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A059871 Number of solutions to the equation p_i = (1+mod(i,2))*p_{i-1} +- p_{i-2} +- p_{i-3} +- ... +- 2 +- 1, where p_i is the i-th prime number (where p_1 = 2 and the "zeroth prime" p_0 is defined to be 1). 5
1, 1, 1, 1, 1, 3, 3, 4, 6, 12, 16, 31, 46, 90, 140, 276, 449, 877, 1443, 2834, 4725, 9395, 16153, 32037, 55872, 110288, 190815, 380488, 672728, 1342395, 2434797, 4808180, 8579625, 17070112, 30858078, 61271317, 110926277, 220979544, 402354848 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

In Burton's book it is said that it is "known" that each prime can be represented as such sum. However, I do not know whether that means it has been proved.

This is Scherk's theorem, which was conjectured by Scherk in 1833 and proved by Pillai in 1928. [T. D. Noe, Oct 03 2008]

REFERENCES

D. M. Burton, Elementary Number Theory.

S. S. Pillai, "On some empirical theorem of Scherk", J. Indian Math. Soc. 17 (1927-28), pp. 164-171.

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

LINKS

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 250 terms from Alois P. Heinz)

J. L. Brown, Proof of Scherk's Conjecture on the Representation of Primes, Amer. Math. Monthly 74 (1967), 31-33.

William Y. Lee, On the representation of integers, Math. Mag. 47 (1974), 150-152.

H. F. Scherk, Bemerkungen über die Bildung der Primzahlen aus einander, Journal für die reine und angewandte Mathematik 10 (1883), pp. 201-208.

H. F. Scherk, Bemerkungen über die Bildung der Primzahlen aus einander, Journal für die reine und angewandte Mathematik 10 (1883), pp. 201-208.

EXAMPLE

For the first five primes we have only one solution for each: 2 = 2*1, 3 = 1*2 + 1*1, 5 = 2*3 - 1*2 + 1*1, 7 = 1*5 + 1*3 - 1*2 + 1*1, 11 = 2*7 - 1*5 + 1*3 - 1*2 + 1*1 and for the next prime 13, we have 3 solutions: 13 = 11-7+5+3+2-1 = 11+7-5-3+2+1 = 11+7-5+3-2-1.

MAPLE

map(nops, primesums_primes_mult(16)); primesums_primes_mult := proc(upto_n) local a, b, i, n, p, t; a := []; for n from 1 to upto_n do b := []; p := ithprime(n); for i from (2^(n-1)) to ((2^n)-1) do t := bin_prime_sum(i); if(t = p) then b := [op(b), i]; fi; od; a := [op(a), b]; print(a); od; RETURN(a); end;

# second Maple program

p:= n-> `if`(n<0, 0, `if`(n=0, 1, ithprime(n))):

sp:= proc(n) sp(n):= `if`(n<0, 0, p(n)+sp(n-1)) end:

b := proc(n, i) option remember; `if`(n>sp(i), 0, `if`(i<0, 1,

        b(n+p(i), i-1)+ b(abs(n-p(i)), i-1)))

     end:

a:= n-> b(p(n) -(1+irem(n, 2))*p(n-1), n-2):

seq(a(n), n=1..40);  # Alois P. Heinz, Aug 05 2012

MATHEMATICA

nmax = 40; d = {1}; a1 = {}; pp = 1;

Do[

  p = Prime[n];

  i = Ceiling[Length[d]/2] +  Abs[p - (1 + Mod[n, 2])*pp];

  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];

  d = PadLeft[d, Length[d] + 2 pp] + PadRight[d, Length[d] + 2 pp];

  pp = p;

  , {n, nmax}];

a1 (* Ray Chandler, Mar 11 2014 *)

CROSSREFS

See A059872 for the table of all solutions encoded as binary vectors and A059873-A059875 for specific sequences. A059876 gives the function bin_prime_sum.

Cf. A022894, A083309.

Sequence in context: A080013 A152949 A058660 * A273096 A076619 A318140

Adjacent sequences:  A059868 A059869 A059870 * A059872 A059873 A059874

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 05 2001

EXTENSIONS

More terms from Naohiro Nomoto, Sep 11 2001

More terms from Larry Reeves (larryr(AT)acm.org), Nov 20 2003

a(33)-a(39) from Donovan Johnson, Oct 01 2010

STATUS

approved

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Last modified December 9 17:51 EST 2018. Contains 318023 sequences. (Running on oeis4.)