A258126 -id:A258126 - cp's OEIS frontend

A258127 Smallest k such that Sum_{i=0..k} binomial(n,i) is prime, or a(n)=0 if there is no such k.

1, 1, 2, 1, 4, 1, 2, 2, 0, 1, 2, 1, 4, 4, 6, 1, 16, 1, 2, 2, 4, 1, 2, 6, 8, 16, 2, 1, 0, 1, 4, 6, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 10, 0, 48, 2, 1, 36, 20, 6, 2, 8, 1, 10, 1, 16, 13, 2, 2, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 8, 74, 64, 1, 16

Offset: 1

Vladimir Shevelev, May 21 2015

nonn

a(n)=0 for n=9,29,33,34,37,38,39,41,45,49,...;
records a(n) are  1,2,4,6,16,48,74,...
at positions 1,3,5,15,17,50,80,...

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A007318, A258126.

a(n) = {my(k = 0); while(! isprime(sum(i=0, k, binomial(n,i))), k++; if ((k>n) && !isprime(binomial(n,k)), return (0);)); k;} \\ Michel Marcus, May 23 2015

a(n) <= n-1.

More terms from Peter J. C. Moses, May 21 2015

A258193 Define a<+>b = odd part(odd part(a) + odd part(b)), where odd part(n) = A000265(n); a(n) is the smallest prime of the form <+>_{0<=i<=k} binomial (n,i), or a(n)=0 if there is no such a prime (see comment).

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 11, 0, 5, 3, 3, 17, 7, 23, 53, 29, 13, 5, 5, 3, 11, 3, 3, 823, 13, 7, 7, 457, 109, 109, 233, 2267, 17, 59, 151, 5, 19, 5, 5, 3, 113, 11, 11, 3, 23, 3, 3, 71, 43, 13, 13, 7, 179, 7, 7, 193, 29, 2137, 863, 443, 31, 5498157739, 977, 163

Offset: 1

Vladimir Shevelev, May 23 2015

nonn

f(n)=<+>_{0<=i<=n} c(i) is defined as the following: f(0)=c(0), f(n)=f(n-1)<+>c(n).

a(n)=0 for 1,2,3,4,8,82,107,...(cf. A258194)

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000265, A007318, A258126, A258127, A258194.

Cf. A034868, A010051.

import Data.Function (on)
a258193 n = head $ (filter ((== 1) . a010051'') $
                    scanl1 (<+>) (a034868_row n)) ++ [0]
            where (<+>) = (a000265 .) . on (+) a000265
-- Reinhard Zumkeller, Jun 20 2015

vSum[a_,b_]:=#[#[a]+#[b]]&[#/2^IntegerExponent[#,2]&];
Table[
First[Select[FoldList[vSum,First[#],Rest[#]]&[Map[Binomial[n,#]&,Range[0,n]]],PrimeQ]/.{}->{0}],{n,100}] (*Peter J. C. Moses, May 23 2015*)

More terms from Peter J. C. Moses, May 23 2015

A258483 Numbers n such that partial sums of the n-th row of Pascal's triangle (A007318) are all nonprime.

Original entry on oeis.org

0, 9, 29, 33, 34, 37, 38, 39, 41, 45, 49, 65, 67, 69, 73, 74, 77, 85, 86, 92, 97, 98, 101, 105, 113, 114, 115, 117, 118, 121, 129, 133, 134, 137, 139, 141, 145, 146, 151, 154, 155, 158, 160, 161, 164, 165, 170, 173, 177, 182, 185, 186, 193, 194, 195, 199, 202, 205, 206, 209, 213, 214, 225, 227, 229, 230, 234, 241, 247, 257, 258, 259, 261, 263, 265, 266, 269, 272, 273, 277, 281

Offset: 0

Vladimir Shevelev, May 31 2015

nonn

It appears that a(n) ~ c*n, as n goes to infinity, where c is constant < 2.17. For heuristic argument, see Robert Israel link.

Peter J. C. Moses, Table of n, a(n) for n = 0..2499
Robert Israel, Re: a surprising property, SeqFan list, Jun 01 2015.

Cf. A007318, A008949, A258126, A258127.

T:= <1>:
res:= 0:
for n from 1 to 1000 do
  T:= <1, T[1..-2] + T[2..-1], 2^n>;
  if not ormap(isprime,T) then res:= res, n fi
od:
res; # Robert Israel, Jun 04 2015

isok(n) = {my(s = 0); for (k=0, n, s += binomial(n, k); if (isprime(s), return (0))); return (1);} \\ Michel Marcus, May 31 2015

cp's OEIS Frontend

A258127 Smallest k such that Sum_{i=0..k} binomial(n,i) is prime, or a(n)=0 if there is no such k.

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A258193 Define a<+>b = odd part(odd part(a) + odd part(b)), where odd part(n) = A000265(n); a(n) is the smallest prime of the form <+>_{0<=i<=k} binomial (n,i), or a(n)=0 if there is no such a prime (see comment).

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A258483 Numbers n such that partial sums of the n-th row of Pascal's triangle (A007318) are all nonprime.

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