A354814 -id:A354814

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A346242

Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

+10
20

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1

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

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A346243(n) - A324198(n).

From Antti Karttunen, Jun 09 2022: (Start)

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A324198(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A008966(n).

For all n >= 1, a(A045344(n)) = -1.

(End)

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.

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };

v346242 = DirInverseCorrect(vector(up_to, n, A324198(n)));

A346242(n) = v346242[n];

CROSSREFS

Cf. A276086, A324198, A346243.

Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).

Cf. also A354347, A354348, A354823, A354824.

KEYWORD

sign,base

AUTHOR

Antti Karttunen, Jul 13 2021

STATUS

approved

A354810

Positions of zeros in A346242.

+10
3

4, 8, 16, 24, 28, 32, 40, 44, 48, 52, 64, 68, 76, 80, 88, 92, 96, 104, 116, 121, 124, 128, 136, 144, 148, 152, 160, 164, 169, 172, 176, 184, 188, 192, 208, 212, 232, 236, 240, 244, 248, 256, 268, 272, 284, 288, 289, 292, 296, 304, 312, 316, 320, 328, 332, 338, 344, 356, 361, 364, 368, 376, 384, 388, 404, 408, 412, 416

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..68.

Index entries for sequences related to primorial base

PROG

(PARI) isA354810(n) = (0==A346242(n));

CROSSREFS

Cf. A346242, A354820 (characteristic function).

Cf. also A354811, A354812, A354813, A354814, A354822.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 07 2022

STATUS

approved

A354812

Positions of +2's in A346242.

+10
3

132, 156, 204, 228, 276, 348, 372, 444, 492, 516, 564, 636, 708, 732, 804, 852, 876, 948, 996, 1068, 1164, 1212, 1236, 1284, 1308, 1356, 1524, 1572, 1644, 1668, 1788, 1812, 1884, 1956, 2004, 2076, 2148, 2172, 2292, 2316, 2364, 2388, 2532, 2676, 2724, 2748, 2796, 2868, 2892, 3012, 3084, 3156, 3228, 3252, 3324, 3372

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

OFFSET

1,1

COMMENTS

Question: Are all terms even?

LINKS

Table of n, a(n) for n=1..56.

PROG

(PARI) isA354812(n) = (2==A346242(n));

CROSSREFS

Cf. A346242.

Cf. also A354814, A354822.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 07 2022

STATUS

approved

A372564

Positions of -4's in A346242.

+10
2

72, 260, 380, 620, 740, 860, 1220, 1340, 1460, 1580, 1940, 2060, 2180, 2540, 2780, 3020, 3140, 3260, 3620, 3860, 3980, 4220, 4460, 4580, 4820, 5420, 5540, 5660, 6140, 6260, 6620, 6740, 6980, 7340, 7460, 7580, 7940, 8180, 8420, 8660, 8780, 9140, 9260, 9740, 9980, 10460, 10820, 10940, 11420, 11540, 12020, 12140, 12260

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

OFFSET

1,1

COMMENTS

Term a(94)=23256 is the first term after 72 that is not a multiple of 20. The first four odd terms 71825, 942667, 1322035, 3613027 occur at indices n=273, 3355, 4619, 11995.

Of the first 20000 terms, 13422 are multiples of 20, and apparently, for any n > 1, 20*A002476(n) is included in this sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to primorial base

PROG

(PARI)

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };

memoA346242 = Map();

A346242(n) = if(1==n, 1, my(v); if(mapisdefined(memoA346242, n, &v), v, v = -sumdiv(n, d, if(d<n, A324198(n/d)*A346242(d), 0)); mapput(memoA346242, n, v); (v)));

isA372564(n) = (-4==A346242(n));

CROSSREFS

Cf. A002476, A346242.

Cf. also A354814.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 24 2024

STATUS

approved

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