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A319351 Filter sequence which records the number of cyclotomic cosets of 2 mod p^k for powers of odd primes p, and for any other number assigns a unique number. 2
1, 2, 3, 4, 3, 5, 6, 7, 6, 8, 3, 9, 3, 10, 11, 12, 6, 13, 3, 14, 15, 16, 6, 17, 6, 18, 19, 20, 3, 21, 22, 23, 24, 25, 26, 27, 3, 28, 29, 30, 6, 31, 19, 32, 33, 34, 6, 35, 36, 37, 38, 39, 3, 40, 41, 42, 43, 44, 3, 45, 3, 46, 47, 48, 49, 50, 3, 51, 52, 53, 6, 54, 55, 56, 57, 58, 59, 60, 6, 61, 36, 62, 3, 63, 64, 65, 66, 67, 55, 68, 69, 70, 71, 72, 73, 74, 6, 75, 76, 77, 3, 78, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to the number of cyclotomic cosets of 2 mod p^k, while all other numbers occur in singular equivalence classes of their own.

Restricted growth sequence transform of function f defined as f(n) = A006694((n-1)/2) when n is an odd prime power > 1, otherwise -n.

For all i, j: a(i) = a(j) => A305976(i) = A305976(j). (See also A305975).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

EXAMPLE

a(7) = a(9) = a(17) = a(23) = a(25) = a(41) = ... because n = 7, 9, 17, 23, 25, 41, ... are such powers of odd primes for which A006694((n-1)/2) = 4.

PROG

(PARI)

up_to = 100000;

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

A006694(n) = (sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1); \\ From A006694

A319351aux(n) = if((n<=2)||!(n%2)||!isprimepower(n), n, -(A006694((n-1)/2)));

v319351 = rgs_transform(vector(up_to, n, A319351aux(n)));

A319351(n) = v319351[n];

CROSSREFS

Cf. A001917, A006694, A286573, A305801, A319350.

Sequence in context: A324533 A141128 A160180 * A323373 A319350 A078908

Adjacent sequences:  A319348 A319349 A319350 * A319352 A319353 A319354

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 26 2018

STATUS

approved

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Last modified May 21 02:48 EDT 2019. Contains 323434 sequences. (Running on oeis4.)