A172246 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,6

LINKS

Table of n, a(n) for n=2..106.

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

Cf. A023022, A172245, A172246, A172247, A172248.

Sequence in context: A337346 A294902 A321516 * A087893 A079169 A345060

Adjacent sequences: A172243 A172244 A172245 * A172247 A172248 A172249

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