A072143 -id:A072143 - cp's OEIS frontend

A292859 Numbers k such that 10 applications of 'Reverse and Subtract' lead to k, whereas fewer than 10 applications do not lead to k.

101451293600894707746789, 105292253210898548706399, 245973964471725640521348, 274359478651754026035528, 551171141805402848917944, 597151082055448828858194

Offset: 1

Meritxell Vila Miñana, Sep 25 2017

There are 10 twenty-four-digit terms in the sequence. Terms of derived sequences can be obtained either by inserting at the center of their digits any number of 9's or by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures.

			105292253210898548706399 -> 888315592687113803586102 -> 686630284375327508072214 -> 274359478651754026035528 -> 551171141805402848917944 -> 101451293600894707746789 -> 886196413897111684407312 -> 672491927785313369715624 -> 245973964471725640521348 -> 597151082055448828858194 -> 105292253210898548706399

Ray Chandler, Table of n, a(n) for n = 1..10
J. H. E. Cohn, Palindromic differences, Fibonacci Quart. 28 (1990), no. 2, 113-120.

Cf. A072142, A072143, A072718, A072719, A215669, A292634, A292635, A292846, A292856, A292857, A292858.

n = f^10(n), n <> f^k(n) for k < 10, where f: x -> |x - reverse(x)|.

Terms ordered by Ray Chandler, Sep 27 2017

A292634 Numbers n such that 4 iterations of 'Reverse and Subtract' lead to n, whereas fewer than 4 iterations do not lead to n.

Original entry on oeis.org

169140971830859028, 312535222687464777, 464929563535070436, 651817066348182933

Offset: 1

Meritxell Vila Miñana, Sep 20 2017

There are 4 eighteen-digit terms in the sequence. Terms of derived sequences can be obtained either by inserting at the center of their digits any number of 9's or by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures.

			|169140971830859028 - 820958038179041961| = 651817066348182933
|651817066348182933 - 339281843660718156| = 312535222687464777
|312535222687464777 - 777464786222535213| = 464929563535070436
|464929563535070436 - 634070535365929464| = 169140971830859028

J. H. E. Cohn, Palindromic differences, Fibonacci Quart. 28 (1990), no. 2, 113-120.

Cf. A072142, A072143, A072718, A072719, A215669, A292635, A292846, A292856, A292857, A292858, A292859.

n = f^4(n), n <> f^k(n) for k < 4, where f: x -> |x - reverse(x)|.

Terms ordered by Ray Chandler, Sep 27 2017

A292635 Numbers n such that 5 applications of "Reverse and Subtract" lead to n, whereas fewer than 5 applications do not lead to n.

Original entry on oeis.org

10591266563195008940873343680499, 27547681086656717245231891334328, 54795638726597554520436127340244, 68723845538328853127615446167114, 88817367774609971118263222539002

Offset: 1

Meritxell Vila Miñana, Sep 20 2017

There are 5 thirty-two-digit terms in the sequence. Further sequences can be obtained by inserting at the center of these terms any number of 9's and by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures.

			10591266563195008940873343680499 -> 88817367774609971118263222539002 -> 68723845538328853127615446167114 -> 27547681086656717245231891334328 -> 54795638726597554520436127340244 -> 10591266563195008940873343680499

J. H. E. Cohn, Palindromic differences, Fibonacci Quart. 28 (1990), no. 2, 113-120.

Cf. A072142, A072143, A072718, A072719, A215669, A292634, A292846, A292856, A292857, A292858, A292859.

n = f^5(n), n <> f^k(n) for k < 5, where f: x -> |x - reverse(x)|.

Terms ordered by Ray Chandler, Sep 27 2017

A073142 List of smallest solutions for some k of x = f^k(x), n <> f^j(n) for j < k, where f: m -> |m - reverse(m)|.

Original entry on oeis.org

0, 2178, 11436678, 108811891188, 118722683079

Offset: 1

Klaus Brockhaus, Jul 17 2002

In the definition, j can be restricted to proper divisors of k. A073143 gives the corresponding values of k. A073144(n) gives the smallest m such that the 'Reverse and Subtract' trajectory of m leads to a(n). Presumably a(6) = 1186781188132188 with k = 17.

			a(3) = 11436678 is the smallest solution of x = f^14(x) and there is no k such that x = f^k(x) has a smallest solution between a(2) = 2178 and a(3).

Cf. A072137, A072141, A072142, A072143, A072718, A072719, A073143, A073144.

Offset changed by N. J. A. Sloane, Dec 01 2007

A073143 Numbers k such that A073142(n) = f^k(A073142(n)), where f: m -> |m - reverse(m)|.

Original entry on oeis.org

1, 2, 14, 22, 12

Offset: 1

Klaus Brockhaus, Jul 17 2002

Presumably a(6) = 17. a(n) is the length of the periodic part (cf. A072137) of the trajectory of A073142(n). Question: Does every k > 0 appear in this sequence?

			a(3) = 14 since A073142(2) = 11436678 is the smallest solution of x = f^14(x).

Cf. A072137, A072141, A072142, A072143, A072718, A072719, A073142, A073144.

Offset changed by N. J. A. Sloane, Dec 01 2007

A073144 Smallest m such that the 'Reverse and Subtract' trajectory (cf. A072137) of m leads to A073142(n).

Original entry on oeis.org

0, 1012, 10001145, 100000114412, 100010505595

Offset: 1

Klaus Brockhaus, Jul 17 2002

Presumably a(6) = 1000000011011012.

			1012 -> 1089 -> 8712 -> 6534 -> 2178 = A073142(1) and no m < 1012 leads to 2178.

Cf. A072137, A072141, A072142, A072143, A072718, A072719, A073142, A073143.

Offset changed by N. J. A. Sloane, Dec 01 2007

A292992 Numbers n such that 13 applications of 'Reverse and Subtract' lead to n, whereas fewer than 13 applications do not lead to n.

Original entry on oeis.org

1195005230033599502088049947699664004979, 1381092199992389193086189078000076108069, 1417996648846699605185820033511533003948, 2845548027720844548271544519722791554517

Offset: 1

Ray Chandler, Sep 28 2017

There are 13 forty-digit terms in the sequence. Terms of derived sequences can be obtained either by inserting at the center of their digits any number of 9's or by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures.

			1195005230033599502088049947699664004979 -> 8598999439933899906714010005600661000932 -> 6208997779868899802537910022201311001974 -> 1417996648846699605185820033511533003948 -> 7075006702306600680629249932976933993193 -> 3161013305514201251368389866944857987486 -> 3686884278982488587263131157210175114127 -> 3527231431145022726364727685688549772736 -> 2845548027720844548271544519722791554517 -> 4309003944558309903456909960554416900965 -> 1381092199992389193086189078000076108069 -> 8226924500016320623717730754999836793762 -> 5552948110021750246544470518899782497534 ->
  1195005230033599502088049947699664004979.

Ray Chandler, Table of n, a(n) for n = 1..13
J. H. E. Cohn, Palindromic differences, Fibonacci Quart. 28 (1990), no. 2, 113-120.

Cf. A072142, A072143, A072718, A072719, A215669, A292634, A292635, A292846, A292856, A292857, A292858, A292859, A292993.

n = f^13(n), n <> f^k(n) for k < 13, where f: x -> |x - reverse(x)|.

A292993 Numbers n such that 15 applications of 'Reverse and Subtract' lead to n, whereas fewer than 15 applications do not lead to n.

Original entry on oeis.org

10695314508256806604321090888649339244708568530399, 11787342277647023379656208735392766826312885522179, 14638655404662283607788118901219361883250644206458, 26730889210860738952361172793674105293199801097128

Offset: 1

Ray Chandler, Sep 28 2017

There are 15 fifty-digit terms in the sequence. Further terms are obtained (a) by inserting at the center of these terms either any number of 0's (for 10695314508256806604321090888649339244708568530399, 26730889210860738952361172793674105293199801097128, 29899105876561459824028272726867015583422139910097, 49102887245877091252834454555175879833145710289795, 55448121688278511195278554322651878413601497706634, 68315444154984874470735536347381553142144945548514, 88608272072487486790367718123691321620571972829202) or any number of 9's (for the other eight terms) and (b) by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures. Method (b) may be applied recursively to all terms. - Ray Chandler, Oct 15 2017

			10695314508256806604321090888649339244708568530399 -> 88608272072487486790367718123691321620571972829202 -> 68315444154984874470735536347381553142144945548514 -> 26730889210860738952361172793674105293199801097128 -> 55448121688278511195278554322651878413601497706634 -> 11787342277647023379656208735392766826312885522179 -> 85335216543715843349697571530304565248364338856532 -> 61769333197331586809394053950610230396629777603174 -> 14638655404662283607788118901219361883250644206458 -> 70821589200576532783422862287551276343389811477183 -> 32644177302242165567844635465112552775889512964376 -> 34702744296615559953311818179764003348330864180247 -> 39505402506768770093485363631571992203338380540496 -> 29899105876561459824028272726867015583422139910097 -> 49102887245877091252834454555175879833145710289795.

Ray Chandler, Table of n, a(n) for n = 1..15
J. H. E. Cohn, Palindromic differences, Fibonacci Quart. 28 (1990), no. 2, 113-120.

Cf. A072142, A072143, A072718, A072719, A215669, A292634, A292635, A292846, A292856, A292857, A292858, A292859, A292992.

n = f^15(n), n <> f^k(n) for k < 15, where f: x -> |x - reverse(x)|.

A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 180, 189, 198, 270, 279, 297, 360, 369, 396, 450, 459, 495, 540, 549, 594, 630, 639, 693, 720, 729, 792, 810, 819, 891, 900, 909, 990, 999, 1089, 1179, 1188, 1269, 1278, 1359, 1368, 1449, 1458, 1539, 1548, 1629, 1638, 1719, 1728, 1800, 1809, 1818, 1890, 1908, 1980, 1989, 1998, 2079

Offset: 1

Michael Greaney, Jul 03 2020

All terms are divisible by 9.
Let f(k) = k - reverse(k). Then f(reverse(k)) = -f(k), since f(reverse(k)) = reverse(k) - reverse(reverse(k)) = reverse(k) - k = - (k - reverse(k)) = -f(k).
Iteration of the function f(k) = k - reverse(k) leads to A072140, A072141, A072142, and A072143.

Michael P. Greaney, Remarkable Reversible Numbers, +plus magazine, (September 29, 2015).

Cf. A056965, A067030, A072140, A072141, A072142, A072143.

Dividing by 9 gives A334145.

A072144 Numbers n such that the period length of the 'Reverse and Subtract' trajectory of n is greater than 2.

Original entry on oeis.org

10001145, 10001827, 10002179, 10002289, 10002894, 10003037, 10003268, 10003378, 10004412, 10004698, 10006304, 10007624, 10007734, 10007965, 10008108, 10008713, 10008823, 10009175, 10009857, 10010022, 10010484

Offset: 1

Klaus Brockhaus, Jun 24 2002

'Reverse and Subtract' (cf. A072137) is defined by x -> |x - reverse(x)|. - Subsequence of A072140.

			10001145 -> 44108856 -> 21771288 -> 66446424 -> 23981958 and 23981958 as a term of A072142 is the first term of the periodic part of the trajectory of 10001145, period length is 14.

Cf. A072137, A072140, A072142, A072143, A072145.

cp's OEIS Frontend

A292859 Numbers k such that 10 applications of 'Reverse and Subtract' lead to k, whereas fewer than 10 applications do not lead to k.

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A292634 Numbers n such that 4 iterations of 'Reverse and Subtract' lead to n, whereas fewer than 4 iterations do not lead to n.

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A292635 Numbers n such that 5 applications of "Reverse and Subtract" lead to n, whereas fewer than 5 applications do not lead to n.

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A073142 List of smallest solutions for some k of x = f^k(x), n <> f^j(n) for j < k, where f: m -> |m - reverse(m)|.

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A073143 Numbers k such that A073142(n) = f^k(A073142(n)), where f: m -> |m - reverse(m)|.

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A073144 Smallest m such that the 'Reverse and Subtract' trajectory (cf. A072137) of m leads to A073142(n).

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A292992 Numbers n such that 13 applications of 'Reverse and Subtract' lead to n, whereas fewer than 13 applications do not lead to n.

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A292993 Numbers n such that 15 applications of 'Reverse and Subtract' lead to n, whereas fewer than 15 applications do not lead to n.

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A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

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