A267000 a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists.

2, 3, 4, 5, 0, 7, 0, 0, 30, 11, 60, 13, 70, 105, 240, 17, 0, 19, 220, 231, 0, 23, 0, 650, 286, 1755, 476, 29, 2730, 31, 1824, 627, 3570, 805, 4788, 37, 646, 897, 1160, 41, 8778, 43, 1276, 11385, 8970, 47, 1776, 36309, 10850, 1581, 41860, 53, 2322, 4070, 2408, 45885, 16530, 59

Offset: 2

Author

Michel Marcus, Jan 08 2016

Keywords

• nonn

Comments

The offset is 2 like A056240 since there is no number m with A001414(m) = 1

Alladi and Erdős state that there is only a finite number of zeros in this sequence.

When a(n) is not zero, A056240(n) <= a(n); a(n) <= A000792(n).

Examples

			a(10) = 30 since A001414(30)=10 and 30 is divisible by 10, and 30/10=3 is squarefree and prime to 10.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..3000
• K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

Crossrefs

Cf. A000792, A001414, A036844, A056240, A064364.

Programs

PARI

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }
first(n) = {my(k=1); while (sopfr(k) != n, k++); k;}
last(n) = polcoeff((1+x+2*x^2+x^4)/(1-3*x^3) + O(x^(n + 3)), n);
a(n) = {na = first(n); nb = last(n); for (m=na, nb, if ((sopfr(m) == n) && (! (m % n)) && issquarefree(m/n) && (gcd(m/n, n) == 1), return(m)););}

Formula

a(p) = p, for p prime.