A101704 -id:A101704 - cp's OEIS frontend

A101705 Numbers n such that n = 12*reversal(n).

0, 540, 5940, 54540, 59940, 540540, 599940, 5400540, 5454540, 5945940, 5999940, 54000540, 54594540, 59405940, 59999940, 540000540, 540540540, 545454540, 545994540, 594005940, 594545940, 599459940, 599999940, 5400000540, 5405940540, 5454054540, 5459994540, 5940005940, 5945945940, 5994059940, 5999999940

Offset: 1

Farideh Firoozbakht, Jan 02 2005

60 divides all terms of the sequence. For all nonnegative integers m and n all numbers of the form f(m,n) = (100*(6*10^m - 1)+ 40)*(10^((m + 2)*n) - 1)/(10^(m + 2) - 1) are in the sequence, in fact f(m,n) = (5.(9)(m))(n).0 where dot between numbers means concatenation and "(r)(t)" means number of r's is t. f(m,1) = 100*(6*10^m - 1)+ 40 = 5.(9)(m).40; f(0,1) = 540, f(1,1) = 5940, f(2,1)=59940, etc. f(m,2) = 5.(9)(m).50(9)(m).40; f(0,2) = 54540, f(1,2) = 5945940, etc. Let g(s,t,r) = s*(10^((L+t)(1+r))-1)/(10^(L+t)-1) where L = number of digits of s. If s is in the sequence then all numbers of the form g(s,t,r) for nonnegative integers t and r are in the sequence (the function g is the same function that has been defined in the sequence A101704). If n and m are nonnegative integers then g(n,0,m) = (n)(m+1) for example g(13,0,3) = (13)(4) = 13131313.

			g(540,0,5)= (540)(6) = 540540540540540540 is in the sequence because reversal(540540540540540540) = 45045045045045045 and 12*45045045045045045 = 540540540540540540.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A101704, A101706, A001232, A008918.

Do[If[n == 12*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 0, 6000000000, 60}]
Select[Range[0,6*10^9,60],#==12IntegerReverse[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 12 2017 *)

def A101705(n):
    if n == 1: return 0
    a = 1<Chai Wah Wu, Jul 23 2024

a(n) = 540*A057148. - Ray Chandler, Oct 09 2017

a(25)-a(31) from Max Alekseyev, Aug 18 2013

A101706 Numbers n such that reversal(n)=(7/3)*n.

Original entry on oeis.org

0, 3267, 32967, 329967, 3299967, 32673267, 32999967, 326703267, 329999967, 3267003267, 3296732967, 3299999967, 32670003267, 32967032967, 32999999967, 326700003267, 326732673267, 329670032967, 329967329967, 329999999967, 3267000003267, 3267329673267, 3296700032967, 3299670329967, 3299999999967

Offset: 1

Farideh Firoozbakht, Jan 01 2005

If m is in the sequence then all numbers of the form g(m,s,t) for nonnagative integers s and t are in the sequence (the function g has been defined in the sequence A101704), for example g(3267,1,1)= 326703267 is in the sequence. If n=0 or n>1 then 33*(10^n-1) is in the sequence.

There are Fibonacci(floor((n-2)/2)) terms with n digits, n>1 (this is essentially A103609). - Ray Chandler, Oct 12 2017

			g(3267,10,2) = 32670000000000326700000000003267 is in the sequence
because reversal(32670000000000326700000000003267) =
76230000000000762300000000007623 =
(7/3)*32670000000000326700000000003267, g(3267,0,4) =
32673267326732673267 is in the sequence because
reversal(32673267326732673267) = 76237623762376237623 =
(7/3)*32673267326732673267.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A101704, A101705, A001232, A008918.

Do[If[FromDigits[Reverse[IntegerDigits[n]]] == (7/3)*n, Print[n]], {n, 100000000}]

Terms a(8) onward from Max Alekseyev, Aug 18 2013

A101703 Numbers n such that reversal(n) = (2/3)*n - 2.

Original entry on oeis.org

21, 291, 885, 2991, 29991, 234651, 299991, 2340651, 2999991, 8221845, 23400651, 29346591, 29999991, 234000651, 293406591, 299999991, 2340000651, 2346534651, 2934006591, 2993465991, 2999999991, 23400000651, 23465934651, 29340006591, 29934065991, 29999999991, 82277815845, 234000000651

Offset: 1

Farideh Firoozbakht, Dec 31 2004

Numbers of the form 3*(10^n-3) are in the sequence, so A086947 is an infinite subsequence of this sequence. Also A101700 is a subsequence of this sequence.
Let f(r,s,t,z) = 2.(9)(r+s).(34.(0)(t).65)(z).(9)(s).1 where the dot between numbers means concatenation and "(m)(n)" means number of m's is n, for example f(0,2,1,3)= 299340653406534065991, it is interesting that all numbers of the form f(r,s,t,z) where r, s, t & z are nonnegative integers and r*z=0 are in this sequence.
Except for 885 & 8221845 all known terms of this sequence are of the form f(r,s,t,z).
For all r, s & t we have f(r,s,t,0)=f(r,s,0,0)=f(r+2s,0,0,0)=A086947(r+2s+1)= 3*(10^(r+2s+1)-3).
a(1) = 21 = f(0,0,0,0), a(2) = 291 = f(1,0,0,0), a(4) = 2991 = f(2,0,0,0) = f(0,1,0,0), a(5) = 29991 = f(3,0,0,0) = f(1,1,0,0), a(6) = 234651 = f(0,0,0,1), a(7) = 299991 = f(4,0,0,0) = f(0,2,0,0), a(8) = 2340651 = f(0,0,1,1), etc. Next term is greater than 11*10^8.
From David Wasserman, Mar 27 2008: (Start)
234653406534651 is a term that doesn't fit the f(r,s,t,z) format.
We may redefine f so that t is a vector of length z, which must be symmetrical to produce a member. For example f(0,0,[0,1,0],3) = 234653406534651 is a member, but f(0,0,[1,0,0],3) = 234065346534651 is not a member.
23465934651 is another member that doesn't fit the pattern. In general there may be any number of 9's between a 5 and a 3, provided that the 9's are symmetrical. So 2346593465934651 is a member, but 23465993465934651 is not. (End)

			f(0,1,2,3) = 2934006534006534006591 is in the sequence because reversal(2934006534006534006591) = 1956004356004356004392 = (2/3)*2934006534006534006591-2.

Giovanni Resta, Table of n, a(n) for n = 1..56 (terms < 10^14)

Cf. A086947, A069215, A101700, A101704.

Do[If[FromDigits[Reverse[IntegerDigits[n]]] == 2/3*n - 2, Print[n]], {n, 1100000000}]

More terms from David Wasserman, Mar 27 2008

A102277 Numbers n such that n = 15*reversal(n).

Original entry on oeis.org

0, 65340, 659340, 6599340, 65999340, 653465340, 659999340, 6534065340, 6599999340, 65340065340, 65934659340, 65999999340, 653400065340, 659340659340, 659999999340, 6534000065340, 6534653465340, 6593400659340, 6599346599340, 6599999999340

Offset: 1

Farideh Firoozbakht, Jan 04 2005

30 divides all terms of the sequence. For all nonnegative integers m and n all numbers of the form f1(m,n) = 660(10^(m + 2) - 1)*(10^((m + 4)*n) - 1)/(10^(m + 4) - 1) are in the sequence, in fact f1(m,n) = (65.(9)(m).34)(n).0 where dot between numbers means concatenation and "(r)(t)" means number of r's is t. With this definition a(1) = 0 = f1(0,0), a(2) = 65340 = f1(0,1), a(3) = 659340 = f1(1,1), a(4) = 6599340 = f1(2,1), a(5) = 65999340 = f1(3,1), a(6) = 653465340 = f1(0,2), a(7) = 659999340 = f1(4,1), a(9) = 6599999340 = f1(5,1), etc. f1(m,1) = 660(10^(m + 2) - 1) = 65.(9)(m).340, f1(m,2) = 65.(9)(m).34.65.(9)(m).340, etc. Let g(s,t,r) = s*(10^((L+t)*(1+r))-1)/(10^(L+t)-1) where L = number of digits of s, in fact g(s,t,r) = (s.(0)(t))(r).s so the function g is the same function that has been defined in the sequence A101704. If s is in the sequence then all numbers of the form g(s,t,r) for nonnegative integers t and r are in the sequence. Next term is greater than 11*10^9. It seems that the eleven next terms are 65340065340, 65934659340, 65999999340, 653400065340, 659340659340 659999999340, 6534000065340, 6534653465340, 6593400659340, 6599346599340 and 6599999999340. Is it true that, all terms of this sequence are of the form g(f1(m,n),r,t)?

			g(65340,0,2)= (65340)(3) = 653406534065340 is in the sequence because reversal(653406534065340) = 43560435604356 = (1/15)*653406534065340.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A001232, A008918, A101704, A101705, A101706.

Do[If[n == 15*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 0, 11000000000, 30}]

a(n) = 10*A101704(n) = 20*A101706(n). - Ray Chandler, Oct 09 2017

More terms from Ray Chandler, Oct 09 2017

A285040 Numbers n such that three-halves of n equals the reverse of n.

Original entry on oeis.org

4356, 43956, 439956, 4399956, 43564356, 43999956, 435604356, 439999956, 4356004356, 4395643956, 4399999956, 43560004356, 43956043956, 43999999956, 435600004356, 435643564356, 439560043956, 439956439956, 439999999956

Offset: 1

Harvey P. Dale, Apr 08 2017

There are Fibonacci(floor((n-2)/2)) terms with n digits (this is essentially A103609). - Ray Chandler, Oct 12 2017

			439956 times 3/2 equals 659934 which is the reverse of 439956.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 158.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A101704, A101705, A101706, A001232, A008918.

Select[2 Range[10^7], 3(#/2) == FromDigits@ Reverse@ IntegerDigits@ # &] (* Giovanni Resta, Apr 08 2017 *)

isok(n) = 3*n/2 == fromdigits(Vecrev(digits(n))); \\ Michel Marcus, Apr 09 2017

Data corrected by Giovanni Resta, Apr 08 2017

cp's OEIS Frontend

A101705 Numbers n such that n = 12*reversal(n).

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A101706 Numbers n such that reversal(n)=(7/3)*n.

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A101703 Numbers n such that reversal(n) = (2/3)*n - 2.

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A102277 Numbers n such that n = 15*reversal(n).

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A285040 Numbers n such that three-halves of n equals the reverse of n.

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