Amarnath Murthy - cp's OEIS frontend

A344848 Fixed points of A219360.

1, 2, 3, 4, 5, 45, 47, 48, 63, 70, 139, 163, 191, 194, 206, 242, 310, 422, 457, 668, 983, 1018, 1022, 1087, 1089, 1194, 1251, 1312, 1460, 1692, 1757, 1795, 1863, 1907, 1956, 2048, 2158, 2169, 2175, 2193, 2285, 2406, 2439, 2513, 2626, 2645, 2697, 2726, 2738, 2788

Offset: 1

Amarnath Murthy, May 30 2021

nonn

Data are obtained by parsing the b-file of A219360, assuming that this is correct up to the maximum index shown here (so no further backtracking steps need to be inserted there at higher indices). - R. J. Mathar, Sep 15 2021

Cf. A219360.

{k: k=A219360(k)}.

A274943 Smallest self-descriptive number in base b, or -1 if no such number exists.

Original entry on oeis.org

-1, -1, 100, 1425, -1, 389305, 8946176, 225331713, 6210001000, 186492227801, 6073061476032, 213404945384449, 8054585122464440, 325144322753909625, 13983676842985394176

Offset: 2

N. J. A. Sloane, Jul 23 2016, following an email from Amarnath Krishnamurthy [Amarnath Murthy] and in addition using data from A108551

A self-descriptive number in base b has b digits, indexed by 0 ... b-1 and for all n, the n-th digit equals the number of n's in the number. In base 10 there is exactly one such number, 6210001000.

			1210_4 = 100, 21200_5 = 1425, 3211000_7 = 389305,
42101000_8 = 8946176, 521001000_9 = 225331713, 6210001000_10,
72100001000_11 = 186492227801, 821000001000_12 = 6073061476032,
9210000001000_13 = 213404945384449, (10)2100000001000_14 =
8054585122464440, (11)21000000001000_15 = 325144322753909625,
(12)21000000001000_16 = 13983676842985394176, etc.

Very similar to A108551.

A178915 Rearrangement of natural numbers so that every partial sum is composite.

Original entry on oeis.org

4, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Amarnath Murthy, Jun 23 2010

a(n) = n for n > 4.

Except for the integers 1 & 4 which are interchanged, the sequence is in order. Proof: Except for the first three triangular numbers (A000217), {0, 1, 3}, they are all composite. - Robert G. Wilson v, Jun 27 2010

			Partial sums are 4,6,9,10,15,21,...

Index entries for linear recurrences with constant coefficients, signature (2, -1).

f[s_List] := Block[{k = 0, t = Plus @@ s}, While[MemberQ[s, k] || PrimeQ[t + k] || t + k < 2, k++ ]; Append[s, k]]; Rest@ Nest[f, {0}, 72] (* Robert G. Wilson v, Jun 27 2010 *)

G.f.: 3 - 3*x^3 + 1/(x-1)^2. - Sergei N. Gladkovskii, Oct 16 2012

a(40)-a(72) from Robert G. Wilson v, Jun 27 2010

A178914 10's complement of nonnegative numbers.

Original entry on oeis.org

10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28

Offset: 0

Amarnath Murthy, Jun 23 2010

Apart from the initial a(0) a duplicate of A089186. - R. J. Mathar, Jun 25 2010

			a(11) = 10's complement of 11 = 89

Cf. A089186.

Join[{10},Table[10^IntegerLength[n]-n,{n,80}]] (* Harvey P. Dale, Feb 06 2015 *)

a(n) = 10^k - n where k is the number of digits in n.

A164883 Cubes with the property that the sum of the cubes of the digits is also a cube.

Original entry on oeis.org

0, 1, 8, 1000, 8000, 474552, 1000000, 1643032, 8000000, 13312053, 27818127, 125751501, 474552000, 1000000000, 1015075125, 1121622319, 1256216039, 1501123625, 1643032000, 3811036328, 8000000000, 11000295424, 13312053000

Offset: 1

Amarnath Murthy, Apr 21 2001

It is known (Murthy 2001) that the sequence is infinite. (1) The number {3(10^(k+2)+1)}^3 for all k produces such numbers. (2) Less trivially, {10^(n+2) - 4}^3 is a member of this sequence for n = 4*{(10^(3k)-1)/27}-1, for all k, for which the sum of the cubes of the digits is {6*10^k}^3.

			474552 = 78^3 is a term since 4^3+7^3+4^3+5^3+5^3+2^3 = 729 = 9^3.

Amarnath Murthy, Smarandache Fermat Additive Cubic Sequence, 2011. (To be published in the Smarandache Notions Journal.)

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

R:= NULL: count:= 0:
for x from 0 while count < 100 do
  v:= x^3;
  t:= add(s^3,s=convert(v,base,10));
  if surd(t,3)::integer then
       R:= R, v; count:= count+1;
  fi;
od:
R; # Robert Israel, Apr 15 2025

Select[Range[0,2500]^3,IntegerQ[Total[IntegerDigits[#]^3]^(1/3)]&] (* Harvey P. Dale, Jun 03 2012 *)

Corrected and extended by Gaurav Kumar, Aug 29 2009

A135566 Least prime not already in the sequence such that the n-th partial concatenation is a multiple of the n-th prime.

Original entry on oeis.org

2, 7, 5, 17, 71, 23, 37, 137, 79, 241, 3, 173, 67, 31, 347, 433, 127, 47, 571, 1069, 107, 227, 229, 853, 647, 271, 83

Offset: 1

Amarnath Murthy, Nov 13 2005

See A114025 for another version.

More terms from David Wasserman, Mar 04 2008

A138561 Start with the list {1}; for each n >= 1, append p(n) primes followed by c(n) composite numbers, where p(n) is the n-th prime and c(n) is the n-th composite number.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 5, 7, 11, 10, 12, 14, 15, 16, 18, 13, 17, 19, 23, 29, 20, 21, 22, 24, 25, 26, 27, 28, 31, 37, 41, 43, 47, 53, 59, 30, 32, 33, 34, 35, 36, 38, 39, 40, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 109, 113, 127, 131

Offset: 1

Amarnath Murthy, Mar 25 2008

A rearrangement of the natural numbers.

			We start with 1; p(1) = 2 is the first prime and the next two terms are 2,3 while c(1)=4 is the first composite number and the next four terms are 4,6,8,9 and so on.

More terms from Sean A. Irvine, Apr 13 2010

A121664 Inverse permutation to A096114.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 11, 10, 9, 7, 8, 12, 23, 22, 21, 19, 20, 18, 13, 14, 15, 17, 16, 24, 47, 46, 45, 43, 44, 42, 37, 38, 39, 41, 40, 36, 25, 26, 27, 29, 28, 30, 35, 34, 33, 31, 32, 48, 95, 94, 93, 91, 92, 90, 85, 86, 87, 89, 88, 84, 73, 74, 75, 77, 76, 78, 83, 82, 81, 79, 80, 72

Offset: 1

Antti Karttunen, Aug 25 2006, based on Amarnath Murthy's comments on A096114

nonn

Inverse: A096114.

A117622 Minesweeper sequence: a(n) is the first nonprime number, k, not occurring previously in the sequence nor the absolute value of its first forward difference among the first differences and a(1)=1.

Original entry on oeis.org

1, 4, 6, 10, 9, 14, 8, 15, 24, 12, 20, 30, 16, 27, 40, 18, 33, 49, 21, 38, 56, 22, 42, 63, 25, 44, 68, 26, 51, 28, 54, 81, 32, 62, 91, 34, 65, 98, 35, 70, 102, 36, 72, 111, 39, 76, 116, 45, 86, 129, 46, 90, 135, 48, 94, 141, 50, 100, 52, 104, 155, 55, 108, 162, 57, 112, 168

Offset: 1

Amarnath Murthy, Apr 08 2006

nonn

Let there be mines under prime numbers in the sequence of natural numbers (on the number line). A man starts from 1 and moves on the line to cover all composite numbers once. He can take a jump of length k only once for every k. He can jump to either side. He moves so that he gives priority to touch the smallest composite number not covered earlier.

Does every composite number get touched?

			Beginning with 1 he takes a jump of 3 to touch 4 then a jump of 2 to touch 6, then a jump of 4 to touch 10 then a jump of 1 in the other direction to touch 9 and so on.

Cf. A117623 (values of k), A081145 (if nothing is mined).

f[s_] := Block[{k = 2, d = Abs[Most@s - Rest@s], l = Last@s}, While[ PrimeQ[k] || MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; Nest[f, {1}, 66] (* Robert G. Wilson v *)

Edited and corrected by Robert G. Wilson v, Jun 13 2006

A117623 Values of k associated with A117622.

Original entry on oeis.org

3, 2, 4, -1, 5, -6, 7, 9, -12, 8, 10, -14, 11, 13, -22, 15, 16, -28, 17, 18, -34, 20, 21, -38, 19, 24, -42, 25, -23, 26, 27, -49, 30, 29, -57, 31, 33, -63, 35, 32, -66, 36, 39, -72, 37, 40, -71, 41, 43, -83, 44, 45, -87, 46, 47, -91, 50, -48, 52, 51, -100, 53, 54, -105, 55, 56, -110, 59, 58, -115, 60

Offset: 1

Amarnath Murthy, Apr 08 2006

sign

Does every positive number appear as an absolute value?

f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[ PrimeQ[k] || MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; t = Nest[f, {1}, 71]; Rest@t - Most@t (* Robert G. Wilson v, Jun 13 2006 *)

More terms from Jonathan Vos Post, Apr 10 2006

Corrected and extended by Robert G. Wilson v, Jun 13 2006

cp's OEIS Frontend

User: Amarnath Murthy

A344848 Fixed points of A219360.

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A274943 Smallest self-descriptive number in base b, or -1 if no such number exists.

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A178915 Rearrangement of natural numbers so that every partial sum is composite.

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Mathematica

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Extensions

A178914 10's complement of nonnegative numbers.

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A164883 Cubes with the property that the sum of the cubes of the digits is also a cube.

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A135566 Least prime not already in the sequence such that the n-th partial concatenation is a multiple of the n-th prime.

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A138561 Start with the list {1}; for each n >= 1, append p(n) primes followed by c(n) composite numbers, where p(n) is the n-th prime and c(n) is the n-th composite number.

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A121664 Inverse permutation to A096114.

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A117622 Minesweeper sequence: a(n) is the first nonprime number, k, not occurring previously in the sequence nor the absolute value of its first forward difference among the first differences and a(1)=1.

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Mathematica

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A117623 Values of k associated with A117622.

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