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A172246
Consider all ways of writing n = a + b where a<=b, gcd(a,b,n)=1, and having the same value of the function N(a,b,n) defined as product of the distinct prime divisors of a*b*n. Let w be the list of frequencies of distinct terms N(a, b, n) found among these partitions. If no frequency is more than 1 then a(n) = 0, otherwise a(n) = [the sum of elements of w] - [the number of elements of w] + 1.
4
0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 3, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 3, 0, 2, 2, 3, 0, 3, 0, 0, 0, 5, 0, 4, 0, 0, 0, 4, 0, 4, 0, 2, 2, 3, 0, 5, 0, 2, 0, 4, 0, 4, 0, 0, 0, 7, 0, 7, 0, 0, 0, 5, 0, 6, 0, 0, 0, 4, 0, 3, 0, 0, 0, 5, 0, 5, 0, 3, 0, 4, 0, 5, 0, 3, 0, 5, 0, 6, 0, 0, 0
OFFSET
2,6
EXAMPLE
a(7)=2 because we have two partitions 7=1+6 and 7=3+4 with different values of N(a,b,n) respectively 1*2*3*7=42 and 2*3*7=42.
From David A. Corneth, Aug 25 2020: (Start)
a(23) = 4 as the terms occurring more than once (so omitting those that occur just once) listed with repetitions is (690, 690, 690, 966, 966) and so the list of frequencies is (3, 2) as 690 occurs thrice and 966 occurs twice.
As at least one of these multiplicities is larger than 1, a(n) = [sum of numbers in (3, 2)] - [number of numbers in (3, 2)] + 1 = (3 + 2) - 2 + 1 = 4. (End)
PROG
(PARI) a(n) = {l = List(); m = Map(); my(res = 0); for(i = 1, n\2, if(gcd([i, (n-i), n]) == 1, c = factorback(factor(i*(n-i)*n)[, 1]); listput(l, c); if(mapisdefined(m, c), mapput(m, c, mapget(m, c) + 1); , mapput(m, c, 1) ) ) ); l = Set(l); for(i = 1, #l, if(mapget(m, l[i]) > 1, res+=mapget(m, l[i]); ) ); w = vector(#l, i, mapget(m, l[i])); w = select(x -> x > 1, w); if(#w == 0, return(0)); w = vecsort(Vec(w)); return(vecsum(w) - #w + 1); } \\ David A. Corneth, Aug 25 2020
CROSSREFS
KEYWORD
nonn,less
AUTHOR
Artur Jasinski, Jan 29 2010
EXTENSIONS
Edited by David A. Corneth, Aug 25 2020, and N. J. A. Sloane, Sep 05 2020
STATUS
approved