A035132 -id:A035132 - cp's OEIS frontend

A075804 Differences between adjacent palindromic nonsquarefree numbers A035132.

4, 1, 35, 44, 11, 22, 50, 41, 20, 10, 10, 20, 20, 41, 10, 20, 41, 10, 10, 20, 20, 20, 41, 50, 10, 31, 20, 20, 10, 10, 10, 10, 51, 61, 20, 20, 20, 20, 21, 90, 332, 550, 231, 220, 220, 110, 110, 220, 671, 110, 220, 11, 110, 110, 220, 110, 110, 220, 341, 220, 330, 341

Offset: 1

Jani Melik, Oct 13 2002

			a(1) = 8 - 4 = 4, a(2) = 9 - 8 = 1, a(3) = 44 - 9 = 35.

Cf. A037010, A035132.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=0; end; s := []; for n from 1 to 7000 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Differences[Select[Range[10000],PalindromeQ[#]&&!SquareFreeQ[#]&]] (* Harvey P. Dale, Dec 28 2024 *)

Edited by Dean Hickerson, Oct 21 2002

A035133 Cubeful (i.e., not cubefree) palindromes.

Original entry on oeis.org

8, 88, 232, 272, 343, 424, 464, 616, 656, 686, 696, 808, 848, 888, 999, 1331, 2112, 2552, 2662, 2992, 3773, 3993, 4224, 4664, 6336, 6776, 8008, 8448, 8888, 14641, 18981, 19791, 21112, 21312, 21512, 21712, 21912, 23032, 23232, 23432, 23632

Offset: 1

Patrick De Geest, Nov 15 1998

Chourasiya & Johnston show that there are infinitely many palindromes not in this sequence. - Charles R Greathouse IV, May 14 2025

			E.g., 34643 = 7^3 * 101.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..627 from R. J. Mathar)
Shashi Chourasiya and Daniel R. Johnston, Power-free palindromes and reversed primes, arXiv preprint (2025). arXiv:2503.21136 [math.NT]
Eric Weisstein's World of Mathematics, Cubefree.

Cf. A002113, A004709, A035132, A046099.

Select[Range[25000],PalindromeQ[#]&&Max[FactorInteger[#][[;;,2]]]>2&] (* Harvey P. Dale, Feb 18 2024 *)

A046099 INTERSECT A002113. - R. J. Mathar, Dec 08 2015

A071251 Squarefree palindromes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 22, 33, 55, 66, 77, 101, 111, 131, 141, 151, 161, 181, 191, 202, 222, 262, 282, 303, 313, 323, 353, 373, 383, 393, 434, 454, 474, 494, 505, 515, 535, 545, 555, 565, 595, 606, 626, 646, 707, 717, 727, 737, 757, 767, 777, 787, 797, 818, 838

Offset: 1

Amarnath Murthy, May 21 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A005117.

Cf. A002385, A069217, A032350, A035132.

ffpal := proc(n) local i,j,k,s,aa,nn,bb,flag; s := n; aa := convert(s,string); nn := length(aa); bb := ``; for i from nn by -1 to 1 do bb := cat(bb,substring(aa,i..i)); od; flag := 0; for j from 1 to nn do if substring(aa,j..j)<>substring(bb,j..j) then flag := 1 fi; od; RETURN(flag); end: ts_ndk_pal := proc(i) if ffpal((i)) = 0 then if (numtheory[issqrfree](i) = 'true' ) then RETURN((i)) fi fi end: andkpal := [seq(ts_ndk_pal(i), i=1..10000)]: andkpal;

Select[ Range[ 1000], # == FromDigits[ Reverse[ IntegerDigits[ # ]]] && Max[ Transpose[ FactorInteger[ # ]] [[2]]] < 2 &]
Select[Range[1000],PalindromeQ[#]&&SquareFreeQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 17 2018 *)

Edited by Robert G. Wilson v, Jun 03 2002

A035134 Squarefree composite palindromes.

Original entry on oeis.org

6, 22, 33, 55, 66, 77, 111, 141, 161, 202, 222, 262, 282, 303, 323, 393, 434, 454, 474, 494, 505, 515, 535, 545, 555, 565, 595, 606, 626, 646, 707, 717, 737, 767, 777, 818, 838, 858, 878, 898, 939, 949, 959, 969, 979, 989, 1001, 1111, 1221, 1441, 1551, 1661

Offset: 1

Patrick De Geest, Nov 15 1998

Palindromes with at least two and all distinct prime factors.

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

Cf. A002113, A005117, A035132, A035135.

N:= 4: # to get all terms with <= N digits
revdigs:= proc(n) local L, j, nL;
  L:= convert(n, base, 10); nL:= nops(L);
  add(L[j]*10^(nL-j), j=1..nL);
end proc:
palis:= $0..9:
for d from 2 to N do
  if d::even then
    palis:= palis, seq(x*10^(d/2)+revdigs(x), x=10^(d/2-1)..10^(d/2)-1)
  else
    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);
  fi
od:
select(t -> not(isprime(t)) and numtheory:-issqrfree(t), [palis][3..-1]): # Robert Israel, Sep 18 2016

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 *)
Select[Range[2000],PalindromeQ[#]&&SquareFreeQ[#]&&CompositeQ[#]&] (* Harvey P. Dale, Apr 10 2022 *)

isA002113(n)=n=digits(n);for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1;
is(n) = n>1 && isA002113(n) && issquarefree(n) && !isprime(n) \\ Altug Alkan, Sep 19 2016
\\ in and output digits as a vector.

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])}
\\ all terms up to n digits
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

from itertools import product
from sympy import factorint, isprime
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal): f = factorint(pal); return len(f)>1 and all(f[p]<2 for p in f)
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

A077572 Nonsquarefree numbers obtained by repeating a single digit.

Original entry on oeis.org

4, 8, 9, 44, 88, 99, 333, 444, 666, 888, 999, 4444, 8888, 9999, 44444, 88888, 99999, 333333, 444444, 666666, 777777, 888888, 999999, 4444444, 8888888, 9999999, 44444444, 88888888, 99999999, 111111111, 222222222, 333333333, 444444444

Offset: 1

Amarnath Murthy, Nov 11 2002

			666 and 111111111 are members but 66, 1111 etc. are not.

Cf. A077571, A087094.

Subsequence of A013929 and A035132.

Select[Flatten[Table[FromDigits[PadRight[{},i,n]],{i,9},{n,9}]],!SquareFreeQ[#]&] (* Harvey P. Dale, Jan 22 2013 *)

from sympy.ntheory.factor_ import core
def repsthru(maxd):
  yield from (int(di*d) for d in range(1, maxd+1) for di in "123456789")
def okrep(r): return r > 3 and core(r, 2) != r
print(list(filter(okrep, repsthru(9)))) # Michael S. Branicky, Apr 08 2021

Corrected and extended by Ray Chandler, Aug 12 2003

Offset changed by Andrew Howroyd, Sep 29 2024

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A075804 Differences between adjacent palindromic nonsquarefree numbers A035132.

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A035133 Cubeful (i.e., not cubefree) palindromes.

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A071251 Squarefree palindromes.

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A035134 Squarefree composite palindromes.

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A077572 Nonsquarefree numbers obtained by repeating a single digit.

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