A111107 -id:A111107 - cp's OEIS frontend

A384676 Binomial transform of A111107.

2, 5, 13, 37, 101, 271, 727, 1931, 5003, 12547, 30449, 71761, 165037, 372149, 826303, 1813219, 3944921, 8533073, 18393821, 39588071, 85192381, 183479291, 395667617, 854417989, 1847225579, 3996807053, 8650687127, 18721431499, 40496966207, 87538925959, 189076973699

Offset: 0

Alois P. Heinz, Jun 06 2025

nonn

All terms are primes.

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

Cf. A000040, A111107.

a(n) = Sum_{i=0..n} A111107(n-i) * binomial(n,i).

A384674 Lexicographically smallest sequence of distinct primes whose inverse binomial transform consists only of primes.

Original entry on oeis.org

2, 5, 11, 23, 47, 97, 211, 491, 1187, 2857, 6659, 14879, 31891, 65929, 132469, 261059, 510031, 999721, 1988797, 4048339, 8450557, 18014701, 38902439, 84347189, 182269327, 390630769, 828123239, 1735146097, 3594509969, 7369765889, 14975024861, 30200498591, 60537295711

Offset: 0

Alexander R. Povolotsky, Jun 06 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000040, A007442, A111107.

a:= proc(n) option remember; local b, p;
      b:= add(a(n-i)*binomial(n, i)*(-1)^i, i=1..n);
      p:= nextprime(abs(b));
      do if isprime(p+b) then break fi ;
         p:= nextprime(p)
      od; p
    end: a(0):=2:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 06 2025

a[n_] := a[n] = Module[{b, p}, b = Sum[a[n-i]*Binomial[n, i]*(-1)^i, {i, 1, n}]; p = NextPrime[Abs[b]]; While[True, If[PrimeQ[p+b], Break[]]; p = NextPrime[p]]; p];
a[0] = 2;
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Aug 28 2025, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jun 06 2025

A384768 Inverse binomial transform of A384674.

Original entry on oeis.org

2, 3, 3, 3, 3, 5, 11, 13, 7, 5, 7, 5, 3, 17, 29, 11, 11, 17, 13, 7, 29, 3, 3, 23, 3, 17, 37, 5, 223, 5, 37, 59, 19, 23, 433, 13, 89, 7, 7, 43, 3, 61, 5, 3, 191, 61, 149, 43, 89, 71, 13, 43, 41, 79, 31, 61, 23, 73, 53, 11, 157, 197, 83, 163, 3, 47, 7, 109, 5

Offset: 0

Alexander R. Povolotsky, Jun 09 2025

nonn

All terms are primes.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A384674, A384676, A000040, A111107.

a(n) = Sum_{i=0..n} (-1)^(n-i) * A384674(i) * binomial(n,i).

cp's OEIS Frontend

A384676 Binomial transform of A111107.

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A384674 Lexicographically smallest sequence of distinct primes whose inverse binomial transform consists only of primes.

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A384768 Inverse binomial transform of A384674.

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