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%I A055233 #48 Feb 19 2024 01:59:06
%S A055233 10,39,155,371,2935561623745,454539357304421
%N A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.
%C A055233 Composite n such that n = p_1 + p_2 + ... + p_k where the p_i are consecutive primes, p_1 is the smallest prime factor of n and p_k is the largest.
%C A055233 Concerning a(6): 454539357304421 is the product of two primes, 3536123 * 128541727 and also the sum of these two plus all the primes in between: 3536123 + 3536129 + 3536131 + ... + 128541719 + 128541727. I do not know if there are any terms in A055233 between 2935561623745 and 454539357304421. (I have searched for values of N satisfying N=Pa*Pb=Pa+...+Pb as far as 5.98*10^16, but this is not quite the same as A055233 or A055514.) - _Robert Munafo_, Nov 20 2002
%C A055233 This is a subsequence of A055514 where the sum must be divisible by the smallest and largest term, but they need not be its smallest and largest prime factor. Without restriction to composite numbers, all primes would be trivially included: see A169802. - _M. F. Hasler_, Nov 21 2021
%H A055233 Erich Friedman, <a href="https://erich-friedman.github.io/numbers.html">What's Special About This Number?</a>
%H A055233 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://primes.utm.edu/curios/page.php?short=39">Prime Curios! 39</a>
%H A055233 Miroslav Kureš, <a href="https://doi.org/10.7546/nntdm.2019.25.2.8-15">Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 8-15.
%H A055233 Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H A055233 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_098.htm">Puzzle 98. Curio 39</a>, The Prime Puzzles and Problems Connection.
%e A055233 10 = 2*5 = 2 + 3 + 5;
%e A055233 39 = 3*13 = 3 + 5 + 7 + 11 + 13;
%e A055233 371 = 7*53 = 7 + 11 + 13 + ... + 53.
%t A055233 Select[Range[2, 10^3], And[CompositeQ@ #1, #1 == #2] & @@ {#, Total@ Prime[Range @@ PrimePi@ {#[[1, 1]], #[[-1, 1]]} &@ FactorInteger[#]]} &] (* _Michael De Vlieger_, Sep 04 2019 *)
%Y A055233 Subsequence of A055514.
%Y A055233 Cf. A074036 (sum of primes from sfp(n) to gpf(n)), A169802 (n = A074036(n)).
%Y A055233 Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).
%K A055233 nice,nonn
%O A055233 1,1
%A A055233 _Carlos Rivera_, Jun 21 2000
%E A055233 a(5) found by _Jud McCranie_, Jul 03 2000
%E A055233 454539357304421 confirmed to be the 6th term by _Donovan Johnson_, Aug 23 2010
%E A055233 Example: removed last (see A055514). - _Manuel Valdivia_, Nov 19 2011

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