Search: a347082 -id:a347082
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, d<n} a(d) * A129283(n/d).
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PROG
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(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)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A346241
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Dirichlet inverse of pointwise sum of A003415 (arithmetic derivative of n) and A063524 (1, 0, 0, 0, ...).
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+10
11
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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
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text;
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OFFSET
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1,4
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LINKS
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FORMULA
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a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A003415(n/d) * a(d).
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PROG
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(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)));
(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)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A359790
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Dirichlet inverse of function f(n) = 1 + n', where n' stands for the arithmetic derivative of n, A003415(n).
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+10
8
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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
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text;
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A003415(n/d)) * a(d).
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PROG
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(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)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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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)
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OFFSET
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1,2
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LINKS
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FORMULA
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PROG
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(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)));
v347084 = DirInverseCorrect(vector(up_to, n, n+A003415(n)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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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;
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listen;
history;
text;
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OFFSET
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1
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LINKS
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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)));
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CROSSREFS
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Characteristic function of A359825.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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;
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OFFSET
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1,1
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LINKS
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FORMULA
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PROG
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(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, -A168036(n)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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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
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listen;
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text;
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OFFSET
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1,1
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COMMENTS
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The first negative term is a(1408) = -131.
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LINKS
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FORMULA
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PROG
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(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)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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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)
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OFFSET
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1,2
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LINKS
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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