A287915 Indices of primes in A007443.

1, 2, 3, 5, 9, 22, 25, 28, 32, 41, 99, 104, 138, 183, 225, 361, 641, 1636, 1719, 3191, 3590, 4144, 5340, 6372, 6893, 6915, 8429, 10024, 10546, 16401, 21636, 22612, 24813, 31416, 36065

Offset: 1

Author

M. F. Hasler, Jun 02 2017

Keywords

• nonn
• hard
• more

Comments

Sequence A007443, the binomial transform of the primes A000040, is defined as A007443(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). This is also the first column of the infinite square array T(m,n) with T(1,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), as for binomial coefficients. Successive rows result from applying this operation of taking the sum of successive terms. So it would be more natural to use index 0 for the first term of this sequence (which is also the only even term, and results from applying the operation 0 times to the primes). This would yield the sequence 0, 1, 2, 4, 8, 21, 24, 27, 31, 40, 98, 103, 137, 182, ...

The next term, if it exists, is greater than 20000. - Vaclav Kotesovec, Dec 19 2020

Any subsequent terms are > 10^5. - Lucas A. Brown, Mar 18 2024

Links

• Lucas A. Brown, Python program.

Crossrefs

Cf. A007443.

Programs

Mathematica

A007443 = Table[Sum[Binomial[n-1, k-1]*Prime[k], {k, 1, n}], {n, 1, 1000}]; Select[Range[Length[A007443]], PrimeQ[A007443[[#]]]&] (* Vaclav Kotesovec, Dec 19 2020 *)

PARI

for(n=1,199,isprime(A007443(n))&&print1(n","))

Extensions

a(18)-a(22) from Jinyuan Wang, Dec 19 2020

a(23)-a(30) from Vaclav Kotesovec, Dec 19 2020

a(31)-a(35) from Lucas A. Brown, Mar 18 2024