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A079696 Numbers one more than composite numbers. 7

%I #13 Apr 11 2014 02:37:01

%S 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 39,40,41,43,45,46,47,49,50,51,52,53,55,56,57,58,59,61,63,64,65,66,67,

%U 69,70,71,73,75,76,77,78,79,81,82,83,85,86,87,88,89,91,92,93,94,95,96

%N Numbers one more than composite numbers.

%C From _Hieronymus Fischer_, Mar 27 2014: (Start)

%C Numbers m such that m == 1 mod j and m > j^2 for any j > 1.

%C 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 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 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 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 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 (End)

%H Hieronymus Fischer, <a href="/A079696/b079696.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A002808(n) + 1.

%F A239703(a(n)) > 0. - _Hieronymus Fischer_, Apr 10 2014

%Y Cf. A072668.

%Y Cf. A007953, A187813, A239703, A239708.

%K nonn,easy

%O 1,1

%A _Vladeta Jovovic_, Jan 31 2003

%E Edited by _Charles R Greathouse IV_, Mar 19 2010

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)