A336469 -id:A336469

A329697

a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

+10
65

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 0, 3, 1, 3, 2, 3, 3, 3, 1, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 2, 2, 3, 3, 3, 2, 4, 3, 4, 2, 3, 3, 4, 0, 3, 3, 4, 1, 4, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 1, 4, 2, 3, 3, 2, 4, 4, 2, 3, 3, 4, 3, 4, 4, 4, 1, 2, 4, 4, 2

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FORMULA

a(A053575(n)) = A336469(n) = a(n) - A005087(n).

CROSSREFS

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.

A003401

Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.
(Formerly M0505)

+10
42

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285

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CROSSREFS

Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).

A053575

Odd part of phi(n): a(n) = A000265(A000010(n)).

+10
27

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 1, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 1, 5, 1, 3, 3, 9, 9, 3, 1, 5, 3, 21, 5, 3, 11, 23, 1, 21, 5, 1, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 1, 3, 5, 33, 1, 11, 3, 35, 3, 9, 9, 5, 9, 15, 3, 39, 1, 27, 5, 41, 3, 1, 21, 7, 5, 11, 3, 9, 11, 15, 23

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FORMULA

A336466(a(n)) = A336468(n), A329697(a(n)) = A336469(n) = A329697(n) - A005087(n).

CROSSREFS

Cf. A000010, A000265, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.

A336477

a(n) = 1 if a regular n-gon is constructible with ruler (or, more precisely, an unmarked straightedge) and compass, 0 otherwise.

+10
9

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

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FORMULA

a(n) = 1 if A005087(n) is equal to A329697(n) [i.e., if A336469(n)=0], and 0 otherwise.

CROSSREFS

Cf. A000010, A000265, A001221, A005087, A209229, A295297, A329697, A336469, A359578 (Dirichlet inverse).

A336471

Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A336158(i) = A336158(j), for all i, j >= 1.

+10
8

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 5, 1, 2, 4, 6, 2, 7, 3, 6, 2, 4, 3, 8, 3, 6, 5, 6, 1, 7, 2, 7, 4, 6, 6, 7, 2, 3, 7, 9, 3, 10, 6, 9, 2, 11, 4, 5, 3, 6, 8, 7, 3, 12, 6, 9, 5, 6, 6, 13, 1, 7, 7, 9, 2, 12, 7, 9, 4, 6, 6, 10, 6, 12, 7, 9, 2, 14, 3, 6, 7, 5, 9, 12, 3, 6, 10, 12, 6, 12, 9, 12, 2, 3, 11, 13, 4, 6, 5, 6, 3, 15

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COMMENTS

a(i) = a(j) => A336469(i) = A336469(j) => A336477(i) = A336477(j).

This sequence has an ability to see where the terms of A003401 are, as they are the indices of zeros in A336469. Specifically, they are numbers k that satisfy the condition A329697(k) = A001221(A336158(k)), i.e., numbers for which A329697(k) is equal to the number of distinct prime divisors of the odd part of k. See also comments in array A334100.

CROSSREFS

Cf. also A003602, A324400, A336159, A336160, A336396, A336469, A336470, A336472, A336473, A336477.

A336396

a(n) = A329697(n) - A087436(n).

+10
7

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 0, 1, 1, 2, 0, 0, 1, 0, 1, 2, 0, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 1, 3, 1, 0, 2, 3, 0, 2, 0, 0, 1, 2, 0, 1, 1, 2, 2, 3, 0, 2, 2, 1, 0, 1, 1, 3, 0, 2, 1, 3, 0, 2, 2, 0, 2, 2, 1, 3, 0, 0, 1, 2, 1, 0, 3, 2, 1, 2, 0, 2, 2, 2, 3, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1

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

FORMULA

a(n) = A336469(n) - A046660(A000265(n)).

CROSSREFS

Cf. A000265, A046660, A087436, A329697, A336118, A336466, A336469.

A336922

a(n) = A331410(n) - A005087(n).

+10
7

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 2, 1, 0, 1, 1, 0, 3, 1, 2, 0, 3, 1, 0, 0, 1, 2, 1, 1, 3, 2, 1, 1, 2, 0, 2, 1, 2, 1, 1, 0, 1, 3, 2, 1, 3, 2, 2, 0, 2, 3, 3, 1, 1, 0, 1, 0, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 3, 2, 1, 1, 2, 1, 3, 2, 2, 0, 3, 2, 3, 1, 4, 2, 1, 1, 0, 1, 3, 0, 2, 1, 2, 3, 4, 2, 2, 1, 1

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

CROSSREFS

Cf. also A336469.

A336928

a(n) = A329697(sigma(n)), where A329697 is totally additive with a(2) = 0 and a(p) = 1 + a(p-1) for odd primes.

+10
5

0, 1, 0, 2, 1, 1, 0, 2, 2, 2, 1, 2, 2, 1, 1, 3, 2, 3, 1, 3, 0, 2, 1, 2, 3, 3, 1, 2, 2, 2, 0, 4, 1, 3, 1, 4, 3, 2, 2, 3, 3, 1, 2, 3, 3, 2, 1, 3, 4, 4, 2, 4, 3, 2, 2, 2, 1, 3, 2, 3, 3, 1, 2, 5, 3, 2, 1, 4, 1, 2, 2, 4, 3, 4, 3, 3, 1, 3, 1, 4, 4, 4, 3, 2, 3, 3, 2, 3, 3, 4, 2, 3, 0, 2, 2, 4, 4, 5, 3, 5, 2, 3, 2, 4, 1

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

CROSSREFS

Cf. also A336469, A336694, A336927, A336929.

A336468

a(n) = A336466(phi(n)), where A336466 is fully multiplicative with a(p) = A000265(p-1) for prime p, with A000265(k) giving the odd part of k.

+10
3

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 5, 1, 1, 5, 1, 11, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1

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

CROSSREFS

Cf. A000010, A000265, A336466, A336469.