A359790 -id:A359790

A359781

Parity of A359780, where A359780 is the Dirichlet inverse of the characteristic function of the numbers with even arithmetic derivative (A003415).

+10
8

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0

(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

FORMULA

a(n) = A359780(n) mod 2.

a(n) <= A358680(n). [See comments in A359780]

a(A056913(n)) = 1.

PROG

(PARI) A359781(n) = (A359780(n)%2);

CROSSREFS

Parity of A359780 and of A359790.

Characteristic function of A359783, whose complement is A359782.

Cf. A056913, A358680, A359784 (where differs from A358680).

Cf. also A359764 [= a(A003961(n)], A359774.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 13 2023

STATUS

approved

A359789

Dirichlet inverse of A036288, where A036288(n) = 1 + sopfr(n), where sopfr is the sum of prime divisors with repetition, A001414.

+10
7

1, -3, -4, 4, -6, 18, -8, -4, 9, 28, -12, -40, -14, 38, 39, 4, -18, -63, -20, -64, 53, 58, -24, 64, 25, 68, -18, -88, -30, -253, -32, -4, 81, 88, 83, 216, -38, 98, 95, 104, -42, -347, -44, -136, -144, 118, -48, -88, 49, -175, 123, -160, -54, 180, 127, 144, 137, 148, -60, 820, -62, 158, -198, 4, 149, -535

(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 > 1, a(n) = -Sum_{d|n, d<n} A036288(n/d) * a(d).

PROG

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

memoA359789 = Map();

A359789(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359789, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A001414(n/d))*A359789(d), 0)); mapput(memoA359789, n, v); (v)));

CROSSREFS

Cf. A001414, A036288, A359774 (parity of terms).

Cf. also A359773, A359788, A359790, A359791.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jan 15 2023

STATUS

approved

A359791

Dirichlet inverse of function f(n) = 1 + A349905(n), where A349905(n) is the arithmetic derivative of prime shifted n.

+10
7

1, -2, -2, -3, -2, -1, -2, -8, -7, -3, -2, 0, -2, -7, -5, -16, -2, 0, -2, -4, -9, -9, -2, 23, -11, -13, -40, -12, -2, 12, -2, -16, -11, -15, -11, 42, -2, -19, -15, 21, -2, 12, -2, -16, -24, -25, -2, 128, -19, -12, -17, -24, -2, 67, -13, 17, -21, -27, -2, 105, -2, -33, -48, 48, -17, 12, -2, -28, -27, 0, -2, 224

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A349905(n/d)) * a(d).

a(n) = A359790(A003961(n)).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

A349905(n) = A003415(A003961(n));

memoA359791 = Map();

A359791(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359791, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A349905(n/d))*A359791(d), 0)); mapput(memoA359791, n, v); (v)));

CROSSREFS

Cf. A003415, A003961, A349905, A359790.

Cf. A359764 (parity of terms), A359765 (positions of odd terms), A359766 (of even terms).

Cf. also A359169.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jan 13 2023

STATUS

approved

A359783

Positions of odd terms in A359780.

+10
6

1, 4, 8, 9, 12, 15, 20, 21, 24, 25, 28, 32, 33, 35, 36, 39, 40, 44, 48, 49, 51, 52, 55, 56, 57, 60, 64, 65, 68, 69, 72, 76, 77, 80, 84, 85, 87, 88, 91, 92, 93, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 129, 132, 133, 135, 136, 140, 141, 143, 144, 145, 148, 152, 155, 156, 159, 160, 161

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..72.

PROG

(PARI) isA359783(n) = A359781(n);

CROSSREFS

Positions of odd terms in A359780 and in A359790.

Cf. A003415, A056913 (subsequence), A358680, A359780, A359781 (characteristic function), A359782 (complement).

Setwise difference A235992\{0} \ A359784.

Cf. also A359765, A359825.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 13 2023

STATUS

approved

A359782

Positions of even terms in A359780.

+10
5

2, 3, 5, 6, 7, 10, 11, 13, 14, 16, 17, 18, 19, 22, 23, 26, 27, 29, 30, 31, 34, 37, 38, 41, 42, 43, 45, 46, 47, 50, 53, 54, 58, 59, 61, 62, 63, 66, 67, 70, 71, 73, 74, 75, 78, 79, 81, 82, 83, 86, 89, 90, 94, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 122, 125, 126, 127, 128, 130, 131

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..75.

PROG

(PARI) isA359782(n) = !(A359781(n));

CROSSREFS

Positions of even terms in A359780 and in A359790, positions of 0's in A359781.

Cf. A358680, A359783 (complement).

Union of A235991 and A359784.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 13 2023

STATUS

approved

A369978

Dirichlet inverse of sequence b(n) = 1+A083345(n), where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

+10
4

1, -2, -2, 2, -2, 2, -2, -4, 1, 0, -2, 3, -2, -2, -1, 9, -2, 4, -2, 9, -3, -6, -2, -8, 1, -8, 2, 15, -2, 12, -2, -18, -7, -12, -5, -14, -2, -14, -9, -22, -2, 18, -2, 27, 10, -18, -2, 20, 1, 10, -13, 33, -2, -8, -9, -36, -15, -24, -2, -16, -2, -26, 14, 36, -11, 30, -2, 45, -19, 16, -2, 22, -2, -32, 12, 51, -11, 36

(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 > 1, a(n) = -Sum_{d|n, d<n} (1+A083345(n/d)) * a(d).

PROG

(PARI)

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

memoA369978 = Map();

A369978(n) = if(1==n, 1, my(v); if(mapisdefined(memoA369978, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A083345(n/d))*A369978(d), 0)); mapput(memoA369978, n, v); (v)));

CROSSREFS

Cf. A003415, A083345, A369001, A369974, A369975 (parity of terms), A369976 (positions of odd terms).

Cf. A359790 and A366265 for similar sequences.

KEYWORD

sign

AUTHOR

Antti Karttunen, Feb 09 2024

STATUS

approved

A359795

Dirichlet inverse of function f(n) = 1 + A048675(n), where A048675(n) is fully additive with a(p) = 2^(1-PrimePi(p)).

+10
1

1, -2, -3, 1, -5, 8, -9, 0, 4, 14, -17, -7, -33, 26, 23, 0, -65, -16, -129, -13, 43, 50, -257, 2, 16, 98, -4, -25, -513, -84, -1025, 0, 83, 194, 77, 24, -2049, 386, 163, 4, -4097, -160, -8193, -49, -52, 770, -16385, 0, 64, -64, 323, -97, -32769, 24, 149, 8, 643, 1538, -65537, 115, -131073, 3074, -100

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: the only odd term that occurs more than once is 1 = a(1) = a(4).

LINKS

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

Index entries for sequences computed from indices in prime factorization

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A048675(n/d)) * a(d).

For n >= 1, a(3^2n) = 4 and a(3^(2n+1)) = -4.

PROG

(PARI)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

memoA359795 = Map();

A359795(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359795, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A048675(n/d))*A359795(d), 0)); mapput(memoA359795, n, v); (v)));

CROSSREFS

Cf. A000720, A048675, A091428 (positions of odd terms), A359592 (parity of terms).

Cf. also A353348, A359603, A359789, A359790, A359791.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jan 26 2023

STATUS

approved