A059873 -id:A059873 - cp's OEIS frontend

A059876 a(n) = bin_prime_sum(n).

2, 1, 3, 3, 5, 7, 9, -1, 1, 3, 5, 5, 7, 9, 11, 3, 5, 7, 9, 9, 11, 13, 15, 13, 15, 17, 19, 19, 21, 23, 25, -7, -5, -3, -1, -1, 1, 3, 5, 3, 5, 7, 9, 9, 11, 13, 15, 7, 9, 11, 13, 13, 15, 17, 19, 17, 19, 21, 23, 23, 25, 27, 29, -3, -1, 1, 3, 3, 5, 7, 9, 7, 9, 11, 13, 13, 15, 17, 19, 11, 13, 15, 17, 17, 19, 21, 23, 21, 23, 25, 27, 27, 29, 31, 33, 19, 21, 23, 25, 25, 27, 29, 31, 29, 31

Offset: 1

Antti Karttunen, Feb 05 2001

sign

From R. J. Mathar, Nov 12 2011: (Start)
The function bin_prime_sum of an argument n is a sum of three numbers. Let s = A000523(n) be the exponent of the largest power of 2 less than or equal to n and prime=A000040. Then the three terms are:
i) (-1)^(n+1);
ii) sum_{i=1..s} prime(i) * (1 + (-1)^[n/2^i] ); where [..] is the floor bracket;
iii) 1 (if n=1), otherwise prime(s) (if s even) or 0 (if s odd). (End)

Cf. A059871, A059872, A059873, A059877, A059878, A059880.

with(numtheory); bin_prime_sum := proc(n) local i,s; s := floor_log_2(n); RETURN(((-1)^(n+1)) + add( (((-1)^(floor(n/(2^i))+1))*ithprime(i)),i=1..s) + (`if`((1 = n),1,((`mod`((s+1),2))*ithprime(s)))) ); end;

a[n_] := With[{s = Floor[Log[2, n]]}, (-1)^(n+1) + Sum[(-1)^(Floor[n/2^i] + 1)*Prime[i], {i, 1, s}] + If[1 == n, 1, Mod[s+1, 2]*Prime[s]]]; Array[a, 105] (* Jean-François Alcover, Mar 07 2016, adapted from Maple *)

a(A059873(n)) = A000040(n).

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).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 05 2001

nonn

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]

			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.

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.

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.

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.

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

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 *)

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

A059875 The lexicographically last sequence of binary encodings of solutions satisfying the equation given in A059871.

Original entry on oeis.org

1, 3, 5, 13, 21, 52, 84, 210, 392, 905, 1601, 3652, 7173, 15364, 28932, 61952, 122900, 253969, 493572, 1017858, 2031636, 4128801, 8159232, 16547841, 33030657, 66584836, 132251649, 266600448, 532677128, 1069548544, 2139095042

Offset: 1

Antti Karttunen, Feb 05 2001

nonn

Apply bin_prime_sum (see A059876) to this sequence and you get A000040, the prime numbers.

Sean A. Irvine, Table of n, a(n) for n = 1..50

Cf. A059873, A059874.

map(last_term, primesums_primes_mult(16)); last_term := proc(l) local n: n := nops(l); if(0 = n) then ([]) else (op(n,l)): fi: end: # primesums_primes_mult given in A059871.

More terms from Naohiro Nomoto, Sep 12 2001

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

A059872 Solutions to the equation given in A059871, encoded as binary vectors and converted to decimal.

Original entry on oeis.org

1, 3, 5, 13, 21, 46, 51, 52, 78, 83, 84, 175, 181, 205, 210, 303, 309, 333, 338, 390, 392, 639, 698, 726, 728, 737, 822, 824, 846, 851, 852, 903, 905, 1143, 1145, 1197, 1202, 1226, 1232, 1311, 1322, 1328, 1350, 1352, 1409, 1562, 1571, 1572, 1601, 2539, 2540

Offset: 1

Antti Karttunen, Feb 05 2001

The rows of this table have lengths given by A059871.
In binary encodings, the least significant bit (bit-0) stands for the factor of 1, the next bit (bit-1) stands for the factor of 2, bit-2 for the factor of 3, bit-3 for the factor of 5, etc., each bit being 0 if the corresponding factor is -1 and 1 if it is +1 (or +2 if the bit is the most significant bit of the code of odd length).
E.g. we have 2 = 2*1 -> 1 in binary, 3 = 1*2 + 1*1 -> 11 in binary, 5 = 2*3 - 1*2 + 1*1 -> 101 in binary, 7 = 1*5 + 1*3 - 1*2 + 1*1 -> 1101 in binary, 11 = 2*7 - 1*5 + 1*3 - 1*2 + 1*1 -> 10101 in binary. Function bin_prime_sum given in A059876 maps such encodings back to primes.

			Rows are:
  1;
  3;
  5;
  13;
  21;
  46,51,52;
  78,83,84;
  175,181,205,210;
  ...

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A059871, A059873, A059874, A059875, A059876.

map(op, primesums_primes_mult(16)); # primesums_primes_mult given in A059871.

A059874 The lexicographically earliest sequence of binary encodings of solutions satisfying 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.

Original entry on oeis.org

1, 3, 5, 13, 21, 51, 83, 175, 303, 639, 1143, 2539, 4571, 9711, 17903, 36735, 69499, 143339, 270327, 556031, 1080315, 2195431, 4259819, 8646647, 17031163, 34078647, 67632893, 136282091, 270467055, 541064701, 1077935867, 2162163707, 4311220203, 8623484927

Offset: 1

Antti Karttunen, Feb 05 2001

nonn

I.e. like A059873, but the encodings must be "odd", with the least significant bit set. I do not know whether this sequence can be extended infinitely.

Sean A. Irvine, Table of n, a(n) for n = 1..50

Cf. A059873, A059875.

More terms from Naohiro Nomoto, Sep 12 2001

More terms from Sean A. Irvine, Oct 13 2022

cp's OEIS Frontend

A059876 a(n) = bin_prime_sum(n).

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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).

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A059875 The lexicographically last sequence of binary encodings of solutions satisfying the equation given in A059871.

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A059872 Solutions to the equation given in A059871, encoded as binary vectors and converted to decimal.

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A059874 The lexicographically earliest sequence of binary encodings of solutions satisfying 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.

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