A073325 -id:A073325 - cp's OEIS frontend

A073324 Smallest x such that prime(x) mod c(x) = n, where prime(j) is the j-th prime, c(j) is the j-th composite number.

5, 1, 2, 8, 3, 242, 4, 245, 100, 8313, 10, 50190, 23, 8338, 3390, 12, 24, 308926, 13, 49, 25, 15, 26, 12556637, 112, 55, 117, 58, 56, 1400, 59, 265, 122, 267, 31, 12556641, 603, 270, 33, 12556639, 126, 272, 65, 66, 127, 63, 35, 50270, 37, 1413, 129, 1434, 38, 1411

Offset: 1

Labos Elemer, Jul 30 2002

nonn

			x=10: p(10)=29,c(10)=18, Mod[29,18]=11 appears first here, so a(11)=10.

Giovanni Resta, Table of n, a(n) for n = 1..400

Cf. A000040, A002808, A065859, A073325, A073326.

f[x_] := Mod[Prime[x], FixedPoint[x+PrimePi[ # ]+1&, x]] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 400000}]; t
Module[{nn=500000,cmps,prs,len},cmps=Select[Range[nn],CompositeQ];len= Length[ cmps];Table[SelectFirst[Thread[{Range[len],Prime[Range[len]],cmps}],Mod[#[[2]], #[[3]]] ==n&],{n,23}]][[All,1]] (* The program generates the first 23 terms of the sequence. *) (* Harvey P. Dale, Nov 26 2022 *)

isc(n) = (n != 1) && !isprime(n);
lista(nn) = {my(vp = primes(nn), vc = select(x->isc(x), [1..nn])); for (n=1, 50, my(k=1); while((vp[k] % vc[k]) != n, k++; if ((k>#vp) || (k>#vc), return)); print1(k, ", "););} \\ Michel Marcus, Sep 02 2019

a(n) = my(p=2); forcomposite(c=4, oo, if(p % c == n, return(primepi(p))); p = nextprime(p+1)); \\ Daniel Suteu, Sep 02 2019

a(n) = Min{x; A000040(x) mod A002808(x) = n} = Min{x; A065859(x) = n}.

a(24)-a(50) from Michel Marcus, Sep 02 2019

More terms from Giovanni Resta, Sep 03 2019

A360789 Least prime p such that p mod primepi(p) = n.

Original entry on oeis.org

2, 3, 5, 7, 379, 23, 401, 61, 59, 29, 67, 71, 467, 79, 83, 179, 431, 89, 176557, 191, 24419, 491, 97, 101, 499, 1213, 3169, 3191, 523, 229, 3187, 223, 3203, 8609, 3163, 251, 176509, 257, 24509, 263, 3253, 269, 547, 3347, 1304867, 293

Offset: 0

Robert G. Wilson v, Feb 20 2023

nonn

Inspired by A048891.

			For n=0, prime p=2 has p mod primepi(p) = 2 mod 1 = 0 so that a(0) = 2.
For n=4, no prime has p mod primepi(p) = 4 until reaching p=379 which is 379 mod 75 = 4, so that a(4) = 379.

Robert G. Wilson v, Table of n, a(n) for n = 0..19217

Cf. A038625, A004648, A048891, A052013, A073325, A073436, A162567, A171430, A171431, A171432, A171434.

V:= Array(0..100): count:= 0:
p:= 1:
for k from 1 while count < 101 do
  p:= nextprime(p);
  v:= p mod k;
  if v <= 100 and V[v] = 0 then V[v]:= p; count:= count+1 fi;
od:
convert(V,list); # Robert Israel, Feb 28 2023

t[_] := 0; p = 2; pi = 1; While[p < 1400000, m = Mod[p, pi]; If[m < 100 && t[m] == 0, t[m] = p]; p = NextPrime@p; pi++]; t /@ Range[0, 99]

a(n)={my(k=n); forprime(p=prime(n+1), oo, k++; if(p%k ==n, return(p)))} \\ Andrew Howroyd, Feb 21 2023

from sympy import prime, nextprime
def A360789(n):
    p, m = prime(n+1), n+1
    while p%m != n:
        p = nextprime(p)
        m += 1
    return p # Chai Wah Wu, Mar 18 2023

a(n) = prime(A073325(n+1)). - Kevin Ryde, Feb 21 2023

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A073324 Smallest x such that prime(x) mod c(x) = n, where prime(j) is the j-th prime, c(j) is the j-th composite number.

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A360789 Least prime p such that p mod primepi(p) = n.

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