A007459 - OEIS

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)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Named after the British mathematician Denis A. Higgs (1932-2011). - Amiram Eldar, Jun 05 2021

No prime of the form a*b^k + 1 (those in A089200) with a > 0, b > 1 and k > 2 is a Higgs's prime. - Mauro Fiorentini, Aug 08 2023

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

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:

a; # N. J. A. Sloane, Feb 12 2017

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 *)

PROG

(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

(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

CROSSREFS

Cf. A057447, A057448, A057459, A089200, A282027.

Sequence in context: A042987 A089189 A097375 * A129944 A176162 A152900

Adjacent sequences: A007456 A007457 A007458 * A007460 A007461 A007462