Strong and weak primes

Strong and weak primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Definitions   (as per number theory)

•   The   prime(p)   is the   p^th   prime.

•   prime(1)   is   2
•   prime(4)   is   7

•   A   strong   prime   is when     prime(p)   is   >   [prime(p-1) + prime(p+1)] ÷ 2
•   A     weak   prime   is when     prime(p)   is   <   [prime(p-1) + prime(p+1)] ÷ 2

Task

•   Find and display (on one line) the first   36   strong primes.
•   Find and display the   count   of the strong primes below   1,000,000.
•   Find and display the   count   of the strong primes below 10,000,000.
•   Find and display (on one line) the first   37   weak primes.
•   Find and display the   count   of the weak primes below   1,000,000.
•   Find and display the   count   of the weak primes below 10,000,000.
•   (Optional)   display the   counts   and   "below numbers"   with commas.

Show all output here.

Also see

•   The OEIS article:   strong   primes.
•   The OEIS article:     weak   primes.

REXX

<lang rexx>/*REXX program lists a sequence (or a count) of ──strong── or ──weak── primes. */ parse arg N kind _ . 1 . okind; upper kind /*obtain optional arguments from the CL*/ if N== | N=="," then N= 36 /*Not specified? Then assume default.*/ if kind== | kind=="," then kind= 'STRONG' /* " " " " " */ if _\== then call ser 'too many arguments specified.' if kind\=='WEAK' & kind\=='STRONG' then call ser 'invalid 2nd argument: ' okind if kind =='WEAK' then weak= 1; else weak= 0 /*WEAK is a binary value for function.*/ w = linesize() - 1 /*obtain the usable width of the term. */ tell= (N>0); @.=; N= abs(N) /*N is negative? Then don't display. */ !.=0;  !.1=2;  !.2=3;  !.3=5;  !.4=7;  !.5=11;  !.6=13;  !.7=17;  !.8=19;  !.9=23; #= 8 @.=; @.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1; @.17=1; @.19=1; start= # + 1 m= 0; lim= 0 /*# is the number of low primes so far*/ $=; do i=3 for #-2 while lim<=N /* [↓] find primes, and maybe show 'em*/

       call strongWeak i-1;       $= strip($)   /*go see if other part of a KIND prime.*/
       end   /*i*/                              /* [↑]  allows faster loop (below).    */
                                                /* [↓]  N:  default lists up to 35 #'s.*/
  do j=!.#+2  by 2  while  lim<N                /*continue on with the next odd prime. */
  if j // 3 == 0  then iterate                  /*is this integer a multiple of three? */
  parse var  j      -1  _                     /*obtain the last decimal digit of  J  */
  if _      == 5  then iterate                  /*is this integer a multiple of five?  */
  if j // 7 == 0  then iterate                  /* "   "     "    "     "     " seven? */
  if j //11 == 0  then iterate                  /* "   "     "    "     "     " eleven?*/
  if j //13 == 0  then iterate                  /* "   "     "    "     "     "  13 ?  */
  if j //17 == 0  then iterate                  /* "   "     "    "     "     "  17 ?  */
  if j //19 == 0  then iterate                  /* "   "     "    "     "     "  19 ?  */
                                                /* [↓]  divide by the primes.   ___    */
           do k=start  to #  while !.k * !.k<=j /*divide  J  by other primes ≤ √ J     */
           if j // !.k ==0   then iterate j     /*÷ by prev. prime?  ¬prime     ___    */
           end   /*k*/                          /* [↑]   only divide up to     √ J     */
  #= # + 1                                      /*bump the count of number of primes.  */
  !.#= j;                     @.j= 1            /*define a prime  and  its index value.*/
  call strongWeak #-1                           /*go see if other part of a KIND prime.*/
  end   /*j*/
                                                /* [↓]  display number of primes found.*/

if $\== then say $ /*display any residual primes in $ list*/ say if tell then say commas(m)' ' kind "primes found."

        else say commas(m)' '     kind    "primes found below or equal to "    commas(N).

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ add: m= m+1; lim= m; if \tell & y>N then do; lim= y; m= m-1; end; else call app; return 1 app: if tell then if length($ y)>w then do; say $; $= y; end; else $= $ y; return 1 ser: say; say; say '***error***' arg(1); say; say; exit 13 /*tell error message. */ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ strongWeak: parse arg x; Lp= x - 1; Hp= x + 1; y=!.x; s= (!.Lp + !.Hp) / 2

           if weak  then if ys  then return add()               /*is  an strong prime.*/
                                      return 0                   /*not  "   "      "   */</lang>

output   when using the default input of:     36   strong

11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439

36  STRONG primes found.

output   when using the default input of:     -1000000   strong

37,723  STRONG primes found below or equal to  1,000,000.

output   when using the default input of:     -10000000   strong

320,991  STRONG primes found below or equal to  10,000,000.

output   when using the default input of:     37   weak

3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401

37  WEAK primes found.

output   when using the default input of:     -1000000   weak

37,780  WEAK primes found below or equal to  1,000,000.

output   when using the default input of:     -1000000   weak

321,750  WEAK primes found below or equal to  10,000,000.