Daniel Joyce - cp's OEIS frontend

User: Daniel Joyce

Daniel Joyce has authored 2 sequences.

A247657 Partial sums of primes, but with a twist.

5, 23, 53, 101, 359, 487, 631, 1669, 5407, 6959, 7517, 8093, 9341, 10009, 15427, 17191, 21011, 26489, 30089, 32609, 42281, 45293, 51683, 56747, 60251, 73471, 77509, 79561, 90197, 101513, 137209, 142867, 169489, 182047, 191717, 194981, 198301, 201661, 211943

Offset: 1

Daniel Joyce, Oct 01 2014

nonn

We start with the primes, 2 3 5 7 11 13 17 19 ..., and form the partial sums (starting with the first two terms), 2+3 = 5, ..., but whenever the partial sum is a prime, we remove it from the list of primes to be added later. Thus, 5 will not be added, and the next term in the partial sums is 2+3+7 = 12, and then 2+3+7+11 = 23, which is again prime, thus not used (later) in the partial sum. The primes that are removed are 5, 23, 53, 101, 359, 487, 631, 1669,... and the partial sums are A247658.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A247658.

Cf. A000040, A010051, A007504.

a247657 n = a247657_list !! (n-1)
a247657_list = f 0 $ drop 2 a000040_list where
   f z (p:ps) | a010051' z' == 1 = z' : f z' (delete z' ps)
              | otherwise        = f z' ps
              where z' = z + p
-- Reinhard Zumkeller, Oct 01 2014

{omit=[];s=2;forprime(p=3,999,if(vecsearch(omit,p),omit=vecextract(omit,"^1");next); isprime(s+=p)||next;print1(s","); omit=concat(omit,s))} \\ M. F. Hasler, Oct 04 2014

A111107 Lexicographically smallest increasing sequence of primes whose binomial transform consists only of primes.

Original entry on oeis.org

2, 3, 5, 11, 13, 29, 43, 53, 59, 71, 79, 83, 103, 113, 139, 173, 181, 227, 269, 277, 317, 383, 463, 509, 673, 701, 751, 863, 967, 977, 1187, 1201, 1493, 1531, 1609, 1637, 1801, 2153, 2221, 2239, 2371, 2377, 2543, 2557, 2683, 2687, 2791, 2837, 3067, 3229, 3257

Offset: 0

Daniel Joyce, Oct 14 2005

In the standard binomial transform of the primes most of the terms are composite.

			The binomial transform of this sequence gives: 2, 5, 13, 37, 101, 271, 727, 1931, 5003, 12547, 30449, 71761, ... = A384676.
The prime 7 and various larger primes are missing from the new sequence because the transform would not consist of primes. For example,
  2,5,13,33
  3,8,20
  5,12
  7
and 33 is not prime, so we must eliminate 7.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000040, A007443, A384676.

a(n) = Sum_{i=0..n} A384676(n-i)*binomial(n,i)*(-1)^i. - Alois P. Heinz, Jun 06 2025

Offset set to 0 by Alois P. Heinz, Jun 06 2025

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User: Daniel Joyce

A247657 Partial sums of primes, but with a twist.

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