A061205 -id:A061205 - cp's OEIS frontend

A203924 Indices of terms which appear at least twice already earlier in A061205.

372, 441, 483, 492, 651, 672, 861, 2412, 2532, 2652, 3162, 3472, 3612, 3720, 3732, 3782, 3852, 3913, 3972, 4053, 4182, 4221, 4263, 4410, 4431, 4473, 4592, 4623, 4641, 4683, 4704, 4812, 4824, 4830, 4851, 4853, 4893, 4920, 4932, 4944, 6132, 6174, 6231, 6293

Offset: 1

M. F. Hasler, Jan 07 2012

Motivated by a question from Franklin T. Adams-Watters, cf. link.

			A061205(156)=101556=A061205(273)=A061205(372), and this is the first time a term appears for the third time in A061205, therefore a(1)=372.

M. F. Hasler, Table of n, a(n) for n = 1..101
Franklin T. Adams-Watters, Number times reversal, SeqFan list, Jan 07 2012
Hans Havermann, Augmented table of n, a(n) (in bold) and its associated indices, for n = 1..21313

seen=dupe=[];for(n=0,999999, setsearch(seen,t=A061205(n)) | !(seen=setunion(seen,Set(t))) | next; setsearch(dupe,t) & (print1(n",")|next); dupe=setunion(dupe,Set(t)))

a(8)-a(44) from Donovan Johnson, Jan 07 2012

A133019 Product of n-th prime and n-th prime written backwards.

Original entry on oeis.org

4, 9, 25, 49, 121, 403, 1207, 1729, 736, 2668, 403, 2701, 574, 1462, 3478, 1855, 5605, 976, 5092, 1207, 2701, 7663, 3154, 8722, 7663, 10201, 31003, 75007, 98209, 35143, 91567, 17161, 100147, 129409, 140209, 22801, 117907, 58843, 127087

Offset: 1

Omar E. Pol, Oct 27 2007

a(8) = 1729 is the second taxicab number, also called the Hardy-Ramanujan number (see A001235, A011541 and A133029).

			a(8) = 1729 because the 8th prime is 19 and 19 written backwards is 91 and 19*91 = 1729.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001235, A004087, A011541, A061205, A067563, A067087, A133029. Prime numbers: A000040.

#*FromDigits[Reverse[IntegerDigits[#]]] & /@ Prime[Range[1, 50]] (* G. C. Greubel, Oct 02 2017 *)
#*IntegerReverse[#]&/@Prime[Range[40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 29 2021 *)

vector(60, n, prime(n)*subst(Polrev(digits(prime(n))), x, 10)) \\ Michel Marcus, Dec 17 2014

a(n) = A000040(n) * A004087(n)

A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

Original entry on oeis.org

100, 144, 169, 200, 288, 300, 400, 441, 500, 528, 600, 700, 768, 800, 825, 867, 882, 900, 961, 1089, 1100, 1584, 2178, 2200, 3300, 4400, 4851, 5500, 6600, 7700, 8712, 8800, 9801, 9900, 10000, 10100, 10404, 10609, 10989

Offset: 1

Reinhard Zumkeller, May 15 2002

If k is a palindrome (A002113), then 100*k is a term. If k is a term, then 100*k is a term. - Chai Wah Wu, Mar 31 2018
From Bernard Schott, Jan 02-10 2019: (Start)
There are six different families of integers in this sequence.
1) If k and rev(k) do not have the same number of digits:
All these integers are in A322835 where the first four families are explained and detailed.
Family 1: A002113(j) * 100^k
Family 2: A035090(j) * 100^k
Family 3: A082994(j) * 100^k
Family 4: A323061(j) * 10^(2k+1)
2) If k and rev(k) have the same number of digits.
All these integers are in A062917.
Family 5: Non-palindromic squares whose reverse is also square. These integers are in A035090.
Family 6: Non-palindromic numbers k, such that k * rev(k) is a square, with k and rev(k) not both square. These integers are in A082994.
3) Relationships between these different sequences.
  A035090 Union A082994 = A062917 with empty intersection, and,
A062917 Union A322835 = {This sequence} with empty intersection. (End)

			a(2)=144: rev(144)=441, 144*441=(12^2)*(21^2)=(12*21)^2 and 144<>12*21=252.
From _Bernard Schott_, Jan 02 2019: (Start)
Example for family 1: 200 * 2 = 400 = 20^2
Example for family 2: 14400 * 441 = 120^2 * 21^2 = 2520^2
Example for family 3: 28800 * 882 = (2 * 120^2) * (2 * 21^2) = 5040^2
Example for family 4: 5449680 * 869445 = 2176740^2
Example for family 5: 169 * 961 = 13^2 * 31^2 = 403^2
Example for family 6: 528 * 825 = (33 * 4^2) * (33 * 5^2) = 660^2. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Reinhard Zumkeller)

Cf. A002113, A061205, A010052, A035090, A062917, A082994, A322835, A323061.

a070760 n = a070760_list !! (n-1)
a070760_list = [x | x <- [0..], let y = a061205 x,
                    y /= x ^ 2, a010052 y == 1]
-- Reinhard Zumkeller, Apr 10 2012, Apr 29 2011

Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* Jean-François Alcover, Nov 30 2011 *)
sdnQ[n_]:=Module[{c=n*IntegerReverse[n]},c!=n^2&&IntegerQ[Sqrt[c]]]; Select[ Range[11000],sdnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2016 *)

A073805 Numbers k such that 1 + k*R(k) is prime, where R(k) is the reverse of k.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 18, 20, 24, 25, 26, 28, 36, 42, 50, 52, 58, 61, 62, 63, 66, 68, 70, 80, 81, 82, 85, 86, 90, 100, 108, 112, 116, 120, 132, 136, 138, 140, 152, 162, 170, 190, 198, 200, 204, 205, 209, 210, 211, 213, 214, 219, 223, 224, 228, 231, 234, 236, 238

Offset: 1

Shyam Sunder Gupta, Aug 23 2002

			16 is a term because 16*61 + 1 = 977 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004086, A061205.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= t -> isprime(t*rev(t)+1):
select(filter, [$1..1000]); # Robert Israel, Jul 03 2024

Select[Range[250],PrimeQ[# IntegerReverse[#]+1]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2017 *)

A256756 a(n) = bitwise XOR of n and the reverse of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 25, 18, 39, 60, 45, 86, 67, 72, 22, 25, 0, 55, 50, 45, 36, 83, 78, 65, 29, 18, 55, 0, 9, 22, 27, 108, 117, 122, 44, 39, 50, 9, 0, 27, 110, 101, 100, 111, 55, 60, 45, 22, 27, 0, 121, 114, 111, 100, 58, 45, 36, 27, 110, 121

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A003987, A004086, A055483, A056964, A056965, A061205, A068634, A175919, A256754, A256755.

a:= n-> Bits[Xor](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitXor[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,65}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitxor(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A003987(n, A004086(n)).

A305231 Numbers that are the product of some integer and its digit reversal.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 25, 36, 40, 49, 64, 81, 90, 100, 121, 160, 250, 252, 360, 400, 403, 484, 490, 574, 640, 736, 765, 810, 900, 976, 1000, 1008, 1089, 1207, 1210, 1300, 1458, 1462, 1600, 1612, 1729, 1855, 1936, 1944, 2268, 2296, 2430, 2500, 2520, 2668, 2701

Offset: 1

Jon E. Schoenfield, May 27 2018

Terms of A061205, sorted in increasing order, with duplicates removed.

			12*21 = 252, so 252 is a term.
156*651 = 101556, so 101556 is a term. (It can also be written as 273*372; see A203924.)

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)

Cf. A061205, A203924.

Cf. A325148 (squares), A359981 (nonsquares).

a:= proc(n) option remember; local k, d; for k from 1+a(n-1) do
      for d in numtheory[divisors](k) do if k = d*(s-> parse(cat(
      seq(s[-i], i=1..length(s)))))(""||d) then return k fi od od
    end: a(1):=0:
seq(a(n), n=1..60);  # Alois P. Heinz, May 27 2018

a={0}; h=-1; For[k=0, k<=2701, k++, For[m=1, m<=DivisorSigma[0, k], m++, d=Divisors[k]; If[k/Part[d, m] == FromDigits[Reverse[IntegerDigits[Part[d, m]]]] && k>h , AppendTo[a, k]; h=k]]]; a (* Stefano Spezia, Jan 28 2023 *)

isok(n) = if (n==0, return (1), fordiv(n, d, if (n/d == fromdigits(Vecrev(digits(d))), return (1))); return (0)); \\ Michel Marcus, May 28 2018

A256754 a(n) = bitwise AND of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 4, 13, 8, 3, 16, 1, 16, 19, 0, 4, 22, 0, 8, 16, 26, 8, 16, 28, 2, 13, 0, 33, 34, 33, 36, 1, 2, 5, 0, 8, 8, 34, 44, 36, 0, 10, 16, 16, 0, 3, 16, 33, 36, 55, 0, 9, 16, 27, 4, 16, 26, 36, 0, 0, 66, 64, 68, 64, 6, 1, 8, 1, 10

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A004086, A004198, A055483, A056964, A056965, A061205, A068634, A256755, A256756.

a:= n-> Bits[And](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitAnd[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,74}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitand(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A004198(n,A004086(n)).

A256755 a(n) = bitwise OR of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 29, 31, 47, 63, 61, 87, 83, 91, 22, 29, 22, 55, 58, 61, 62, 91, 94, 93, 31, 31, 55, 33, 43, 55, 63, 109, 119, 127, 44, 47, 58, 43, 44, 63, 110, 111, 116, 127, 55, 63, 61, 55, 63, 55, 121, 123, 127, 127, 62, 61, 62, 63, 110

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A003986, A004086, A055483, A056964, A056965, A061205, A068634, A256754, A256756.

a:= n-> Bits[Or](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitOr[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,64}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitor(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A003986(n,A004086(n)).

A342127 Numbers m such that the product of m and the string m in reverse contains m as a substring.

Original entry on oeis.org

0, 1, 5, 6, 10, 47, 50, 60, 75, 78, 100, 125, 152, 457, 500, 600, 750, 1000, 1025, 1052, 1250, 1520, 5000, 5625, 6000, 7500, 10000, 10025, 10052, 10250, 10520, 12266, 12500, 15200, 23258, 43567, 50000, 56250, 60000, 62656, 75000, 82291, 90625, 98254, 100000, 100025, 100052, 100250, 100520

Offset: 1

Scott R. Shannon, Mar 01 2021

Numerous patterns exist in the terms, e.g., all numbers of the form 1*10^k, 5*10^k, 6*10^k, 75*10^k, 10^(k+2)+25, where k>=0, are in the sequence.

			6 is a term as 6*reverse(6) = 6*6 = 36 contains '6' as a substring.
47 is a term as 47*reverse(47) = 47*74 = 3478 contains '47' as a substring.
1052 is a term as 1052*reverse(1052) = 1052*2501 = 2631052 contains '1052' as a substring.

Robert Israel, Table of n, a(n) for n = 1..153

Cf. A061205, A342130 (base 2), A001477, A004086, A181721, A203565, A332795.

filter:= proc(n) local L,d,Lp,r,i;
  L:= convert(n,base,10);
  d:= nops(L);
  r:= add(L[-i]*10^(i-1),i=1..d);
  Lp:= convert(n*r,base,10);
  ormap(t -> Lp[t..t+d-1] = L, [$1..nops(Lp)+1-d])
end proc:
select(filter, [$0..120000]); # Robert Israel, Mar 24 2024

Select[Range[0,110000],SequenceCount[IntegerDigits[# IntegerReverse[#]],IntegerDigits[#]]>0&] (* Harvey P. Dale, Apr 20 2024 *)

isok(m) = #strsplit(Str(m*fromdigits(Vecrev(digits(m)))), Str(m)) > 1; \\ Michel Marcus, Mar 01 2021

def ok(n): return (s:=str(n)) in str(n*int(s[::-1]))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Mar 25 2024

A133022 Product of n-th Fibonacci number and n-th Fibonacci number written backwards.

Original entry on oeis.org

0, 1, 1, 4, 9, 25, 64, 403, 252, 1462, 3025, 8722, 63504, 77356, 291421, 9760, 778743, 12697747, 12537568, 7584334, 38398140, 710406346, 208476181, 2168819074, 4004525952, 3905576425, 47722137553, 160019976838, 37728297243, 474332543035, 33479625520

Offset: 0

Omar E. Pol, Nov 07 2007

			403 = 13*31.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A004091, A061205, A133019.

a:= n-> (f-> (s-> f*parse(cat(s[-i]$i=1..length(s))))(
          ""||f))(((<<0|1>, <1|1>>^n)[1, 2])):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 06 2018

#*FromDigits[Reverse[IntegerDigits[#]]]&/@Fibonacci[Range[0,40]] (* Harvey P. Dale, Oct 12 2012 *)

a(n) = A000045(n) * A004091(n).

Corrected and extended by Harvey P. Dale, Oct 12 2012

cp's OEIS Frontend

A203924 Indices of terms which appear at least twice already earlier in A061205.

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A133019 Product of n-th prime and n-th prime written backwards.

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A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

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A073805 Numbers k such that 1 + k*R(k) is prime, where R(k) is the reverse of k.

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Maple

Mathematica

A256756 a(n) = bitwise XOR of n and the reverse of n.

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Maple

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A305231 Numbers that are the product of some integer and its digit reversal.

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Maple

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A256754 a(n) = bitwise AND of n and the reverse of n.

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Maple

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A256755 a(n) = bitwise OR of n and the reverse of n.

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