These are the songs that reached number one on the Top 100 Best Sellers chart in 1963 as published by Cash Box magazine.
Issue date Song ArtistJanuary 5 Telstar The TornadoesJanuary 12 January 19 Go Away Little Girl Steve LawrenceJanuary 26 February 2 Walk Right In Rooftop SingersFebruary 9 Hey Paula Paul & PaulaFebruary 16 February 23 March 2 March 9 Walk Like A Man The Four SeasonsMarch 16 March 23 Our Day Will Come Ruby & The Romantics March 30 He's So Fine The ChiffonsApril 6 April 13 April 20 April 27 Can't Get Used to Losing You Andy Williams May 4 I Will Follow Him Little Peggy MarchMay 11 May 18 May 25 If You Wanna Be Happy Jimmy Soul June 1 It's My Party Lesley GoreJune 8 June 15 Sukiyaki Kyu SakamotoJune 22 June 29 July 6 July 13 Easier Said Than Done The EssexJuly 20 July 27 Surf City Jan and Dean August 3 Fingertips (pt. 2) Little Stevie WonderAugust 10 August 17 August 24 August 31 Hello Muddah, Hello Faddah! (A Letter From Camp) Allan Sherman September 7 My Boyfriend's Back The AngelsSeptember 14 September 21 Blue Velvet Bobby VintonSeptember 28 October 5 October 12 Be My Baby The Ronettes October 19 Sugar Shack Jimmy Gilmer & The FireballsOctober 26 November 2 November 9 Deep Purple Nino Tempo & April StevensNovember 16 November 23 I'm Leaving It Up To You Dale & GraceNovember 30 Dominique The Singing NunDecember 7 December 14 December 21 December 28
See also 1963 in music List of Hot 100 number-one singles of 1963 (U.S.)
References https://web.archive.org/web/20110818052122/http://cashboxmagazine.com/archives/60s_files/1963.html