ויקרא פרק כג, טו וּסְפַרְ תֶּם לָכֶּם מִמָּחֳּרַת הַשַבָת מִיּוֹם הֲבִיאֲכֶּם אֶּת-עֹּמֶּר הַתְנוּפָה שֶּׁבַע שַׁבָתוֹת תְמִימֹּת תִהְיֶּינָה:
מסכת מנחות דף סו/א אי מיום הביאכם יכול יקצור ויספור ויביא ביום ת"ל שבע שבתות תמימות תהיינה אימתי אתה מוצא שבע שבתות תמימות בזמן שאתה מתחיל לימנות מבערב יכול יקצור ויביא ויספור בלילה ת"ל מיום הביאכם הא כיצד קצירה וספירה בלילה והבאה ביום
The reaping must be prior to the counting. And the bringing of the omer offering to the Bais HaMikdash is done during the daytime.
Tosefos brings in the name of the Baal Halachos Gedolos that if one forgets a complete day to count, then one does not count any further, since the counting has to be complete, counting all the days of the seven weeks. Tosefos questions that this ruling but does not explain what is the difficulty. The obvious difficulty is that the concept of “completeness” applies to the daily counting done in daily periods each night, and not be applied to the entire period 49 days of saying the omer (each day is a separate mitzvah).
Even though Tosefos questions this ruling, most poskim decide that the 49 days and seven weeks need to be complete and thus if one misses an entire day, one no longer counts (with a blessing). Thus, it would seem obvious that a young person that becomes an adult (bar-mitzvah) during this period, or a person that converts during this period, would not count (with a blessing).
However, one might say that a young person could connect the period prior to Bar Mitzvah (he counted when exempt) and now count with a blessing. Support to this concept is from the Gemara (Yevamos 62a), “A person that converts that already had children is exempt from the mitzvah of Piryah U’rivyah after converting due to the period prior to converting.”
