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

Author

Antti Karttunen, Sep 26 2018

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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];