A008347 -id:A008347 - cp's OEIS frontend

A077133 a(n) is the difference between the sum of the first n even-indexed primes and the sum of the first n odd-indexed primes.

1, 3, 5, 7, 13, 19, 21, 27, 29, 33, 39, 45, 49, 53, 57, 61, 63, 65, 71, 77, 79, 81, 83, 95, 97, 103, 113, 119, 121, 125, 135, 139, 143, 149, 151, 157, 163, 167, 175, 183, 185, 187, 191, 199, 201, 213, 217, 221, 233, 251, 261, 267, 273, 279, 281, 287, 289, 299

Offset: 1

Jon Perry, Nov 29 2002

nonn

Some odd numbers such as 11, 17, 23 and 25 never appear.

			a(2) = 3 as the sum of the first 2 even-indexed primes is prime(2) + prime(4) = 3 + 7 = 10, the sum of the first 2 odd-indexed primes is prime(1) + prime(3) = 2 + 5 = 7 and 10 - 7 = 3. [edited by _Paolo Xausa_, Apr 12 2023]

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008347, A077126, A077131.

with(numtheory): A008347 := proc(n) option remember; if n = 0 then 0 else abs(A008347(n-1)-ithprime(n)); fi; end proc:
seq(A008347(2n),n=1..80); # Ridouane Oudra, Aug 31 2019

Table[ Sum[ Prime[2i], {i, 1, n}] - Sum[ Prime[2i - 1], {i, 1, n}], {n, 1, 60}]
A077133[nmax_]:=Accumulate[Prime[Range[2,2nmax,2]]-Prime[Range[1,2nmax,2]]];A077133[100] (* Paolo Xausa, Apr 12 2023 *)

my(pc=1, p1s=0, p2s=0); forprime (p=2, 500, pc=!pc; if (pc, p1s+=p, p2s+=p); if (pc,print1(p1s-p2s, ", ")))

a(n) = Sum_{i=0..n-1} (prime(2*i+2) - prime(2*i+1)).
a(n) = A008347(2n). - Ridouane Oudra, Aug 31 2019
a(n) = A077126(n) - A077131(n). - Michel Marcus, Oct 05 2019

Edited and extended by Robert G. Wilson v, Nov 30 2002

Name clarified by Paolo Xausa, Apr 12 2023

A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

Original entry on oeis.org

1, 4, 7, 9, 13, 17, 19, 23, 25, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Zhi-Wei Sun, Mar 31 2012

nonn

On Mar 28 2012, Zhi-Wei Sun conjectured that a(n) is the (n+1)-th prime p_{n+1} with the only exceptions being a(1)=1, a(2)=4, a(4)=9 and a(9)=25. He has shown that 2(s_k)^2 (k=1,...,n) are indeed pairwise distinct modulo p_{n+1} and hence a(n) does not exceed p_{n+1}.
Note that the sequence 0,s_1,s_2,s_3,... is A008347 introduced by N. J. A. Sloane and J. H. Conway.
Compare a(n) with the sequence A210640.
The conjecture was verified for n up to 2*10^5 by the author in 2018, and for n up to 3*10^5 by Chang Zhang (a student at Nanjing Univ.) in June 2020. - Zhi-Wei Sun, Jun 22 2020

			We have a(4)=9 since 2(s_1)^2=8, 2(s_2)^2=2, 2(s_3)^2=32, 2(s_4)^2=18 are pairwise distinct modulo 9 but not pairwise distinct modulo any of 1,...,8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..600
Zhi-Wei Sun, An amazing recurrence for primes, a message to Number Theory List, March 31, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A008347, A210640, A210393, A210394, A210186, A210144, A208494, A208643, A207982.

s[n_]:=Sum[(-1)^k*Prime[k],{k,1,n}]
f[n_]:=2*s[n]^2
R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,1,Prime[n+1]}];
   Print[n];Label[aa];Continue,{n,1,600}]

A240860 a(n) = Sum_{i=1..n} (-1)^{i+1} prime(i)^2, where prime(k) denotes the k-th prime: alternating sum of the squares of the first n primes.

Original entry on oeis.org

4, -5, 20, -29, 92, -77, 212, -149, 380, -461, 500, -869, 812, -1037, 1172, -1637, 1844, -1877, 2612, -2429, 2900, -3341, 3548, -4373, 5036, -5165, 5444, -6005, 5876, -6893, 9236, -7925, 10844, -8477, 13724, -9077, 15572, -10997, 16892, -13037, 19004, -13757

Offset: 1

Timothy Varghese, May 06 2014

sign

For n even this is the negative of the sum of (3^2 - 2^2) + (7^2 - 5^2) + ... + (prime(n)^2 - prime(n-1)^2). But this is half of the terms in the sum of (3^2 - 2^2) + (5^2 - 3^2) + (7^2 - 5^2) + ... + (prime(n)^2 - prime(n-1)^2) which has a sum that telescopes to prime(n)^2 - 4. Thus a good estimate of a(n) (half the terms) is prime(n)^2/2 (half the square of the n-th prime) which works well up to n = 10000. For odd n, add prime(n)^2 to the estimate for even n.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Timothy Varghese, Alternating sums, MathOverflow, May 2014.

Cf. A000040, A008347, A024450.

a(n) = sum(i=1, n, (-1)^(i+1)*prime(i)^2); \\ Michel Marcus, May 09 2014

A213202 Number of ways to write n = p + sum_{k=1..m}(-1)^(m-k)*p_k, where p is a Sophie Germain prime and p_k is the k-th prime.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 2, 2, 3, 1, 2, 2, 2, 3, 2, 1, 3, 2, 1, 3, 1, 3, 5, 2, 2, 3, 2, 3, 4, 4, 4, 2, 3, 3, 3, 3, 2, 1, 2, 4, 5, 4, 4, 4, 2, 3, 3, 4, 4, 3, 2, 1, 4, 6, 6, 4, 4, 4, 4, 4, 4, 4, 2, 3, 3, 5, 6, 2, 2, 1, 4, 4, 5, 3, 3, 1, 2, 5, 4, 5, 5, 2, 4, 5, 7, 2, 5, 1, 5, 4, 4, 4, 6, 3, 2, 6, 4, 5, 4

Offset: 1

Zhi-Wei Sun, Mar 01 2013

nonn

Conjecture: a(n)>0 for all n>2.
This has been verified for n up to 10^7.
Let s_n=sum_{k=1}^n(-1)^{n-k}p_k for n=1,2,3,... The author also made the following conjectures:
(1) For each n>2, there is an integer k>0 such that 3(n-s_k)-1 and 3(n-s_k)+1 are twin primes.
(2) For each n>3, there is an integer k>0 such that 3(n-s_k)-2 and 3(n-s_k)+2 are cousin primes.
(3) Every n=6,7,... can be written as p+s_k (k>0) with p and p+6 sexy primes.
(4) Any integer n>3 different from 65 and 365 can be written as p+s_k (k>0) with p a term of A210479.
(5) Each integer n>8 can be written as q+s_k (k>0) with q-4, q, q+4 all practical.
(6) Any integer n>1 can be written as j(j+1)/2+s_k with j,k>0.

			a(11)=1 since 11=3+p_5-p_4+p_3-p_2+p_1 with 3 and 2*3+1 both prime.
a(182)=1 since 182=179+(7-5+3-2) with 179 and 2*179+1 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A005384, A008347, A005153, A210479.

sp[n_]:=qq[n]=PrimeQ[n]&&PrimeQ[2n+1]
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
a[n_]:=a[n]=Sum[If[n-s[m]>0&&sp[n-s[m]],1,0],{m,1,n}]
Do[Print[n," ",a[n]],{n,1,100}]

A226913 Not of form sum_{i=0..n-1} (-1)^i prime(n-i), where prime(k) denotes the k-th prime.

Original entry on oeis.org

6, 9, 10, 11, 14, 15, 17, 20, 23, 24, 25, 28, 30, 31, 34, 35, 36, 37, 41, 42, 43, 46, 47, 48, 50, 51, 55, 58, 59, 60, 62, 64, 66, 67, 68, 69, 72, 73, 74, 75, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 93, 94, 98, 99, 100, 101, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 117, 118, 120, 122, 123, 124

Offset: 1

Ralf Stephan, Sep 01 2013

nonn

Complement of A008347.

Robert G. Wilson v, Table of n, a(n) for n = 1..10140

A242266 a(n) = |{0 < g < prime(n): g is a primitive root mod prime(n) with g = sum_{j=1..k} prime(j) for some k > 0}|.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 1, 3, 2, 2, 3, 3, 2, 3, 3, 1, 3, 2, 3, 3, 5, 2, 2, 6, 2, 4, 1, 3, 2, 3, 5, 2, 2, 2, 6, 6, 6, 7, 2, 6, 4, 4, 4, 5, 6, 5, 6, 3, 1, 3, 7, 9, 9, 2, 5, 2, 2, 6, 4, 5, 6, 6, 4, 3, 8, 3, 6, 6, 7, 5, 6, 9, 8, 6, 4, 4

Offset: 1

Zhi-Wei Sun, May 09 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1. In other words, for any odd prime p, there is a positive integer k such that the sum of the first k primes is not only a primitive root modulo p but also smaller than p.
(ii) For any n > 1, there is a number k among 1, ..., n such that sum_{j=1..k}(-1)^(k-j)*prime(j) is a primitive root modulo prime(n).
We have verified parts (i) and (ii) for n up to 700000 and 250000 respectively. Note that prime(700000) > 10^7.

			a(4) = 1 since prime(1) + prime(2) = 2 + 3 = 5 is a primitive root modulo prime(4) = 7 with 5 < 7.
a(7) = 2 since prime(1) = 2 and prime(1) + prime(2) + prime(3) = 2 + 3 + 5 = 10 are not only primitive roots modulo prime(7) = 17 but also smaller than 17.
a(53) = 1 since sum_{j=1..10} prime(j) = 129 is a primitive root modulo prime(53) = 241 with 129 < 241.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A008347, A007504, A236966, A239957, A241476, A241504, A241516, A242248, A242250, A242277.

f[0]=0
f[n_]:=Prime[n]+f[n-1]
dv[n_]:=Divisors[n]
Do[m=0;Do[If[f[k]>=Prime[n],Goto[cc]];Do[If[Mod[f[k]^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,n}];Label[cc];Print[n," ",m];Continue,{n,1,80}]

A121573 Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n).

Original entry on oeis.org

1, 3, 5, 7, 3, 1, 3, 3, 7, 5, 3, 5, 3, 1, 11, 13, 21, 25, 23, 23, 31, 33, 43, 35, 37, 33, 29, 29, 33, 31, 35, 33, 43, 47, 49, 51, 51, 53, 49, 51, 59, 63, 65, 61, 63, 59, 63, 65, 55, 43, 39, 35, 39, 39, 43, 41, 51, 43, 45, 41, 33, 35, 33, 31, 31, 35, 33, 29, 25, 15, 7, 5, 9, 7, 17, 15, 31, 35, 33, 35, 43, 45, 47, 53, 55, 63, 67, 59, 51, 63, 61

Offset: 1

T. D. Noe, Aug 08 2006

This sequence was inspired by seeing two lines in the plot of A008347. It was expected that, on average, the gaps above prime(2n) would be larger than the gaps below prime(2n) and hence a(n) would be a mostly positive sequence. With some exceptions, this is true for the first 6330 terms. However, as the plot shows, over 500000 negative terms follow!

			a(6)=1 because the prime gaps above and below the even-indexed primes (3,7,13,19,29,37) are 2,4,4,4,2,4 and 1,2,2,2,6,6, respectively. The sums of these gaps are 20 and 19, which differ by 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. D. Noe, Plot of 10^6 terms

Cf. A008347 (alternating sum of primes), A036263 (second difference of primes).

a121573 n = a121573_list !! (n-1)
a121573_list = scanl1 (+) $ map a036263 [1, 3 ..]
-- Reinhard Zumkeller, Aug 02 2012

s=0; Table[s=s+Prime[2n-1]+Prime[2n+1]-2*Prime[2n], {n,115}]
With[{g=Transpose[Differences/@Partition[Prime[Range[400]],3,2]]}, Accumulate[g[[2]]]-Accumulate[g[[1]]]](* Harvey P. Dale, May 28 2013 *)

a(n) = Sum_{k=1..n} (prime(2k+1) - prime(2k)) - Sum_{k=1..n} (prime(2k) - prime(2k-1)).
a(n) = Sum_{k=1..n} A036263(2k-1).
a(n) = prime(2n+1) - 2*A008347(2n) - 2.

Typo in Formula fixed by Reinhard Zumkeller, Aug 02 2012

A131196 Numbers n such that 1 + S(n) = 0, where S(n) = (S(n-1) + A000040(n))*(-1)^n; S(0)=0, n=>1.

Original entry on oeis.org

22, 38, 200, 302, 468, 560, 1186, 1208, 2006, 2026, 2106, 23698, 23716, 25968, 25990, 26706, 48316, 311888, 311914, 311938, 313866, 331540, 332002, 377102, 377634, 377670, 377748, 378428, 378452, 378996, 379026, 379090, 387618, 388140, 389398

Offset: 1

Manuel Valdivia, Sep 26 2007

nonn

The terms are equal to A130642 for n/2 even (70 terms) and to A130643 for n/2 odd (91 terms).

			S(21)=(..((((0+2)*-1)+3)*1)+5)*-1)+7)*1)+11)*- 1)+13)*1)+...+71)*1)+73)*-1 = -80, 1 + S(22) =1 + (-80 + 79)*1 = 0, hence 22 is a term.
S(37)=(..((((0+2)*-1)+3)*1)+5)*-1)+7)*1)+11)*- 1)+13)*1)+...+151)*1)+157)*-1 = -164, 1 + S(38) =1 + (-164 + 163)*1 = 0, hence 38 is a term.

Cf. A130642, A130643, A008347, A066033, A000040.

S=0;a=0; Do[S=(S+Prime[n])*(-1)^n; If[1+S==0,a++; Print[a," ",n]], {n, 1, 10^8, 1}]

A131197 Numbers n such that 1 - S(n) = 0, where S(n) = (S(n-1) + A000040(n))*(-1)^n; S(0)=0, n=>1.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 190, 194, 306, 308, 462, 464, 472, 474, 476, 478, 490, 1884, 1890, 1938, 23636, 23656, 23850, 25226, 25834, 25984, 26642, 26650, 26924, 26998, 27000, 311922, 313880, 313946, 331676, 331762, 331782, 332676, 377078, 377518, 377666

Offset: 1

Manuel Valdivia, Sep 26 2007

nonn

The terms are equal to A130642 for n/2 odd (100 terms) and to A130643 for n/2 even (86 terms).

			S(11)=(..((((0+2)*-1)+3)*1)+5)*-1)+7)*1)+11)*- 1)+13)*1)+...+29)*1)+31)*-1 = -36, 1 - S(12)=1 - (-36 + 37)*1 = 0, hence 12 is a term.
S(13)=(..((((0+2)*-1)+3)*1)+5)*-1)+7)*1)+11)*- 1)+13)*1)+...+37)*1)+41)*-1 = -42, 1 - S(14)=1 - (-42 + 43)*1 = 0, hence 14 is a term.

Cf. A130642, A130643, A008347, A066033, A000040.

S=0;a=0; Do[S=(S+Prime[n])*(-1)^n; If[1-S==0,a++; Print[a," ",n]], {n, 1, 10^8, 1}]

A136288 Primes which are the absolute value of the alternating sum and difference of the first n primes.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 29, 53, 61, 71, 79, 83, 97, 103, 113, 139, 149, 151, 157, 163, 167, 191, 199, 233, 251, 281, 337, 347, 353, 397, 421, 433, 461, 563, 599, 643, 719, 773, 797, 811, 859, 883, 953, 977, 1031, 1039, 1061, 1063, 1091, 1097, 1153, 1187, 1201, 1213

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

			5 = abs(2-3+5-7+11-13) (first 6 primes),
7 = abs(2-3+5-7+11-13+17-19) (first 8 primes),
etc.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..10000 (first 144 terms by Paolo P. Lava and Giorgio Balzarotti)

Cf. A000040, A008347, A013918.

P:=proc(n) local i,s; s:=0; for i from 1 by 1 to n do s:=s+(-1)^i*ithprime(i); if isprime(abs(s)) then print(abs(s)); fi; od; end: P(1000);

Select[Abs@Accumulate@Table[(-1)^(k+1)*Prime@k,{k,355}],PrimeQ] (* Giorgos Kalogeropoulos, Sep 22 2021 *)

A000040 INTERSECT A008347. - R. J. Mathar, Apr 04 2008

cp's OEIS Frontend

A077133 a(n) is the difference between the sum of the first n even-indexed primes and the sum of the first n odd-indexed primes.

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A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

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A240860 a(n) = Sum_{i=1..n} (-1)^{i+1} prime(i)^2, where prime(k) denotes the k-th prime: alternating sum of the squares of the first n primes.

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A213202 Number of ways to write n = p + sum_{k=1..m}(-1)^(m-k)*p_k, where p is a Sophie Germain prime and p_k is the k-th prime.

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A226913 Not of form sum_{i=0..n-1} (-1)^i prime(n-i), where prime(k) denotes the k-th prime.

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A242266 a(n) = |{0 < g < prime(n): g is a primitive root mod prime(n) with g = sum_{j=1..k} prime(j) for some k > 0}|.

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A121573 Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n).

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A131196 Numbers n such that 1 + S(n) = 0, where S(n) = (S(n-1) + A000040(n))*(-1)^n; S(0)=0, n=>1.

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