A007459 Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 383, 397, 419, 421, 431, 461, 463, 491, 509, 523, 547, 557, 571
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- Stanley Burris and Simon Lee, Tarski's high school identities, Amer. Math. Monthly, Vol. 100, No. 3 (1993), pp. 231-236.
- Robert G. Wilson v, Note to N. J. A. Sloane with attachment, (Annotated scanned copy of The Am. Math. Mo. Vol. 100 No. 3 pp. 233, Mar. 1993).
- Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993
Programs
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Haskell
a007459 n = a007459_list !! (n-1) a007459_list = f 1 a000040_list where f q (p:ps) = if mod q (p - 1) == 0 then p : f (q * p ^ 2) ps else f q ps -- Reinhard Zumkeller, Apr 14 2013
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Maple
a:=[2]; P:=1; j:=1; for n from 2 to 32 do P:=P*a[n-1]^2; for i from j+1 to 250 do if (P mod (ithprime(i)-1)) = 0 then a:=[op(a),ithprime(i)]; j:=i; break; fi; od: od: a; # N. J. A. Sloane, Feb 12 2017
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Mathematica
f[ n_List ] := (a = n; b = Apply[ Times, a^2 ]; d = NextPrime[ a[ [ -1 ] ] ]; While[ ! IntegerQ[ b/(d - 1) ] || d > b, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ] nxt[{p_,a_}]:=Module[{np=NextPrime[a]},While[PowerMod[p,2,np-1] != 0,np = NextPrime[np]];{p*np,np}]; NestList[nxt,{2,2},60][[All,2]] (* Harvey P. Dale, Jul 09 2021 *) -
PARI
step(v)=my(N=vecprod(v)^2);forprime(p=v[#v]+1,,if(N%(p-1)==0,return(concat(v,p)))) first(n)=my(v=[2]);for(i=2,n,v=step(v));v \\ Charles R Greathouse IV, Jun 11 2015
Extensions
More terms from David W. Wilson
Definition clarified by N. J. A. Sloane, Feb 12 2017
Comments