New at De Gruyter
Editor-in-Chief: Nathanson, Melvyn B. / Nešetril, Jaroslav / Pomerance, Carl
Managing Editor: Landman, Bruce M. / Robertson, Aaron
Editorial Board Member: Nowakowski, Richard J. / Fraenkel, Aviezri / Ruzsa, Imre / Wilf, Herb / Andrews, George / Berlekamp, Elwyn / Brown, Ezra / Brown, Tom / Canfield, E.Rodney / Chung, Fan / Goldston, Daniel A. / Gowers, William Timothy / Graham, Ronald / Granville, Andrew / Griggs, Jerrold / Guy, Richard / Harborth, Heiko / Hindman, Neil / Leader, Imre / Lefmann, Hanno / Ono, Ken / Rödl, Vojtech / Serra, Oriol / Solymosi, Jozsef / Sós, Vera T. / Tichy, Robert F. / Winkler, Peter / Zeilberger, Doron / Prömel, Hans Jürgen
Mathematical Citation Quotient 2011: 0.23
Volume 12 (2012)
Volume 11 (2011)
Volume 10 (2010)
Most Downloaded Articles
- Odd Catalan Numbers Modulo by Lin, Hsueh-Yung
- On the Number of Carries Occurring in an Addition Mod by Flori, Jean-Pierre and Randriam, Hugues
- Counting Depth Zero Patterns in Ballot Paths by Niederhausen, Heinrich and Sullivan, Shaun
- Digital Sums and Functional Equations by Girgensohn, Roland
- Neither (4k 2 + 1) nor (2k(k – 1) + 1) is a Perfect Square by Fang, Jin-Hui
Power Totients with Almost Primes
1Department of Mathematics, University of Missouri, Columbia, Missouri, USA.
2Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México.
Citation Information: Integers. Volume 11, Issue 3, Pages 307–313, ISSN (Print) 1867-0652, DOI: 10.1515/integ.2011.024, June 2011
- Published Online:
A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k ⩾ 2, let ℱk(X) be the number of k-th powers mk ⩽ X such that φ(n) = mk for some squarefree k-almost prime n, where φ(·) is the Euler function. We show that the lower bound ℱk(X) ≫ X 1/k/(log X)2k holds, where the implied constant is absolute and the lower bound is uniform over a certain range of k relative to X. In particular, our results imply that there are infinitely many pairs (p, q) of distinct primes such that (p – 1) (q – 1) is a perfect square.