A364574 -id:A364574

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A364575

a(n) = A364574(A005940(1+n)), where A364574 is the Dirichlet inverse of A005941 [the inverse permutation of A005940].

+20
3

1, -2, -3, 0, -5, 6, 2, 0, -9, 10, 19, 0, 12, -4, 0, 0, -17, 18, 35, 0, 69, -38, -22, 0, 56, -24, -64, 0, -24, 0, 0, 0, -33, 34, 67, 0, 133, -70, -42, 0, 265, -138, -339, 0, -276, 44, 8, 0, 240, -112, -288, 0, -640, 128, 124, 0, -336, 48, 176, 0, 48, 0, 0, 0, -65, 66, 131, 0, 261, -134, -82, 0, 521, -266, -659, 0

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

OFFSET

0,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16383

FORMULA

a(n) = A364574(A005940(1+n)).

PROG

(PARI)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };

A005941(n) = { my(f=factor(n), p, 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]); (1+res) }; \\ (After David A. Corneth's program for A156552)

memoA364574 = Map();

A364574(n) = if(1==n, 1, my(v); if(mapisdefined(memoA364574, n, &v), v, v = -sumdiv(n, d, if(d<n, A005941(n/d)*A364574(d), 0)); mapput(memoA364574, n, v); (v)));

A364575(n) = A364574(A005940(1+n));

CROSSREFS

Cf. A005940, A005941, A085405 (reduced modulo 2), A364574.

Cf. also A324052, A324640 (scatter plots).

KEYWORD

sign,look

AUTHOR

Antti Karttunen, Aug 05 2023

STATUS

approved

A364952

Dirichlet inverse of A364557, which is Möbius transform of A005941.

+10
2

1, -1, -2, -1, -4, 2, -8, -1, 0, 4, -16, 2, -32, 8, 12, -1, -64, 0, -128, 4, 24, 16, -256, 2, 8, 32, 0, 8, -512, -12, -1024, -1, 48, 64, 56, 0, -2048, 128, 96, 4, -4096, -24, -8192, 16, -8, 256, -16384, 2, 48, -8, 192, 32, -32768, 0, 112, 8, 384, 512, -65536, -12, -131072, 1024, -16, -1, 224, -48, -262144, 64, 768

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

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

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} A364557(n/d) * a(d).

a(p) = -A000079(A000720(p)-1) for all primes p.

PROG

(PARI)

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

memoA364952 = Map();

A364952(n) = if(1==n, 1, my(v); if(mapisdefined(memoA364952, n, &v), v, v = -sumdiv(n, d, if(d<n, A364557(n/d)*A364952(d), 0)); mapput(memoA364952, n, v); (v)));

CROSSREFS

Cf. A000079, A000720, A005941, A364557, A364953.

Cf. also A364574.

KEYWORD

sign

AUTHOR

Antti Karttunen, Aug 29 2023

STATUS

approved

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