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

Author

Labos Elemer, Jul 30 2002

Keywords

• nonn

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

PARI

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

Formula

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

Extensions

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

More terms from Giovanni Resta, Sep 03 2019