cp's OEIS Frontend

A073919 Smallest prime p with bigomega(p-1)=n, where bigomega(m)=A001222(m) is the number of prime divisors of m (counted with multiplicity).

%I A073919 #26 Jan 07 2025 10:03:10
%S A073919 2,3,5,13,17,73,97,193,257,769,3457,7681,15361,12289,40961,114689,
%T A073919 65537,737281,1376257,786433,5308417,7340033,14155777,28311553,
%U A073919 104857601,113246209,167772161,469762049,2113929217,1811939329
%N A073919 Smallest prime p with bigomega(p-1)=n, where bigomega(m)=A001222(m) is the number of prime divisors of m (counted with multiplicity).
%H A073919 David W. Wilson and Robert G. Wilson v, <a href="/A073919/b073919.txt">Table of n, a(n) for n = 0..351</a>
%e A073919 a(2) = 5 = 2*2 + 1. a(5) = 73 = 2*2*2*3*3 + 1.
%t A073919 ptns[n_, 0] := If[n==0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n<k, Return[{}]]; ptns[n, k]=1+Union@@Table[PadRight[ #, k]&/@ptns[n-k, r], {r, 0, k}]]; a[n_] := Module[{i, l, v}, v=Infinity; For[i=n, True, i++, l=(Times@@Prime/@#&)/@ptns[i, n]; If[Min@@l>v, Return[v]]; minp=Min@@Select[l+1, ProvablePrimeQ]; If[minp<v, v=minp]]] (* First do <<NumberTheory`PrimeQ`. ptns[n, k] is list of partitions of n into exactly k parts *) Array[a, 30, 0]
%t A073919 With[{x=Table[{Prime[n],PrimeOmega[Prime[n]-1]},{n,104000000}]},Transpose[ Table[ SelectFirst[x,#[[2]]==i&],{i,0,29}]][[1]]] (* _Harvey P. Dale_, Sep 28 2014 *)
%o A073919 (PARI)
%o A073919 almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
%o A073919 a(n) = if(n==0, return(2)); my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, if(isprime(v[k]+1), return(v[k]+1))); x=y+1; y=2*x); \\ _Daniel Suteu_, Jan 07 2025
%Y A073919 Cf. A118883.
%K A073919 nonn
%O A073919 0,1
%A A073919 _Amarnath Murthy_, Aug 18 2002
%E A073919 Edited by _Dean Hickerson_, Nov 12 2002