A052297 -id:A052297 - cp's OEIS frontend

A077218 Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-st prime.

0, 2, 2, 7, 3, 8, 3, 7, 14, 3, 15, 8, 3, 8, 15, 14, 4, 16, 8, 5, 13, 11, 14, 21, 10, 3, 9, 5, 10, 36, 12, 16, 3, 26, 4, 16, 17, 8, 16, 15, 5, 26, 7, 9, 4, 33, 30, 12, 4, 10, 14, 6, 29, 20, 14, 15, 5, 17, 10, 3, 28, 40, 9, 5, 9, 42, 16, 27, 4, 14, 13, 22, 17, 18, 8, 19, 22, 11, 23, 27, 5

Offset: 1

Amarnath Murthy, Nov 03 2002

Also, number of prime factors (with multiplicity) of the product P(n) of the composite numbers between n-th and (n+1)-th prime.

			a(6) = 8. Prime(6) = 13 and prime(7) = 17. 14, 15, and 16 are the composite numbers between 13 and 17. 14 has two prime factors (2 and 7); 15 has two prime factors (3 and 5); and 16 has four prime factors (2, 2, 2, and 2). Thus, a(6) = 2 + 2 + 4 = 8 total prime factors. [corrected by _Harvey P. Dale_, May 25 2011]

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 2000.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A052297.

Total[PrimeOmega[Range[First[#]+1,Last[#]-1]]]&/@Partition[Prime[ Range[90]],2,1] (* Harvey P. Dale, May 25 2011 *)

a(n) = Sum_{k=A000040(n)+1..A000040(n+1)-1} A001222(k). - Reinhard Zumkeller, Nov 29 2002

More terms and better description from Reinhard Zumkeller, Nov 29 2002

A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

0, 2, 5, 10, 5, 17, 5, 28, 30, 10, 45, 42, 12, 44, 47, 76, 10, 72, 57, 5, 97, 51, 117, 150, 28, 22, 83, 5, 65, 321, 66, 131, 28, 298, 10, 108, 172, 145, 109, 205, 10, 276, 5, 127, 16, 441, 582, 130, 24, 80, 232, 10, 276, 195, 270, 256, 10, 218, 187, 52, 388, 701, 162

Offset: 1

Karl-Heinz Hofmann, Mar 26 2023

			a(6): 6th prime = 13 and the (6+1)th prime = 17; the composites between are {14,15,16} and the distinct prime factors of this set are {2,7,3,5} (no duplicates allowed); so a(6) = 2 + 7 + 3 + 5 = 17.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A052297, A077218, A008472, A061214.

a[n_] := Plus @@ Union@ (Join @@ (FactorInteger[#][[;; , 1]] & /@ Range[Prime[n] + 1, Prime[n + 1] - 1])); Array[a, 65] (* Amiram Eldar, Mar 27 2023 *)

a(n) = my(list=List()); for(i=prime(n)+1, prime(n+1)-1, my(f=factor(i)[,1]); for (k=1, #f, listput(list, f[k]))); vecsum(Set(list)); \\ Michel Marcus, Mar 27 2023

from sympy import primefactors, sieve
def A361806(n):
    primeset = []
    for composites in range (sieve[n]+1, sieve[n+1]):
        for p in primefactors(composites): primeset.append(p)
    return(sum(set(primeset)))

a(n) = A008472(A061214(n)).

A219611 a(n) is the smallest omega(A061214(k)) sampled over all indices k of prime gaps prime(k+1) - prime(k) = 2n, where omega = A001221.

Original entry on oeis.org

1, 3, 5, 9, 11, 14, 14, 21

Offset: 1

Naohiro Nomoto, Apr 12 2013

The example demonstrates that the minimum order of the set of primes represented by all composites in the prime gap 2*n is not necessarily obtained by using the smallest prime(k) (that would be A038664).

			For n=8: p_283-p_282 = p_296-p_295 = 2*8=16; omega(A061214(282)) > omega(A061214(295)); omega(A061214(295)) = 21; so a(8) = 21.

Cf. A052297.

cp's OEIS Frontend

A077218 Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-st prime.

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A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

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A219611 a(n) is the smallest omega(A061214(k)) sampled over all indices k of prime gaps prime(k+1) - prime(k) = 2n, where omega = A001221.

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