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Search: a347082 -id:a347082
Displaying 1-8 of 8 results found. page 1
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A347084 Dirichlet inverse of A129283, n + A003415(n). +10
12
1, -3, -4, 1, -6, 13, -8, 1, 1, 19, -12, -6, -14, 25, 25, 1, -18, -5, -20, -8, 33, 37, -24, -5, 1, 43, 2, -10, -30, -87, -32, 1, 49, 55, 49, 6, -38, 61, 57, -7, -42, -113, -44, -14, -8, 73, -48, -4, 1, -5, 73, -16, -54, -9, 73, -9, 81, 91, -60, 51, -62, 97, -10, 1, 85, -165, -68, -20, 97, -163, -72, 2, -74, 115 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, d<n} a(d) * A129283(n/d).
a(n) = A347085(n) - A129283(n).
a(n) = A347082(n) - A347086(n).
PROG
(PARI)
up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347084 = DirInverseCorrect(vector(up_to, n, n+A003415(n)));
A347084(n) = v347084[n];
CROSSREFS
Cf. A003415, A129283, A347082, A347085, A347086, A348995 (positions of 1's).
Cf. also A346241, A348976.
KEYWORD
sign
AUTHOR
Antti Karttunen, Aug 17 2021
STATUS
approved
A346241 Dirichlet inverse of pointwise sum of A003415 (arithmetic derivative of n) and A063524 (1, 0, 0, 0, ...). +10
11
1, -1, -1, -3, -1, -3, -1, -5, -5, -5, -1, -1, -1, -7, -6, -3, -1, -2, -1, -5, -8, -11, -1, 17, -9, -13, -16, -9, -1, 3, -1, 11, -12, -17, -10, 33, -1, -19, -14, 19, -1, 1, -1, -17, -14, -23, -1, 63, -13, -14, -18, -21, -1, 28, -14, 21, -20, -29, -1, 76, -1, -31, -22, 45, -16, -3, -1, -29, -24, -9, -1, 112, -1, -37, -22, -33 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A003415(n/d) * a(d).
PROG
(PARI)
up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415plusA063524(n) = if(n<=1, 1, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v346241 = DirInverseCorrect(vector(up_to, n, A003415plusA063524(n)));
A346241(n) = v346241[n];
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
memoA346241 = Map();
A346241(n) = if(1==n, 1, my(v); if(mapisdefined(memoA346241, n, &v), v, v = -sumdiv(n, d, if(d<n, A003415(n/d)*A346241(d), 0)); mapput(memoA346241, n, v); (v)));
CROSSREFS
Cf. A003415, A354806, A354807, A354808 (positions of negative terms), A354809 (of terms >= 0), A354818 (of even terms).
Cf. also A317835, A347082, A347084, A354821.
KEYWORD
sign
AUTHOR
Antti Karttunen, Jul 13 2021
STATUS
approved
A359790 Dirichlet inverse of function f(n) = 1 + n', where n' stands for the arithmetic derivative of n, A003415(n). +10
8
1, -2, -2, -1, -2, 2, -2, -1, -3, 0, -2, 3, -2, -2, -1, 0, -2, 6, -2, 3, -3, -6, -2, 7, -7, -8, -8, 3, -2, 12, -2, 3, -7, -12, -5, 9, -2, -14, -9, 11, -2, 18, -2, 3, 0, -18, -2, 11, -11, 6, -13, 3, -2, 26, -9, 15, -15, -24, -2, 17, -2, -26, -4, 9, -11, 30, -2, 3, -19, 16, -2, 9, -2, -32, 0, 3, -11, 36, -2, 23, -16, -36 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A003415(n/d)) * a(d).
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
memoA359790 = Map();
A359790(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359790, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A003415(n/d))*A359790(d), 0)); mapput(memoA359790, n, v); (v)));
CROSSREFS
Cf. A003415, A359780, A359781 (parity of terms), A359782 (positions of even terms), A359783 (of odd terms).
Cf. also A346241, A347082, A347084, A359603, A359789, A359791 [= a(A003961(n))] (for similar constructions).
KEYWORD
sign
AUTHOR
Antti Karttunen, Jan 13 2023
STATUS
approved
A347086 Difference between the Dirichlet inverse of -A168036, n - A003415(n) and the Dirichlet inverse of A129283, n + A003415(n), where A003415 is the Arithmetic derivative of n. +10
5
0, 2, 2, 0, 2, -10, 2, 2, 0, -14, 2, 6, 2, -18, -16, 8, 2, 6, 2, 6, -20, -26, 2, 6, 0, -30, 2, 6, 2, 74, 2, 26, -28, -38, -24, 0, 2, -42, -32, 2, 2, 94, 2, 6, 6, -50, 2, 10, 0, 6, -40, 6, 2, 14, -32, -2, -44, -62, 2, -48, 2, -66, 6, 80, -36, 134, 2, 6, -52, 130, 2, 20, 2, -78, 6, 6, -36, 154, 2, -6, 12, -86, 2, -60 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = A347082(n) - A347084(n).
PROG
(PARI)
up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347082 = DirInverseCorrect(vector(up_to, n, n-A003415(n)));
A347082(n) = v347082[n];
v347084 = DirInverseCorrect(vector(up_to, n, n+A003415(n)));
A347084(n) = v347084[n];
A347086(n) = (A347082(n)-A347084(n));
CROSSREFS
Cf. A003415, A129283, A168036, A347082, A347084.
KEYWORD
sign
AUTHOR
Antti Karttunen, Aug 17 2021
STATUS
approved
A359824 Parity of A359823, where A359823 is the Dirichlet inverse of A359820. +10
5
1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359820(n) = ((n+A003415(n))%2);
memoA359823 = Map();
A359823(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359823, n, &v), v, v = -sumdiv(n, d, if(d<n, A359820(n/d)*A359823(d), 0)); mapput(memoA359823, n, v); (v)));
A359824(n) = (A359823(n)%2);
CROSSREFS
Characteristic function of A359825.
Parity of A347082, A347084 and A359823.
Cf. A359820, A359820.
Cf. also A359764 [= a(A003961(n))].
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 14 2023
STATUS
approved
A347083 Sum of -A168036 and its Dirichlet inverse. +10
4
2, 0, 0, 1, 0, 4, 0, -1, 4, 8, 0, -4, 0, 12, 16, -7, 0, -2, 0, -6, 24, 20, 0, -19, 16, 24, 4, -8, 0, -14, 0, -21, 40, 32, 48, -18, 0, 36, 48, -33, 0, -18, 0, -12, 4, 44, 0, -58, 36, 6, 64, -14, 0, -22, 80, -47, 72, 56, 0, -29, 0, 60, 8, -47, 96, -26, 0, -18, 88, -22, 0, -62, 0, 72, 20, -20, 120, -30, 0, -108, -11, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = A347082(n) - A168036(n).
For n > 1, a(n) = Sum_{d|n, 1<d<n} A168036(d) * A347082(n/d).
PROG
(PARI)
up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A168036(n) = (A003415(n)-n);
v347082 = DirInverseCorrect(vector(up_to, n, -A168036(n)));
A347082(n) = v347082[n];
A347083(n) = (A347082(n)-A168036(n));
CROSSREFS
Cf. A003415, A168036, A347082, A347085.
KEYWORD
sign
AUTHOR
Antti Karttunen, Aug 17 2021
STATUS
approved
A347087 Sum of n and the Dirichlet inverse of n - A003415(n). +10
2
2, 1, 1, 5, 1, 9, 1, 11, 10, 15, 1, 12, 1, 21, 24, 25, 1, 19, 1, 18, 34, 33, 1, 25, 26, 39, 31, 24, 1, 17, 1, 59, 54, 51, 60, 42, 1, 57, 64, 35, 1, 23, 1, 36, 43, 69, 1, 54, 50, 51, 84, 42, 1, 59, 96, 45, 94, 87, 1, 63, 1, 93, 59, 145, 114, 35, 1, 54, 114, 37, 1, 94, 1, 111, 75, 60, 138, 41, 1, 68, 97, 123, 1, 87 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The first negative term is a(1408) = -131.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = n + A347082(n).
PROG
(PARI)
up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347082 = DirInverseCorrect(vector(up_to, n, n-A003415(n)));
A347082(n) = v347082[n];
A347087(n) = (n+A347082(n));
CROSSREFS
Cf. A003415, A168036, A347082, A347083, A347086.
KEYWORD
sign
AUTHOR
Antti Karttunen, Aug 17 2021
STATUS
approved
A359825 Positions of odd terms in A359823, where A359823 is the Dirichlet inverse of A359820. +10
2
1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 102, 104, 106, 110, 111, 114, 115, 118, 119, 120, 121, 122, 123, 126, 128, 129 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..73.
PROG
(PARI) isA359825(n) = A359824(n);
CROSSREFS
Positions of odd terms in A347082, A347084 and A359823.
Cf. A003415, A359820, A359824 (characteristic function).
Cf. also A359765, A359783.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 14 2023
STATUS
approved
page 1

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Last modified August 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)