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%I #48 Feb 19 2024 01:59:06
%S 10,39,155,371,2935561623745,454539357304421
%N Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.
%C 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 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 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 Erich Friedman, <a href="https://erich-friedman.github.io/numbers.html">What's Special About This Number?</a>
%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://primes.utm.edu/curios/page.php?short=39">Prime Curios! 39</a>
%H 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 Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_098.htm">Puzzle 98. Curio 39</a>, The Prime Puzzles and Problems Connection.
%e 10 = 2*5 = 2 + 3 + 5;
%e 39 = 3*13 = 3 + 5 + 7 + 11 + 13;
%e 371 = 7*53 = 7 + 11 + 13 + ... + 53.
%t 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 Subsequence of A055514.
%Y Cf. A074036 (sum of primes from sfp(n) to gpf(n)), A169802 (n = A074036(n)).
%Y Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).
%K nice,nonn
%O 1,1
%A _Carlos Rivera_, Jun 21 2000
%E a(5) found by _Jud McCranie_, Jul 03 2000
%E 454539357304421 confirmed to be the 6th term by _Donovan Johnson_, Aug 23 2010
%E Example: removed last (see A055514). - _Manuel Valdivia_, Nov 19 2011
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