A129286 -id:A129286 - cp's OEIS frontend

A129283 a(n) = (arithmetic derivative of n) + n.

0, 1, 3, 4, 8, 6, 11, 8, 20, 15, 17, 12, 28, 14, 23, 23, 48, 18, 39, 20, 44, 31, 35, 24, 68, 35, 41, 54, 60, 30, 61, 32, 112, 47, 53, 47, 96, 38, 59, 55, 108, 42, 83, 44, 92, 84, 71, 48, 160, 63, 95, 71, 108, 54, 135, 71, 148, 79, 89, 60, 152, 62, 95, 114, 256, 83, 127, 68

Offset: 0

Reinhard Zumkeller, Apr 07 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A003415, A051674, A129284, A129285, A129286, A136141.

a129283 n = a003415 n + n  -- Reinhard Zumkeller, Nov 01 2013

A129283 := proc(n)
    n+A003415(n) ;
end proc:
seq(A129283(n),n=0..40) ; # R. J. Mathar, Feb 04 2022

ad[n_] := Switch[n, 0|1, 0, ?PrimeQ, 1, , Sum[Module[{p, e}, {p, e} = pe; n*e/p], {pe, FactorInteger[n]}]];
a[n_] := ad[n] + n;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 23 2023 *)

a(n) = A003415(n) + n.
a(n) = A003415(n*A051674(k)) / A051674(k).
a(A129284(n)) > 1, a(A129285(n)) > 1, a(A129286(n)) > 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + Sum_{p prime} 1/(p*(p-1)) = 1 + A136141 = 1.773156... . - Amiram Eldar, May 14 2025

A129152 The n-th arithmetic derivative of 5^6.

Original entry on oeis.org

15625, 18750, 34375, 37500, 87500, 187500, 475000, 1212500, 2437500, 6362500, 12737500, 25487500, 50987500, 101987500, 206975000, 530037500, 1060087500, 3890025000, 15175012500, 45525375000, 177026512500, 596222100000, 2708984250000, 12765250350000

Offset: 0

Reinhard Zumkeller, Apr 01 2007

nonn

In general, the trajectory of p^(p+1) under A003415 is equal to p^p times the trajectory of p under A129283: n -> n + n'. Here we have the case p = 5 (see A129286 for a(n)/5^5), see A129150 and A129151 for p = 2 and 3. - M. F. Hasler, Nov 28 2019

Cf. A129150, A129151, A068327.

Cf. A099309, A051674, A100716.

a129152 n = a129152_list !! n
a129152_list = iterate a003415 15625  -- Reinhard Zumkeller, Apr 29 2012

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; s = 5^6; Join[{s}, Table[s = dn[s], {18}]] (* T. D. Noe, Mar 07 2013 *)

A129152_upto(N)=vector(N,n,N=if(n>1,A003415(N),5^6)) \\ gives a(0..N-1). To get a(1..N) put A003415() around if() instead inside.  M. F. Hasler, Nov 28 2019

a(n+1) = A003415(a(n)), a(0) = 5^6 = 15625.

a(n) = A129286(n)*5^5; A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

A090635 Trajectory of 63 under the map k -> A003415(k) (taking the arithmetic derivative).

Original entry on oeis.org

63, 51, 20, 24, 44, 48, 112, 240, 608, 1552, 3120, 8144, 16304, 32624, 65264, 130544, 264928, 678448, 1356912, 4979232, 19424016, 58272480, 226593936, 763164288, 3467499840, 16339520448, 65370077568, 295266178368, 1223245608192, 6931725175296, 40582548986112

Offset: 1

N. J. A. Sloane, Dec 14 2003

nonn

A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.

Cf. A003415, A090636, A090637.

A090635(n, a=63)={if(n<0, vector(-n, n, if(n>1, a=A003415(a), a)), for(n=2, n, a=A003415(a)); a)}  \\ For n<0 return the vector a[1..-n]. - M. F. Hasler, Nov 27 2019

a(n+1) = A003415(a(n)), a(1) = 63. a(n) = 4*A129286(n-3) for n > 2. - M. F. Hasler, Nov 27 2019

More explicit name from M. F. Hasler, Nov 27 2019

A129284 a(n) = A129150(n) / 4, where A129150(n) = n-th arithmetic derivative of 2^3.

Original entry on oeis.org

2, 3, 4, 8, 20, 44, 92, 188, 380, 856, 2148, 5024, 17616, 58768, 176320, 755904, 3305920, 13885184, 69634816, 348174336, 2385273856, 14652403712, 102566830080, 849285738496, 6035962949632, 44017806979072, 308166534991872, 2380768960708608, 23410894780694528

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2, see A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Paolo P. Lava, Table of n, a(n) for n = 0..75

Cf. A129285, A129286, A051674.

a129284 n = a129150 n `div` 4  -- Reinhard Zumkeller, Nov 01 2013, corrected by M. F. Hasler, Nov 29 2019

A129284_upto(n)=A129150_upto(n)\4 \\ M. F. Hasler, Nov 29 2019

a(n+1) = A129283(a(n)), a(0) = 2.

a(18)-a(28) from Paolo P. Lava, Apr 16 2012

Edited by M. F. Hasler, Nov 27 2019

A129285 a(n) = A129151(n) / 27.

Original entry on oeis.org

3, 4, 8, 20, 44, 92, 188, 380, 856, 2148, 5024, 17616, 58768, 176320, 755904, 3305920, 13885184, 69634816, 348174336, 2385273856, 14652403712, 102566830080, 849285738496, 6035962949632, 44017806979072, 308166534991872, 2380768960708608, 23410894780694528

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

Cf. A129284 (essentially the same), A129283 (n + n'), A129286, A051674, A129150, A090636 (trajectory of 15 under arithmetic derivative A003415).

a129285 n = a129151 n `div` 27  -- Reinhard Zumkeller, Nov 01 2013 [Moved here from A129284, first erroneous line deleted. - M. F. Hasler, Nov 27 2019]

a(n+1) = A129283(a(n)), a(0) = 3.
a(n) = A129284(n+1). - Eric M. Schmidt, Oct 22 2013
Thus a(n) = A129150(n+1) / 4 = A090636(n+3) / 4. - M. F. Hasler, Nov 27 2019

cp's OEIS Frontend

A129283 a(n) = (arithmetic derivative of n) + n.

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A129152 The n-th arithmetic derivative of 5^6.

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A090635 Trajectory of 63 under the map k -> A003415(k) (taking the arithmetic derivative).

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A129284 a(n) = A129150(n) / 4, where A129150(n) = n-th arithmetic derivative of 2^3.

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A129285 a(n) = A129151(n) / 27.

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