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.

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

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 A332826 A353840

Adjacent sequences: A319348 A319349 A319350 * A319352 A319353 A319354

Keyword

nonn

Author

Antti Karttunen, Sep 26 2018

Status

approved