A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

10, 39, 155, 371, 2935561623745, 454539357304421

Offset: 1

Author

Carlos Rivera, Jun 21 2000

Keywords

• nice
• nonn

Comments

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.

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

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

Examples

			10 = 2*5 = 2 + 3 + 5;
39 = 3*13 = 3 + 5 + 7 + 11 + 13;
371 = 7*53 = 7 + 11 + 13 + ... + 53.
		

Links

• Erich Friedman, What's Special About This Number?
• G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 39
• Miroslav Kureš, Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 8-15.
• Robert Munafo, Notable Properties of Specific Numbers
• Carlos Rivera, Puzzle 98. Curio 39, The Prime Puzzles and Problems Connection.

Crossrefs

Subsequence of A055514.

Cf. A074036 (sum of primes from sfp(n) to gpf(n)), A169802 (n = A074036(n)).

Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).

Programs

Mathematica

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 *)

Extensions

a(5) found by Jud McCranie, Jul 03 2000

454539357304421 confirmed to be the 6th term by Donovan Johnson, Aug 23 2010

Example: removed last (see A055514). - Manuel Valdivia, Nov 19 2011