Geodesic
Ramanujan Congruences for p-k (n)
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 67-78

Voir la notice de l'article provenant de la source Cambridge University Press

Let 1 2 Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation. 
Atkin, A. O. L. Ramanujan Congruences for p-k (n). Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 67-78. doi: 10.4153/CJM-1968-009-6
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[1] 1. Atkin, A. O. L., Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14–32. Google Scholar

[2] 2. Atkin, A. O. L. and O'Brien, J. N., Some properties of pin) and c(n) modulo powers of 13, Trans. Amer. Math. Soc, 126 (1967), 442–459. Google Scholar

[3] 3. Ramanathan, K. G., Identities and congruences of the Ramanujan type, Can. J. Math., 2 1950), 168–178. Google Scholar

[4] 4. Ramanujan, S., Some properties of pin), the number of partitions of n, Proc. Cambridge Phil. Soc, 19 (1919), 207–210. Google Scholar

[5] 5. Watson, G. N., Ramanujans Vermutung ûber Zerfâllungsanzahlen, J. fiir Math., 179 (1938), 97–128. Google Scholar

[6] 6. Weber, H., Lehrbuch der Algebra, Vol. 3 (Brunswick, 1908; reprinted Chelsea, N.Y.). Google Scholar

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