cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A046447 Apart from initial term, composite numbers with the property that the concatenation of their prime factors is a palindrome.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 39, 49, 64, 69, 81, 119, 121, 125, 128, 129, 159, 219, 243, 249, 256, 259, 329, 339, 343, 403, 429, 469, 507, 512, 625, 669, 679, 729, 795, 1024, 1207, 1309, 1329, 1331, 1533, 1547, 1587, 1589, 1703, 2023, 2048, 2097, 2187, 2319
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Comments

Prime factors considered with multiplicity. - Harvey P. Dale, Apr 20 2025

Examples

			81 is a term because 81 = 3 * 3 * 3 * 3 -> 3333 is palindromic.

Links

Crossrefs

Cf. A046448, A046449, A046450.
Cf. A027746, A018252, A136522, A002113.

Programs

  • Haskell
    a046447 n = a046447_list !! (n-1)
a046447_list = 1 : filter f [1..] where
   f x = length ps > 1 && ps' == reverse ps'
         where ps' = concatMap show ps; ps = a027746_row x
-- Reinhard Zumkeller, May 02 2014
  • Mathematica
    concat[n_]:=Flatten[Table[IntegerDigits[First[n]],{Last[n]}]]; palQ[n_]:= Module[{x=Flatten[concat/@FactorInteger[n]]},x==Reverse[x]&&!PrimeQ[n]]; Select[Range[2500],palQ] (* Harvey P. Dale, May 24 2011 *)
cpfpQ[n_]:=PalindromeQ[FromDigits[Flatten[IntegerDigits/@Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]]]]; Join[{1},Select[Range[2500],CompositeQ[ #]&&cpfpQ[#]&]] (* Harvey P. Dale, Apr 20 2025 *)
  • Python
    from sympy import factorint, isprime
A046447_list = [1]
for n in range(4, 10**6):
    if not isprime(n):
        s = ''.join([str(p)*e for p, e in sorted(factorint(n).items())])
        if s == s[::-1]:
            A046447_list.append(n) # Chai Wah Wu, Jan 03 2015

Extensions

Definition slightly modified by Harvey P. Dale, Apr 20 2025

A046450 Concatenation of prime factors of palindromic composite is a palindrome.

Original entry on oeis.org

4, 8, 9, 121, 343, 1331, 10001, 10201, 14641, 36763, 1030301, 1037301, 1226221, 9396939, 104060401, 12467976421, 14432823441, 93969696939, 119092290911, 1030507050301, 1120237320211, 1225559555221, 1234469644321, 1334459544331, 100330272033001, 101222252222101
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Examples

			E.g., 14432823441 = 3 * 3 * 281 * 313 * 18233 -> 3328131318233 is palindromic.

Crossrefs

Cf. A046447, A046448, A046449.

Programs

  • Mathematica
    d[n_] := IntegerDigits[n]; co[n_, k_] := Nest[Flatten[d[{#, n}]] &, n, k - 1]; t = {}; Do[If[! PrimeQ[n] && Reverse[x = d[n]] == x && Reverse[y = Flatten[d[co @@@ FactorInteger[n]]]] == y, AppendTo[t, n]], {n, 2, 10^7}]; t (* Jayanta Basu, Jun 24 2013 *)

Extensions

Corrected by Charles R Greathouse IV, Apr 23 2010
a(18)-a(24) from Donovan Johnson, May 03 2010
a(25)-a(26) from Donovan Johnson, Aug 09 2011

A046448 Squarefree nonprimes with property that the concatenation of the prime factors is a palindrome.

Original entry on oeis.org

1, 39, 69, 119, 129, 159, 219, 249, 259, 329, 339, 403, 429, 469, 669, 679, 795, 1207, 1309, 1329, 1533, 1547, 1589, 1703, 2319, 2321, 2359, 2649, 2701, 3039, 3421, 3503, 3629, 3633, 3639, 3729, 3899, 4303, 4607, 4839, 5289, 5295, 5565, 5603, 5739, 6209
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Examples

			E.g., 429 = 3 * 11 * 13 -> 31113 is palindromic.

Crossrefs

Cf. A046447, A046449, A046450, A002113.

Programs

  • Mathematica
    Select[Range[6210], ! PrimeQ[#] && SquareFreeQ[#] && Reverse[x = Flatten[IntegerDigits[First /@ FactorInteger[#]]]] == x &] (* Jayanta Basu, Jun 24 2013 *)
Select[Range[6500],!PrimeQ[#]&&SquareFreeQ[#]&&PalindromeQ[ FromDigits[ Flatten[ IntegerDigits/@ FactorInteger[#][[All,1]]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2020 *)

Extensions

Definition clarified by N. J. A. Sloane, Oct 25 2020

A249300 Composite numbers whose concatenation of their aliquot parts, in ascending order, is a palindrome.

Original entry on oeis.org

93, 121, 393, 497, 755, 842, 961, 993, 1042, 1255, 1293, 1642, 1681, 1893, 1897, 3721, 3755, 3997, 4043, 4061, 4442, 5041, 5755, 6797, 8197, 8842, 9993, 11042, 11255, 16593, 17309, 17642, 22255, 23221, 23597, 26242, 26493, 26797, 29793, 30043, 30242, 30383
Offset: 1

Views

Author

Paolo P. Lava, Oct 24 2014

Keywords

Examples

			Aliquot parts of 157442 are 1, 2, 78621; their concatenation in ascending order is concat(1,2,78621) = 1278621, which is a palindrome.

Links

Crossrefs

Cf. A046449.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 2 to q do if not isprime(n) then a:=sort([op(divisors(n))]); b:=0;
for k from nops(a)-1 by -1 to 1 do b:=b*10^(ilog10(a[k])+1)+a[k]; od; a:=0; c:=b;
for k from 1 to ilog10(b)+1 do a:=10*a+(c mod 10); c:=trunc(c/10); od;
if a=b then print(n); fi; fi; od; end: P(10^9);
  • PARI
    isok(n) = {d = vecsort(divisors(n)); if (#d > 2, s = ""; for (i=1, #d-1, s = concat(s, Str(d[i]));); d = digits(eval(s)); d == Vecrev(d););} \\ Michel Marcus, Oct 25 2014

A249301 Composite numbers whose concatenation of their aliquot parts, in descending order, is a palindrome.

Original entry on oeis.org

39, 119, 121, 169, 254, 289, 361, 393, 411, 417, 755, 785, 1211, 1253, 1703, 2554, 3503, 3629, 4197, 6401, 7555, 10001, 12131, 12287, 12439, 14803, 15563, 17147, 17363, 23701, 24202, 24322, 24646, 24686, 24746, 25514, 25838, 25918, 25958, 26827, 30383, 30521
Offset: 1

Views

Author

Paolo P. Lava, Oct 24 2014

Keywords

Examples

			Aliquot parts of 24332 are 1, 2, 121661; their concatenation in descending order is concat(12166,2,1) = 1216621, which is a palindrome.

Links

Crossrefs

Cf. A046449.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 2 to q do if not isprime(n) then a:=sort([op(divisors(n))]); b:=0;
for k from 1 to nops(a)-1 do b:=b*10^(ilog10(a[k])+1)+a[k]; od; a:=0; c:=b;
for k from 1 to ilog10(b)+1 do a:=10*a+(c mod 10); c:=trunc(c/10); od;
if a=b then print(n); fi; fi; od; end: P(10^9);
  • PARI
    isok(n) = {d = vecsort(divisors(n),,4); if (#d > 2, s = ""; for (i=2, #d, s = concat(s, Str(d[i]));); d = digits(eval(s)); d == Vecrev(d););} \\ Michel Marcus, Oct 25 2014

A273776 Smallest composite number with n distinct prime factors with property that the concatenation of its distinct prime factors, in descending order, is a palindrome.

Original entry on oeis.org

4, 46, 138, 690, 197890, 5444670, 156719940, 4941906970, 135969743910
Offset: 1

Views

Author

Paolo P. Lava, Jul 06 2016

Keywords

Comments

4 = 2 * 2
46 = 2 * 23
138 = 2 * 3 * 23
690 = 2 * 3 * 5 * 23
197890 = 2 * 5 * 7 * 11 * 257
5444670 = 2 * 3 * 5 * 7 * 11 * 2357
156719940 = 2 * 2 * 3 * 5 * 13 * 17 * 53 * 223
4941906970 = 2 * 5 * 7 * 11 * 13 * 17 * 113 * 257
135969743910 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 113 * 2357

Examples

			Prime factors of 4 are 2, 2 and concat(2,2) = 22 is palindromic.
Prime factors of 46 are 2, 23 and concat(23,2) = 232 is palindromic.
Prime factors of 5444670 are 2, 3, 5, 7, 11, 2357 and concat(2357,11,7,5,3,2) = 2357117532 is palindromic.

Crossrefs

Cf. A046449.

Programs

  • Maple
    with(numtheory): T:=proc(w) local x, y, z; x:=0; y:=w; for z from 1 to ilog10(w)+1 do x:=10*x+(y mod 10); y:=trunc(y/10); od; x; end;
P:=proc(q) local a,b,c,i,j,k,n; c:=1; for j from 1 to q do for n from c to q do
if not isprime(n) then a:=ifactors(n)[2]; b:=[]; if nops(a)=j then for k from 1 to nops(a) do
for i from 1 to a[k][2] do b:=[op(b),a[k][1]]; od; od; b:=sort(b); a:=b[nops(b)];
for k from nops(b)-1 by -1 to 1 do a:=a*10^(ilog10(b[k])+1)+b[k]; od;
if T(a)=a then c:=n; print(n); break; fi; fi; fi; od; od; end: P(10^9);

Extensions

a(7)-a(9) from Giovanni Resta
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.