A309225 -id:A309225 - cp's OEIS frontend

A008348 a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).

0, 2, 5, 0, 7, 18, 5, 22, 3, 26, 55, 24, 61, 20, 63, 16, 69, 10, 71, 4, 75, 2, 81, 164, 75, 172, 71, 174, 67, 176, 63, 190, 59, 196, 57, 206, 55, 212, 49, 216, 43, 222, 41, 232, 39, 236, 37, 248, 25, 252, 23, 256, 17, 258, 7, 264, 1, 270, 541, 264, 545, 262, 555

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) < 2*prime(n). Conjecture: a(n) > 0 for n > 3. - Thomas Ordowski, Dec 03 2016 [This conjecture is false, because a(369019)=0. The next counterexample occurs at n = 22877145. - Dmitry Kamenetsky, Feb 14 2017. (Cf. A309225.)]

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A008344, A022831, A022837, A309225.

A008348 := proc(n) option remember; if n = 0 then 0 elif A008348(n-1)>=ithprime(n) then A008348(n-1)-ithprime(n); else A008348(n-1)+ithprime(n); fi; end;
# Maple from N. J. A. Sloane, Aug 31 2019 (Start)
# Riecaman transform
Riecaman := proc(a,s,M)
# Start with s, add or subtract a[n], get M terms. If a has w terms, can get M=w+1 terms.
local b,M2,n,t;
if whattype(a) <> list then ERROR("First argument should be a list"); fi;
if a[1]=0 then ERROR("a[1] should not be zero"); fi;
M2 := min(nops(a),M-1);
b:=[s]; t:=s;
for n from 1 to M2 do
   if a[n]>t then t:=t+a[n] else t:=t-a[n]; fi; b:=[op(b),t]; od:
b; end;
# Riecaman transform of primes, starting at s=0
p1:=[seq(ithprime(i),i=1..100)];
q0:=Riecaman(p1,0,99);
# End

a := {0}; For[n = 2, n < 100, n++, If[a[[n - 1]] >= Prime[n - 1], b := a[[n - 1]] - Prime[n - 1], b := a[[n - 1]] + Prime[n - 1];]; a = Append[a, b]]; a (* Stefan Steinerberger, May 02 2006 *)
nxt[{n_,a_}]:={n+1,If[aHarvey P. Dale, Sep 13 2024 *)

lista(nn) = {print1(a=0, ", "); for (n=1, nn, if (a < (p=prime(n)), a += p, a -= p); print1(a, ", "););} \\ Michel Marcus, Dec 04 2016

a(n) = c(1)p(1) + ... + c(n)p(n), where c(i) = 1 if a(i-1) > p(i) and c(i) = -1 if a(i-1) <= p(i) (p(i) = primes). - Clark Kimberling

More terms from Clark Kimberling

Name edited by Dmitry Kamenetsky, Feb 14 2017

A324787 Index of n-th low point in A022837.

Original entry on oeis.org

2, 5, 12, 29, 78, 199, 508, 1355, 3592, 9589, 25752, 70579, 194228, 539961, 1507602, 4228745, 11913940, 33690443, 95581182, 272003821, 776082524, 2219823175, 6363074656

Offset: 0

N. J. A. Sloane, Sep 04 2019

A "low point" in a sequence is a term which is less than the previous term (this condition is skipped for the initial term) and which is followed by two or more increases.

Rémy Sigrist, PARI program for A324787

Cf. A022837, A324788, A324789, A324790.

If the basic sequence (A022837) began with 0 instead of 1 we would get A008348, A309226, A324782, A324783, A309225.

Riecaman := proc(a,s,M)
# Start with s, add or subtract a[n], get M terms. If a has w terms, can get M=w+1 terms.
local b,M2,n,t;
if whattype(a) <> list then ERROR("First argument should be a list"); fi;
if a[1]=0 then ERROR("a[1] should not be zero"); fi;
M2 := min(nops(a),M-1);
b:=[s]; t:=s;
for n from 1 to M2 do
   if a[n]>t then t:=t+a[n] else t:=t-a[n]; fi; b:=[op(b),t]; od:
b; end;
blocks := proc(a,S) local b,c,d,M,L,n;
# Given a list a, whose leading term has index S, return [b,c,d], where b lists the indices of the low points in a, c lists the values of a at the low points, and d lists the length of runs between the low points.
b:=[]; c:=[]; d:=[]; L:=1;
# if a[1] a low point?
   n:=1;
   if( (a[n+1]>a[n]) and (a[n+2]>a[n+1]) ) then
   b:=[op(b),n+S-1]; c:=[op(c),a[n]]; d:=[op(d), n-L]; L:=n; fi;
for n from 2 to nops(a)-2 do
# if a[n] a low point?
   if( (a[n-1]>a[n]) and (a[n+1]>a[n]) and (a[n+2]>a[n+1]) ) then
   b:=[op(b),n+S-1]; c:=[op(c),a[n]]; d:=[op(d), n-L]; L:=n; fi; od;
[b,c,d]; end;
p0:=[seq(ithprime(n),n=1..100001)]:
q1:=Riecaman(p0,1,100000):
blocks(q1,0); # produces [the present sequence, A324788, A324789]

PARI
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See Links section.
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Modified definition to make offset 0. - N. J. A. Sloane, Oct 02 2019

a(12)-a(22) from Rémy Sigrist, Oct 18 2020

A324790 Indices of zeros in A022837.

Original entry on oeis.org

2, 12, 508, 3592, 25752, 1507602

Offset: 1

N. J. A. Sloane, Sep 04 2019

For k > 0, A022837(2*k-1) is odd and A022837(2*k) is even. So a(n) is even. - Seiichi Manyama, Sep 06 2019

If it exists, a(7) > 10^10. - Michael S. Branicky, Sep 02 2024

Cf. A022837, A309225. See A324787 for further information.

a(5)-a(6) from Seiichi Manyama, Sep 05 2019

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A008348 a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).

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A324787 Index of n-th low point in A022837.

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Maple

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A324790 Indices of zeros in A022837.

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