A035134 - OEIS

%I #36 Apr 10 2022 09:18:40

%S 6,22,33,55,66,77,111,141,161,202,222,262,282,303,323,393,434,454,474,

%T 494,505,515,535,545,555,565,595,606,626,646,707,717,737,767,777,818,

%U 838,858,878,898,939,949,959,969,979,989,1001,1111,1221,1441,1551,1661

%N Squarefree composite palindromes.

%C Palindromes with at least two and all distinct prime factors.

%H Robert Israel, <a href="/A035134/b035134.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%p N:= 4: # to get all terms with <= N digits

%p revdigs:= proc(n) local L, j, nL;

%p   L:= convert(n, base, 10); nL:= nops(L);

%p   add(L[j]*10^(nL-j), j=1..nL);

%p end proc:

%p palis:= $0..9:

%p for d from 2 to N do

%p   if d::even then

%p     palis:= palis, seq(x*10^(d/2)+revdigs(x), x=10^(d/2-1)..10^(d/2)-1)

%p   else

%p     palis:= palis, seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+revdigs(x), y=0..9), x=10^((d-3)/2)..10^((d-1)/2)-1);

%p   fi

%p od:

%p select(t -> not(isprime(t)) and numtheory:-issqrfree(t), [palis][3..-1]): # _Robert Israel_, Sep 18 2016

%t sqfQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]<=1; palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; Select[Range[2,1662],!PrimeQ[#] && sqfQ[#] && palQ[#] &] (* _Jayanta Basu_, May 12 2013 *)

%t Select[Range[2000],PalindromeQ[#]&&SquareFreeQ[#]&&CompositeQ[#]&] (* _Harvey P. Dale_, Apr 10 2022 *)

%o (PARI) isA002113(n)=n=digits(n);for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1;

%o is(n) = n>1 && isA002113(n) && issquarefree(n) && !isprime(n) \\ _Altug Alkan_, Sep 19 2016

%o \\ in and output digits as a vector.

%o (PARI) nxtA002113(n)={my(d=n); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); d}\\sum(i=1, #d, 10^(#d-i)*d[i])}

%o \\ all terms up to n digits

%o lista(n) = {my(p = [6],l=List(), sp, i); while(#p <= n, sp = sum(i=1,#p,p[i]*10^(#p-i)); if(issquarefree(sp)&&!isprime(sp), listput(l,sp)); p=nxtA002113(p));l} \\ _David A. Corneth_, Sep 19 2016

%o (Python)

%o from itertools import product

%o from sympy import factorint, isprime

%o def pals(d, base=10): # all d-digit palindromes

%o     digits = "".join(str(i) for i in range(base))

%o     for p in product(digits, repeat=d//2):

%o         if d > 1 and p[0] == "0": continue

%o         left = "".join(p); right = left[::-1]

%o         for mid in [[""], digits][d%2]: yield int(left + mid + right)

%o def ok(pal): f = factorint(pal); return len(f)>1 and all(f[p]<2 for p in f)

%o print(list(filter(ok, (p for d in range(1, 5) for p in pals(d) if ok(p))))) # _Michael S. Branicky_, Jun 22 2021

%Y Cf. A002113, A005117, A035132, A035135.

%K nonn,base

%O 1,1

%A _Patrick De Geest_, Nov 15 1998