A035133 -id:A035133 - cp's OEIS frontend

A035132 Nonsquarefree palindromes.

4, 8, 9, 44, 88, 99, 121, 171, 212, 232, 242, 252, 272, 292, 333, 343, 363, 404, 414, 424, 444, 464, 484, 525, 575, 585, 616, 636, 656, 666, 676, 686, 696, 747, 808, 828, 848, 868, 888, 909, 999, 1331, 1881, 2112, 2332, 2552, 2662, 2772, 2992, 3663, 3773

Offset: 1

Patrick De Geest, Nov 15 1998

			4114 = 2 * 11^2 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A002113 and A013929.
Supersequence of A077572.
Cf. A005117, A035133, A071251, A075804 (first differences).

sfQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]>1; palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; Select[Range[2,3776],sfQ[#] && palQ[#] &] (* Jayanta Basu, May 12 2013 *)
Select[Range[4000], PalindromeQ[#] && !SquareFreeQ[#] &] (* Amiram Eldar, Feb 25 2021 *)

from itertools import product
from sympy.ntheory.factor_ import core
def palsthru(maxdigits):
  midrange = [[""], [str(i) for i in range(10)]]
  for digits in range(1, maxdigits+1):
    for p in product("0123456789", repeat=digits//2):
      left = "".join(p)
      if len(left) and left[0] == '0': continue
      for middle in midrange[digits%2]: yield int(left+middle+left[::-1])
def okpal(p): return p > 3 and core(p, 2) != p
print(list(filter(okpal, palsthru(4)))) # Michael S. Branicky, Apr 08 2021

A035135 Cubefree composite palindromes.

Original entry on oeis.org

4, 6, 9, 22, 33, 44, 55, 66, 77, 99, 111, 121, 141, 161, 171, 202, 212, 222, 242, 252, 262, 282, 292, 303, 323, 333, 363, 393, 404, 414, 434, 444, 454, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 626, 636, 646, 666, 676, 707, 717

Offset: 1

Patrick De Geest, Nov 15 1998

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cubefree.

Intersection of A002113, A002808 and A004709.

Cf. A035133, A035134.

cufQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]<=2; palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; Select[Range[2,718],!PrimeQ[#] && cufQ[#] && palQ[#] &] (* Jayanta Basu, May 12 2013 *)

A133514 Biquadrateful (i.e., not biquadrate-free) palindromes.

Original entry on oeis.org

272, 464, 656, 848, 2112, 2992, 4224, 6336, 8448, 14641, 21312, 21712, 23232, 23632, 25152, 25552, 25952, 27072, 27472, 27872, 29392, 29792, 31213, 40304, 40704, 42224, 42624, 44144, 44544, 44944, 46064, 46464, 46864, 48384, 48784, 61216, 61616, 62426, 63136

Offset: 1

Jonathan Vos Post, Nov 30 2007

This is to A035133 as 4th powers are to cubes. To make an analogy between analogies, the preceding sentence is to "A130873 is to 4th powers as A120398 is to cubes" as palindromes are to sums of two distinct prime powers.

			a(10) = 14641 = 11^4 (the smallest odd value in this sequence).
a(11) = 21312 = 2^6 * 3^2 * 37.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002113, A046101.

isA046101 := proc(n) local ifs,f ; ifs := ifactors(n)[2] ; for f in ifs do if op(2,f) >= 4 then RETURN(true) ; fi ; od: RETURN(false) ; end: isA002113 := proc(n) local digs,i ; digs := convert(n,base,10) ; for i from 1 to nops(digs) do if op(i,digs) <> op(-i,digs) then RETURN(false) ; fi ; od: RETURN(true) ; end: isA133514 := proc(n) isA046101(n) and isA002113(n) ; end: for n from 1 to 100000 do if isA133514(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 12 2008
# second Maple program:
q:= n->StringTools[IsPalindrome](""||n) and max(map(i->i[2], ifactors(n)[2]))>3:
select(q, [$1..70000])[];  # Alois P. Heinz, Sep 27 2023

a = {}; For[n = 2, n < 100000, n++, If[FromDigits[Reverse[IntegerDigits[n]]] == n, b = 0; For[l = 1, l < Length[FactorInteger[n]] + 1, l++, If[FactorInteger[n][[l,2]] > 3, b = 1]]; If[b == 1, AppendTo[a, n]]]]; a (* Stefan Steinerberger, Dec 26 2007 *)
Select[Range@100000,PalindromeQ@#&&3Hans Rudolf Widmer, Sep 27 2023 *)

A002113 INTERSECTION A046101.

More terms from Stefan Steinerberger, Dec 26 2007

More terms from R. J. Mathar, Jan 12 2008

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A035132 Nonsquarefree palindromes.

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A035135 Cubefree composite palindromes.

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A133514 Biquadrateful (i.e., not biquadrate-free) palindromes.

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