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 A245907 Let D = {d(n,i)}, i = 1..q, denote the set of divisors of n; then a(n) = number of multiplicative groups G(n,p) = D/kZ, 1 < k < n. 1
 0, 1, 1, 3, 0, 4, 3, 3, 1, 7, 2, 7, 1, 3, 4, 8, 1, 11, 2, 3, 2, 9, 2, 7, 1, 9, 2, 15, 0, 13, 7, 3, 1, 7, 2, 11, 3, 3, 2, 19, 1, 15, 2, 5, 2, 11, 2, 13, 1, 3, 3, 15, 2, 7, 4, 5, 2, 15, 1, 15, 1, 6, 8, 7, 1, 15, 5, 3, 1, 29, 3, 14, 2, 5, 4, 9, 2, 23, 3, 13, 1, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 COMMENTS We introduce the structure of a finite group in order to study the divisors of each integer. We see that the study of the classification of the divisors is dependent on the values k. The trivial group {1} is counted. The principle of the algorithm is to compute all the products d(n,i)/kZ * d(n,j)/kZ and also the inverse of each element such that if x is in the group, then there exists x’ in the group with x*x’ = 1. An interesting property: a(n)= 0 for n = 2, 6, 30, 186, 366, 426, 606, 786, 1266, 1446, 1626, 1686, ... where n>30 is of the form n = 6*q with q prime of the form (10*k + 1) => q = 31, 61, 71, 101, 131, 211, 241, 271, 281, 311, 421, 491, ... a(n) = 1 for n = 3, 4, 10, 14, 18, 26, 34, 42, 50, 60, 62, 66, ... LINKS Michel Lagneau, Table of n, a(n) for n = 2..2000 Eric Weisstein's World of Mathematics, Finite Group Wikipedia, Finite group EXAMPLE a(133) = 11 because there exist eleven finite groups formed from the four divisors {1,7,19,133} of 133. The eleven finite groups G(133,p) are: G(133,2) = {1} G(133,3} = {1} G(133,4} = {1,3} G(133,5} = {1,2,3,4} G(133,6} = {1} G(133,8} = {1,3,5,7} G(133,10} = {1,3,7,9} G(133,12} = {1,7} G(133,15} = {1,4,7,13} G(133,24} = {1,7,13,19} G(133,30} = {1,7,13,19} MAPLE with(numtheory): for n from 2 to 100 do:   x:=divisors(n):n1:=nops(x):ind:=0:     for p from 2 to n-1 do:       lst:={}:         for i from 1 to n1 do:          lst:=lst union {irem(x[i], p)}:         od:          n2:=nops(lst):lst1:={}:           for a from 1 to n2 do:             for b from 1 to n2 do:              lst1:=lst1 union {irem(lst[a]*lst[b], p)}:             od:           od:           if lst1=lst           then           n3:=nops(lst1):lst2:={}:             for c from 1 to n3 do:               for d from 1 to n3 do:                if irem(lst1[c]*lst1[d], p)=1                then lst2:=lst2 union {lst1[c]}:                else                fi:               od:              od:              if lst2=lst               then               ind:=ind+1:               else              fi:            fi:          od:         printf(`%d, `, ind):       od: CROSSREFS Sequence in context: A113486 A108572 A322335 * A104686 A301428 A104514 Adjacent sequences:  A245904 A245905 A245906 * A245908 A245909 A245910 KEYWORD nonn AUTHOR Michel Lagneau, Nov 13 2014 STATUS approved

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Last modified January 28 06:29 EST 2020. Contains 331317 sequences. (Running on oeis4.)