan 27 2024

A304527 Difference between A064664 (the inverse of EKG-sequence) and its Möbius-transform.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 3, 5, 11, 1, 5, 1, 15, 14, 8, 1, 5, 1, 10, 18, 21, 1, 12, 10, 29, 6, 14, 1, 8, 1, 17, 24, 34, 23, 15, 1, 38, 32, 23, 1, 12, 1, 20, 12, 44, 1, 25, 14, 23, 37, 28, 1, 28, 29, 31, 41, 58, 1, 34, 1, 62, 16, 31, 37, 18, 1, 33, 47, 22, 1, 39, 1, 68, 25, 37, 33, 26, 1, 49, 22, 75, 1, 50, 42, 82, 61, 46, 1, 58, 41, 43, 65, 90, 46, 59, 1

Offset: 1

Antti Karttunen, May 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A000040, A008683, A064664, A064955, A304526.

A304527(n) = -sumdiv(n, d, (dA064664(d));

a(n) = A064664(n) - A304526(n).
a(n) = Sum_{d|n, dA304526(d).
a(n) = -Sum_{d|n, dA008683(n/d)*A064664(d).
For n >= 2, a(2*A000040(n))-1 = a(4*A000040(n)) = A064955(n). - Antti Karttunen, Dec 04 2022

A323411 Dirichlet inverse of A064664, the inverse permutation of EKG-sequence.

Original entry on oeis.org

1, -2, -5, 1, -10, 16, -14, -4, 19, 31, -20, -21, -28, 43, 89, 4, -33, -98, -37, -42, 125, 61, -43, 48, 76, 85, -87, -58, -57, -409, -61, -1, 179, 100, 255, 203, -67, 112, 251, 98, -74, -573, -81, -85, -559, 130, -89, -100, 146, -370, 296, -107, -100, 548, 347, 145, 332, 172, -107, 846, -115, 184, -783, 3, 506, -825, -128

Offset: 1

Antti Karttunen, Jan 13 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A323412.

Cf. also A304526, A304527, A317843, A318664, A318665.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From precomputed file.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];

A323412 Sum of the inverse permutation of EKG-sequence, A064664, and its Dirichlet inverse, A323411.

Original entry on oeis.org

2, 0, 0, 4, 0, 20, 0, 4, 25, 40, 0, -14, 0, 56, 100, 21, 0, -86, 0, -24, 140, 80, 0, 64, 100, 112, -65, -32, 0, -386, 0, 30, 200, 132, 280, 233, 0, 148, 280, 138, 0, -538, 0, -44, -520, 172, 0, -55, 196, -324, 330, -60, 0, 596, 400, 194, 370, 228, 0, 898, 0, 244, -732, 67, 560, -766, 0, -70, 430, -1068, 0, -380, 0, 268, -1040, -78, 560

Offset: 1

Antti Karttunen, Jan 13 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A064664, A323411.

Cf. also A304527, A323365.

A323412(n) = (A064664(n)+A323411(n)); \\ Uses also the program code given in A323411.

A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

Original entry on oeis.org

1, 5, 6, 11, 10, 24, 39, 22, 15, 14, 25, 69, 21, 20, 53, 130, 76, 51, 29, 28, 54, 112, 97, 50, 78, 96, 34, 33, 84, 209, 232, 38, 37, 85, 153, 44, 43, 111, 156, 179, 109, 58, 57, 142, 383, 140, 148, 352, 281, 124, 249, 299, 93, 118, 218, 62, 61, 143, 172, 68, 67, 173, 641, 696, 162, 75, 74, 210, 227, 238, 191, 535, 82, 81

Offset: 1

Antti Karttunen, May 31 2018

nonn

Permutation of A064957.

Cf. A305294 (a left inverse).

Cf. A003961, A064413, A064423, A064664, A064955, A064957.

a(n) = A064664(A003961(A064413(n))).
Other identities. For all n >= 1:
A305294(a(n)) = n.
a(A064955(n)) = A064955(1+n).
For all n >= 2, a(A064423(n)) = 1+A064955(1+n).

A349617 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 2, -1, 5, -6, 7, 2, -9, -11, 9, 2, 15, -15, -29, 1, 16, 18, 18, 5, -41, -21, 20, -4, -26, -29, 4, 7, 28, 64, 30, -3, -61, -34, -80, 9, 30, -38, -81, -6, 33, 92, 38, 14, 51, -44, 42, 10, -48, 53, -99, 6, 47, 4, -102, -17, -111, -58, 48, -4, 54, -62, 69, 2, -151, 146, 61, 18, -131, 157, 63, -3, 65, -68, 92, 18

Offset: 1

Antti Karttunen, Nov 23 2021

Dirichlet convolution of this sequence with A000010 (Euler phi) is A304526 (Möbius transform of the inverse permutation of EKG-sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A055615, A064413, A064664, A349616 (Dirichlet inverse).

Cf. also A000010, A304526, A349614.

A055615(n) = (n*moebius(n));
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
A349617(n) = sumdiv(n,d,A064664(d)*A055615(n/d));

a(n) = Sum_{d|n} A064664(d) * A055615(n/d).

A255482 a(n) = A064664(n)-A255479(n).

Original entry on oeis.org

0, 0, 2, -1, 0, -1, 1, 2, -1, 1, -1, -2, 4, 2, -1, 1, 2, -2, -1, 0, 0, -4, 0, -4, -6, 5, 5, 1, 6, -5, 2, 4, 2, 3, -7, -7, 0, 0, 3, 6, -4, 0, 0, 2, -3, 1, -1, 1, -2, 0, 1, -1, -1, 1, -5, -1, -2, 7, -1, -3, -4, 3, -3, 0, -6

Offset: 1

N. J. A. Sloane, Feb 28 2015

sign

Gives an idea of how similar A255582 is to A064413.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A064413, A255582, A064664, A255479.

a255482 n = a064664 n - a255479 n  -- Reinhard Zumkeller, Mar 10 2015

A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 1, 5, 5, 4, 3, 10, 10, 9, 2, 1, 5, 14, 14, 13, 8, 4, 6, 11, 10, 20, 20, 19, 3, 1, 28, 28, 27, 9, 33, 33, 32, 7, 5, 14, 37, 37, 36, 2, 6, 20, 8, 10, 24, 18, 4, 15, 21, 11, 43, 43, 42, 13, 57, 57, 56, 3, 1, 28, 61, 61, 60, 12, 19, 33, 5, 67, 67, 66, 17, 9, 25, 14, 74, 74, 73, 7, 29, 34, 6, 37, 81, 81, 80, 2

Offset: 1

Antti Karttunen, May 31 2018

nonn

Cf. A064413, A064423, A064955, A064664, A064989, A305293.

a(n) = A064664(A064989(A064413(n))).
For all n >= 1, a(A305293(n)) = n.
For all n >= 2, a(A064423(n)) = a(A064955(n)) = A064955(n-1).
For all n >= 3, a(1+A064955(n)) = A064423(n-1).

A318664 Numerators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.

Original entry on oeis.org

1, 1, 5, 1, 5, -1, 7, 3, -1, -1, 10, 3, 14, -1, -7, 5, 33, 59, 37, 9, -10, -1, 43, -1, -1, -1, 181, 13, 57, 89, 61, 15, -29, -1, -45, 31, 67, -1, -41, 1, 37, 129, 81, 11, 301, -1, 89, 21, 1, 26, -97, 10, 50, -93, -47, -5, -109, -1, 107, -33, 115, -1, 411, 15, -43, 201, 64, 33, -127, 56, 67, 181, 69, -1, 283, 35, -31, 255, 151, 7

Offset: 1

Antti Karttunen, Sep 01 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to EKG sequence

Cf. A064664, A304526, A304527, A305293, A305294, A318665 (denominators).

Cf. also A317929, A317930.

v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From b-file of A064413 prepared beforehand.
A064413(n) = v064413[n];
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
up_to = (2^14);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318664_65 = DirSqrt(vector(up_to, n, A064664(n)));
A318664(n) = numerator(v318664_65[n]);
A318665(n) = denominator(v318664_65[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064664(n) - Sum_{d|n, d>1, d 1.

For n >= 2, a(2*A000040(n)) = -1.

cp's OEIS Frontend

A349614 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation, A064413) with the Dirichlet inverse of A064413.

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A304526 Möbius transform of A064664, the inverse of EKG-sequence.

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A304527 Difference between A064664 (the inverse of EKG-sequence) and its Möbius-transform.

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A323411 Dirichlet inverse of A064664, the inverse permutation of EKG-sequence.

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A323412 Sum of the inverse permutation of EKG-sequence, A064664, and its Dirichlet inverse, A323411.

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A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

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A349617 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation) with A055615 (Dirichlet inverse of n).

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A255482 a(n) = A064664(n)-A255479(n).

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A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

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A318664 Numerators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.