A306986 Number of primitive abundant numbers (A071395) < 10^n.

0, 3, 14, 98, 441, 1734, 8667, 41653, 213087, 1123424

Offset: 1

Author

Amiram Eldar, Mar 18 2019

Keywords

• nonn
• more

Examples

			There are 3 terms of A071395 below 100 (20, 70, and 88), thus a(2) = 3.
		

Links

• Michael R. Avidon, On the distribution of primitive abundant numbers, Acta Arithmetica, Vol. 77, No. 2 (1996), pp. 195-205.
• Paul Erdős, On primitive abundant numbers, J. London Math. Soc., Volume s1-10, Issue 1 (1935), pp. 49-58, alternative link.
• Aleksandar Ivić, The distribution of primitive abundant numbers, Studia Sci. Math. Hungar., Vol. 20 (1985), pp. 183-187.

Crossrefs

Cf. A071395, A302992.

Programs

Mathematica

paQ[n_] := DivisorSigma[1, n] > 2n && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ n] == 1;  c = 0; k = 1;  seq={}; Do[ While[ k < 10^n, If[ paQ[k], c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq (* after Michael De Vlieger at A071395 *)

PARI

ispab(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, 1, 0);}
lista(nmax) = {my(c = 0, r = 10); for(k = 1, 10^nmax, if(ispab(k), c++); if(k+1 == r, print1(c, ", "); r *= 10));} \\ Amiram Eldar, Mar 26 2023

Extensions

a(10) from Amiram Eldar, Mar 26 2023