A079551 a(n) = Sum_{primes p <= n} d(p-1), where d() = A000005.

0, 0, 1, 3, 3, 6, 6, 10, 10, 10, 10, 14, 14, 20, 20, 20, 20, 25, 25, 31, 31, 31, 31, 35, 35, 35, 35, 35, 35, 41, 41, 49, 49, 49, 49, 49, 49, 58, 58, 58, 58, 66, 66, 74, 74, 74, 74, 78, 78, 78, 78, 78, 78, 84, 84, 84, 84, 84, 84, 88, 88, 100, 100, 100, 100, 100, 100, 108, 108, 108, 108

Offset: 0

Author

N. J. A. Sloane, Jan 24 2003

Keywords

• nonn

References

• Yuri V. Linnik, The dispersion method in binary additive problems, American Mathematical Society, 1963, chapter VIII.
• József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, section II.11, p. 49.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Enrico Bombieri, John B. Friedlander, and Henryk Iwaniec, Primes in arithmetic progressions to large moduli, Acta Mathematica, Vol. 156, No. 1 (1986), pp. 203-251.
• Heini Halberstam, Footnote to the Titchmarsh-Linnik divisor problem, Proceedings of the American Mathematical Society, Vol. 18, No. 1 (1967), pp. 187-188.
• Yurii Vladimirovich Linnik, New versions and new uses of the dispersion methods in binary additive problems, Doklady Akademii Nauk SSSR, Vol. 137, No. 6. (1961), pp. 1299-1302 (in Russian).
• Gaetano Rodriquez, Sul problema dei divisori di Titchmarsh, Bollettino dell'Unione Matematica Italiana, Vol. 20, No. 3 (1965), pp. 358-366.
• E. C. Titchmarsh, A divisor problem, Rendiconti del Circolo Matematico di Palermo (1884-1940), December 1930, Volume 54, Issue 1, pp. 414-429.

Crossrefs

Cf. A000005, A079552, A082695.

Row sums of triangle A143540. - Gary W. Adamson, Aug 23 2008

Programs

Mathematica

a[n_] := Sum[DivisorSigma[0, p-1], {p, Select[Range[n], PrimeQ]}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 26 2015 *)

PARI

a(n) = sum(p=1, n, if (isprime(p), numdiv(p-1))); \\ Michel Marcus, Aug 03 2018

Formula

Several asymptotic estimates are known: see Sándor et al.

a(n) ~ (zeta(2)*zeta(3)/zeta(6)) * n. - Amiram Eldar, Jul 22 2019