cp's OEIS Frontend

A079696 Numbers one more than composite numbers.

%I A079696 #16 Mar 19 2025 17:41:47
%S A079696 5,7,9,10,11,13,15,16,17,19,21,22,23,25,26,27,28,29,31,33,34,35,36,37,
%T A079696 39,40,41,43,45,46,47,49,50,51,52,53,55,56,57,58,59,61,63,64,65,66,67,
%U A079696 69,70,71,73,75,76,77,78,79,81,82,83,85,86,87,88,89,91,92,93,94,95,96
%N A079696 Numbers one more than composite numbers.
%C A079696 From _Hieronymus Fischer_, Mar 27 2014: (Start)
%C A079696 Numbers m such that m == 1 mod j and m > j^2 for any j > 1.
%C A079696 Example: m == 6 mod 10 is a term for m > 6, since m = 6 + 10k = 1 + (2k+1)*5, and m > (2k+1)^2 (for k := 1, m = 16), and m > 5^2 (for k > 1, m > 16).
%C A079696 A187813 and this sequence have no terms in common; this means that for each term a(n) there exists a base b such that the base-b digit sum is b.
%C A079696 Example: m = 1 + 3k, k > 3, is a term, since m > 3(1+3) > 3^2, thus the base-b-digit sum of (m) is = b for any b > 1 (here the base b is k+1 since 1+3k = 2(k+1) + k-1).
%C A079696 In general: Given a term a(n) there are p and q with p >= q > 1 such that a(n) = 1 + p*q. With b := p + 1 we get a(n) = (q-1)*b + b - (q-1), where 1 <= q-1 < b, which implies that the base-b digital sum of a(n) is = q-1 + b - (q-1) = b.
%C A079696 This sequence is the complement of the disjunction of A187813 with A239708. This means that a number m is a term if and only if there is a base b > 2 such that the base-b digit sum of m is b.
%C A079696 (End)
%H A079696 Hieronymus Fischer, <a href="/A079696/b079696.txt">Table of n, a(n) for n = 1..10000</a>
%F A079696 a(n) = A002808(n) + 1.
%F A079696 A239703(a(n)) > 0. - _Hieronymus Fischer_, Apr 10 2014
%o A079696 (Python)
%o A079696 from sympy import composite
%o A079696 def A079696(n): return composite(n)+1 # _Chai Wah Wu_, Mar 19 2025
%Y A079696 Cf. A072668.
%Y A079696 Cf. A007953, A187813, A239703, A239708.
%K A079696 nonn,easy
%O A079696 1,1
%A A079696 _Vladeta Jovovic_, Jan 31 2003
%E A079696 Edited by _Charles R Greathouse IV_, Mar 19 2010