Luca Pezzullo - cp's OEIS frontend

User: Luca Pezzullo

Luca Pezzullo has authored 2 sequences.

A274140 Sum of primes dividing n-th triangular number, counted with multiplicity.

0, 3, 5, 7, 8, 10, 11, 10, 11, 16, 16, 18, 20, 15, 14, 23, 23, 25, 26, 17, 21, 34, 30, 17, 23, 22, 18, 38, 37, 39, 39, 22, 31, 29, 20, 45, 56, 35, 25, 50, 51, 53, 56, 24, 34, 70, 56, 23, 24, 30, 35, 68, 62, 25, 27, 33, 51, 88, 69, 71, 92, 44, 23, 28, 32, 81, 86, 45, 38, 83, 81, 83, 110, 50, 34, 39, 34, 95, 90

Offset: 1

Luca Pezzullo, Jun 11 2016

nonn

			a(4) = 7; the 4th triangular number is 10, the prime factors of 10 are 2 and 5, and 2+5 = 7.
a(6) = 10; the 6th triangular number is 21, the prime factors of 21 are 3 and 7, and 3+7 = 10.

Eric Weisstein's World of Mathematics, Sum of prime factors
Eric Weisstein's World of Mathematics, Triangular number

Cf. A000217 (triangular numbers), A001414 (sum of primes dividing n).

a[1]=0; a[n_] := Plus @@ Times @@@ FactorInteger[n (n+1)/2]; Array[a, 80] (* Giovanni Resta, Jun 12 2016 *)
Join[{0},Rest[Total[Times@@@FactorInteger[#]]&/@Accumulate[Range[100]]]] (* Harvey P. Dale, May 06 2024 *)

a(n) = my(f=factor(n*(n+1)/2));sum(i=1,matsize(f)[1],f[i,1]*f[i,2]) \\ David A. Corneth, Jun 12 2016

For any integer coefficient C(n) of the polynomial generated by the Triangular Numbers generating function f(x)=x/((1-x)^3), if C(n) = Product (p_j^k_j) then a(n) = Sum (p_j * k_j).

a(n) = A001414(A000217(n)).

a(30) and a(38) corrected by Giovanni Resta, Jun 12 2016

A227346 Distance between consecutive pairs of primes differing by 6 (p, p+6).

Original entry on oeis.org

2, 4, 2, 4, 6, 8, 6, 4, 6, 6, 8, 6, 6, 10, 14, 4, 2, 4, 24, 20, 6, 10, 6, 18, 2, 30, 4, 6, 18, 6, 6, 8, 6, 30, 4, 20, 16, 6, 14, 6, 10, 50, 10, 14, 4, 42, 38, 16, 6, 8, 16, 6, 8, 6, 6, 28, 6, 6, 24, 50, 6, 18, 70, 2, 30, 4, 20, 4, 60, 6, 24, 6, 14, 22, 20, 30

Offset: 1

Luca Pezzullo, Jul 08 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A023201 (n and n+6 are primes).

Cf. A053320 (differences for Cousin primes).

with(numtheory): pre:=0: for n from 1 to 3000 do if isprime(n) and isprime(n+6) then if pre<>0 then printf("%d, ", n-pre) fi: pre:=n fi od: # adapted from original program by C. Ronaldo for A053320

Differences[Select[Prime[Range[200]], PrimeQ[# + 6] &]] (* T. D. Noe, Jul 09 2013 *)

a(n) = A023201(n+1) - A023201(n). - Zak Seidov, Sep 20 2013

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User: Luca Pezzullo

A274140 Sum of primes dividing n-th triangular number, counted with multiplicity.

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