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 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. (Formerly M0660) 8

%I M0660

%S 2,3,5,7,11,13,19,23,29,31,37,43,47,53,59,61,67,71,79,101,107,127,131,

%T 139,149,151,157,173,181,191,197,199,211,223,229,263,269,277,283,311,

%U 317,331,347,349,367,373,383,397,419,421,431,461,463,491,509,523,547,557,571

%N Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007459/b007459.txt">Table of n, a(n) for n = 1..1000</a>

%H S. Burris and S. Lee, <a href="http://www.jstor.org/stable/2324454">Tarski's high school identities</a>, Amer. Math. Monthly 100 (1993), 231-236.

%H R. G. Wilson, V, <a href="/A007459/a007459.pdf">Note to N. J. A. Sloane with attachment</a>, (Annotated scanned copy of The Am. Math. Mo. Vol. 100 No. 3 pp. 233, Mar. 1993).

%H R. G. Wilson, V, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>

%p a:=[2]; P:=1; j:=1;

%p for n from 2 to 32 do

%p P:=P*a[n-1]^2;

%p for i from j+1 to 250 do

%p if (P mod (ithprime(i)-1)) = 0 then

%p a:=[op(a),ithprime(i)]; j:=i; break; fi;

%p od:

%p od:

%p a; # _N. J. A. Sloane_, Feb 12 2017

%t 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 ]

%o a007459 n = a007459_list !! (n-1)

%o a007459_list = f 1 a000040_list where

%o f q (p:ps) = if mod q (p - 1) == 0 then p : f (q * p ^ 2) ps else f q ps

%o -- _Reinhard Zumkeller_, Apr 14 2013

%o (PARI) step(v)=my(N=vecprod(v)^2);forprime(p=v[#v]+1,,if(N%(p-1)==0,return(concat(v,p))))

%o first(n)=my(v=[2]);for(i=2,n,v=step(v));v \\ _Charles R Greathouse IV_, Jun 11 2015

%Y Cf. A057447, A057448, A057459, A282027.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_

%E Definition clarified by _N. J. A. Sloane_, Feb 12 2017

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Last modified December 13 01:33 EST 2017. Contains 295954 sequences.