A367021 Numbers that can be written as both the sum of two or more consecutive nonprimes and the sum of two or more consecutive primes.

5, 10, 17, 18, 23, 26, 28, 31, 36, 39, 41, 49, 53, 58, 59, 60, 67, 68, 71, 75, 77, 78, 83, 84, 90, 95, 97, 101, 102, 109, 112, 121, 124, 127, 128, 129, 131, 132, 138, 139, 143, 150, 152, 155, 156, 158, 159, 160, 161, 162, 168, 169, 172, 173, 180, 181, 184, 187, 192

Offset: 1

Author

Tamas Sandor Nagy, Nov 01 2023

Keywords

• nonn

Comments

It seems that more than one consecutive number set from one kind or the other may exist for a term. Also, for some terms, an equal number of addends from each kind may correspond.

Examples

			5 is a term because 5 = 1 + 4 = 2 + 3, which is the sum of two consecutive nonprimes and also the sum of two consecutive primes.
17 is a term because 17 = 8 + 9 = 2 + 3 + 5 + 7, the sum of two consecutive nonprimes and also the sum of four consecutive primes.
		

Links

• David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A018252, A050936, A366976.

Programs

Python

from sympy import isprime
primes = [x for x in range(2,3000) if isprime(x)]
comps = [x for x in range(1,3000) if not isprime(x)]
psums = set(sum(primes[p:p+pn]) for pn in range(2,100) for p in range(len(primes)-pn))
csums = set(sum(comps[c:c+cn]) for cn in range(2,100) for c in range(len(comps)-cn))
terms = sorted(list(psums.intersection(csums)))
print(terms)
# David Consiglio, Jr., Dec 18 2023

Extensions

More terms from David Consiglio, Jr., Dec 18 2023