A302690 -id:A302690 - cp's OEIS frontend

A302694 a(n) is the smallest integer k such that A002828(k*n) = 3.

3, 3, 1, 3, 6, 1, 2, 3, 3, 3, 1, 1, 6, 1, 2, 3, 3, 3, 1, 6, 1, 1, 2, 1, 3, 3, 1, 2, 6, 1, 2, 3, 1, 3, 1, 3, 6, 1, 2, 3, 3, 1, 1, 1, 6, 1, 2, 1, 3, 3, 1, 6, 6, 1, 2, 1, 1, 3, 1, 2, 6, 1, 2, 3, 3, 1, 1, 3, 1, 1, 2, 3, 3, 3, 1, 1, 1, 1, 2, 6, 3, 3, 1, 1, 6, 1, 2, 1, 3, 3

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2018

nonn

All terms are squarefree.

			a(2) = 3 because A002828(1*2) = 2, A002828(2*2) = 1,..., and 3 is the smallest multiplier leading to A002828(3*2) = 3.

Cf. A002828, A005117, A007521, A007913, A302690.

A302694 := proc(n)
    for k from 1 do
        if A002828(k*n) = 3 then
            return k;
        end if;
    end do:
end proc:
seq(A302694(n),n=1..100) ; # R. J. Mathar, Apr 16 2018

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1;
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7;
a002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ A002828
a(n) = {my(m=1); while(a002828(m*n)!=3, m++); m; } \\ Michel Marcus, Apr 12 2018

a(n^2) = 3.

Conjecture: a(n) <= 6.

Name corrected and more terms added by Michel Marcus, Apr 12 2018

A302972 a(n) is the smallest integer r such that A002828(r*n) = 4.

Original entry on oeis.org

7, 14, 5, 7, 3, 10, 1, 14, 7, 6, 5, 5, 3, 2, 1, 7, 7, 14, 5, 3, 3, 10, 1, 10, 7, 6, 5, 1, 3, 2, 1, 14, 7, 14, 5, 7, 3, 10, 1, 6, 7, 6, 5, 5, 3, 2, 1, 5, 7, 14, 5, 3, 3, 10, 1, 2, 7, 6, 5, 1, 3, 2, 1, 7, 7, 14, 5, 7, 3, 10, 1, 14, 7, 6, 5, 5, 3, 2, 1, 3, 7, 14, 5, 3, 3

Offset: 1

Juri-Stepan Gerasimov, Apr 16 2018

nonn

All terms are squarefree.

			a(1) = 7 because A002828(1*1) = 1, A002828(2*1) = 2, A002828(3*1) = 3, A002828(5*1) = 2, A002828(6*1) = 3, ..., and 7 is the smallest positive multiplier leading to A002828(7*1) = 7.

Cf. A002828, A005117, A007913, A302690, A302694.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1;
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7;
a002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ A002828
a(n) = {my(m=1); while(a002828(m*n)!=4, m++); m; } \\ Michel Marcus, Apr 17 2018

a(n^2) = 7.
a(n^2) + A302694(n^2) + A302690(n^2) + A007913(n^2) = 13.
a(n^2)*A302694(n^2)*A302690(n^2)*A007913(n^2) = 42.

More terms from Michel Marcus, Apr 17 2018

cp's OEIS Frontend

A302694 a(n) is the smallest integer k such that A002828(k*n) = 3.

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