A066341 Sum of distinct terms in n-th row of Fermat's triangle.

1, 1, 1, 1, 8, 1, 1, 1, 12, 1, 14, 1, 16, 17, 1, 1, 20, 1, 22, 23, 24, 1, 26, 1, 28, 1, 30, 1, 94, 1, 1, 35, 36, 37, 38, 1, 40, 41, 42, 1, 130, 1, 46, 47, 48, 1, 50, 1, 52, 53, 54, 1, 56, 57, 58, 59, 60, 1, 184, 1, 64, 65, 1, 67, 202, 1, 70, 71, 214, 1, 74, 1, 76, 77, 78, 79, 238, 1

Offset: 2

Author

Wouter Meeussen, Jan 01 2002

Keywords

• easy
• nonn

Examples

			Fermat's triangle (A066340) is {1}, {1, 1}, {1, 0, 1}, {1, 1, 1, 1}, {1, 4, 3, 4, 1}, ... and the distinct terms in each row are {1}, {1}, {0, 1}, {1}, {1, 3, 4}, ... with sums 1, 1, 1, 1, 8, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..13737 (terms 2..2000 from Muniru A Asiru)

Crossrefs

Cf. A066340, A001221, A007875, A087893.

Programs

GAP

List(List(List([2..80],n->List([1..n-1],m->PowerMod(m,Phi(n),n))),Set),Sum); # Muniru A Asiru, Aug 06 2018

Mathematica

Plus@@@(Union/@Table[ (PowerMod[ #, EulerPhi[ k ], k ])&/@ Range[ k-1 ], {k, 2, 256} ]) or equivalently Table[ w=Length[ FactorInteger[ k ]];(2^(w-1)-1)*k+2^(w-1), {k, 2, 256} ]

PARI

A066341(n) = { my(ph = eulerphi(n),m=Map(),t,s=0); for(k=1,n-1,t = ((k^ph)%n); if(!mapisdefined(m, t), s += t; mapput(m,t,t))); (s); }; \\ Antti Karttunen, Aug 06 2018

Formula

Conjectures from Ridouane Oudra, Apr 05 2025: (Start)

a(n) = (n+1)*2^(omega(n)-1) - n, where omega(n) = A001221(n).

a(n) = (n+1)*A007875(n) - n.

a(n) = (n/2)*A087893(n) + A007875(n). (End)