A351257 -id:A351257 - cp's OEIS frontend

A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130

Offset: 1

T. D. Noe, Oct 12 2004

nonn

The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.

For all n, A003415(a(n)) is also a term of the sequence. A351255 gives the nonzero terms as ordered by their position in A276086. - Antti Karttunen, Feb 14 2022

			18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths).
Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1 - charfun).
Cf. A276086, A328116, A351255 (permutation of nonzero terms), A351257, A351259, A351261, A351072 (number of prime(k)-smooth terms > 1).
Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
Union of A359544 and A359545.

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]

\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Åhlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022

For all n >= 0, A328309(a(n)) = n. - Antti Karttunen, Feb 14 2022

A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 3, 2, 6, 4, 3, 2, 5, 2, 5, 6, 6, 2, 5, 2, 7, 4, 3, 2

Offset: 0

Antti Karttunen, Oct 07 2019

The terms of this sequence are currently known only up to n=23, with the value of a(24) still being uncertain. For the tentative values of the later terms, see sequence A328324 which gives upper bounds for these terms, many of which are very likely also exact values for them.
As A051903(A003415(n)) >= A051903(n)-1, it means that it takes always at least A051903(n) steps to a prime if iterating solely with A003415.
Some known values and upper bounds from n=24 onward:
a(24) <= 11.
a(25) = 4.
a(26) = 7.
a(27) <= 22.
a(33) = 4.
a(39) = 4.
a(40) = 5.
a(42) = 3.
a(44) <= 10.
a(45) = 5.
a(46) = 5.
a(48) = 9.
a(49) = 6.
a(50) = 6.
a(55) = 7.
a(74) = 5.
a(77) = 6.
a(80) <= 18.
a(111) = 6.
a(112) = 8.
a(125) <= 9.
a(240) = 7.
a(625) <= 10.
a(875) = 8.
From Antti Karttunen, Feb 20 2022: (Start)
a(2556) <= 20.
a(5005) <= 19.
What is the value of a(128), and is A328324(128) well-defined?
When I created this sequence, I conjectured that by applying two simple arithmetic operations "arithmetic derivative" (A003415) and "primorial base exp-function" (A276086) in some combination, and starting from any positive integer, we could always reach zero (via a prime and 1).
At the first sight it seems almost certain that the conjecture holds, as it is always possible at every step to choose from two options (which very rarely meet, see A351088), leading to an exponentially growing search tree, and also because A276086 always jumps out of any dead-end path with p^p-factors (dead-end from the arithmetic derivative's point of view). However, it should be realized that one can reach the terms of either A157037 or A327978 with a single step of A003415 only from squarefree numbers (or respectively, cubefree numbers that are not multiples of 4, see A328234), and in general, because A003415 decreases the maximal exponent of the prime factorization (A051903) at most by one, if the maximal exponent in the prime factorization of n is large, there is a correspondingly long path to traverse if we take only A003415-steps in the iteration, and any step could always lead with certain probability to a p^p-number. Note that the antiderivatives of primorials with a square factor seem quite rare, see A351029.
And although taking a A276086-step will always land us to a p^p-free number (which a priori is not in the obvious dead-end path of A003415, although of course it might eventually lead to one), it (in most cases) also increases the magnitude of number considerably, that tends to make the escape even harder. Particularly, in the majority of cases A276086 increases the maximal exponent (which in the preimage is A328114, "maximal digit value used when n is written in primorial base"), so there will be even a longer journey down to squarefree numbers when using A003415. See the sequences A351067 and A351071 for the diminishing ratios suggesting rapidly diminishing chances of successfully reaching zero from larger terms of A276086. Also, the asymptotic density of A276156 is zero, even though A351073 may contain a few larger values.
On the other hand, if we could prove that by (for example) continuing upwards with any p^p-path of A003415 we could eventually reach with a near certainty a region of numbers with low values of A328114 (i.e., numbers with smallish digits in primorial base, like A276156), then the situation might change (see also A351089). However, a few empirical runs seemed to indicate otherwise.
For all of the above reasons, I now conjecture that there are natural numbers from which it is not possible to reach zero with any combination of steps. For example 128 or 5^5 = 3125.
(End)

			Let -A> stand for an application of A003415 and -B> for an application of A276086, then, we have for example:
a(8) = 6 as we have 8 -A>  12 -B>  25 -A> 10 -A>  7 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(15) = 6 as we have 15 -B> 150 -A> 185 -A> 42 -A> 41 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(20) = 7, as 20 -B> 375 -A> 350 -A> 365 -A> 78 -A> 71 -A> 1 -A> 0, and there are no shorter paths.
For n=112, we know that a(112) cannot be larger than eight, as A328099^(8)(112) = 0, so we have a path of length 8 as 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. Checking all 32 combinations of the paths of lengths of 5 starting from 112 shows that none of them or their prefixes ends with a prime, thus there cannot be any shorter path, and indeed a(112) = 8.
a(24) <= 11 as A328099^(11)(24) = 0, i.e., we have 24 -A> 44 -A> 48 -A> 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. On the other hand, 24 -B> 625 -B> 17794411250 -A> 41620434625 -A> 58507928150 -A> 86090357185 -A> 54113940517 -A> 19982203325 -A> 12038411230 -A> 8426887871 -A> 1 -A> 0, thus offering another path of length 11.

Cf. A003415, A046099, A051674, A051903, A068346, A276086, A276087, A327859, A327860, A328099, A328112, A328114, A328116, A328307.
Cf. A328324 (a sequence giving upper bounds, computed with restricted search space).
Sequences for whose terms k, value a(k) has a guaranteed constant upper bound: A000040, A002110, A143293, A157037, A192192, A327978, A328232, A328233, A328239, A328240, A328243, A328249, A328313.
Sequences for whose terms k, it is guaranteed that a(k) has finite value > 0, even if not bound by a constant: A099308, A328116.
Cf. also A256750, A327966, A328110, A351029, A351088, A351067, A351071, A351073, A351089 and A351255, A351256, A351257, A351258, A351261.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327969(n,searchlim=0) = if(!n,n,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, print("n=", n, " k=", k, " xs=", xs); newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A003415(u); if(0==a, return(k)); if(isprime(a), return(k+2)); b = A276086(u); if(isprime(b), return(k+1+(u>2))); newxs = setunion([a],newxs); if(!searchlim || (b<=searchlim),newxs = setunion([b],newxs))); xs = newxs));

a(0) = 0, a(p^p) = 1 + a(A276086(p^p)) for primes p, and for other numbers, a(n) = 1+min(a(A003415(n)), a(A276086(n))).
a(p) = 2 for all primes p.
For all n, a(n) <= A328324(n).
Let A stand the transition x -> A003415(x), and B stand for x -> A276086(x). The following sequences give some constant upper limits, because it is guaranteed that the combination given in brackets (the leftmost A or B is applied first) will always lead to a prime:
For all n, a(A157037(n)) = 3.  [A]
For n > 1, a(A002110(n)) = 3.  [B]
For all n, a(A192192(n)) <= 4. [AA]
For all n, a(A327978(n)) = 4.  [AB]
For all n, a(A328233(n)) <= 4. [BA]
For all n, a(A143293(n)) <= 4. [BB]
For all n, a(A328239(n)) <= 5. [AAA]
For all n, a(A328240(n)) <= 5. [BAA]
For all n, a(A328243(n)) <= 5. [ABB]
For all n, a(A328313(n)) <= 5. [BBB]
For all n, a(A328249(n)) <= 6. [BAAA]
For all k in A046099, a(k) >= 4, and if A328114(k) > 1, then certainly a(k) > 4.

A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 30, 25, 150, 375, 750, 5625, 7, 14, 21, 42, 126, 70, 105, 315, 350, 1575, 3150, 1750, 2625, 49, 98, 882, 490, 735, 4410, 2450, 3675, 11025, 12250, 30625, 61250, 183750, 686, 3430, 5145, 25725, 77175, 385875, 1929375, 3858750, 4802, 72030, 120050, 180075, 33614, 100842, 117649, 705894, 26471025

Offset: 1

Antti Karttunen, Feb 10 2022

Equal to nonzero terms of A099308 when sorted into ascending order. In this order, which is dictated by the primorial base expansion of n (A049345) and mapped to products of prime powers by A276086, all terms of A099308 that are prime(k)-smooth appear before the terms that are not prime(k)-smooth.
Number of terms whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n-1)-smooth] is given by A351071(n): 1, 4, 8, 44, 216, 1474, 11130, ...
For all n > 1, A003415(a(n)) is also a term of the sequence.
Note that only 451 of the first 105367 terms (all 19-smooth terms) are such that there occurs a 19-smooth number (A080682) larger than 1 on the path before 1 is encountered, when starting from x = a(n) and iterating with map x -> A003415(x).

Antti Karttunen, Table of n, a(n) for n = 1..12878 (all 17-smooth terms of this sequence)
Antti Karttunen, 105368 initial terms, without indices (all 19-smooth terms of this sequence, and also A276086(9699690) = 23, the first 23-smooth term)
Index entries for sequences related to primorial base

Cf. A003415, A049345, A099307, A099308, A276086, A328116, A351071, A351072 (number of prime(n)-smooth terms).

Cf. A351256 [= A051903(a(n))], A351257 [= A099307(a(n))], A351258, A351259 [= A351078(a(n))], A351261 [= A351079(a(n))].

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(u, ", ")));

a(n) = A276086(A328116(n)).

A351261 a(n) is the largest term encountered on the path to 0 when iterating the map x -> x', and starting from x = A351255(n). Here x' means the arithmetic derivative of x, A003415.

Original entry on oeis.org

1, 2, 3, 6, 9, 21, 5, 10, 31, 25, 185, 375, 1075, 12575, 7, 14, 21, 42, 165, 70, 105, 318, 365, 1905, 5385, 2175, 2825, 49, 98, 1281, 490, 735, 7287, 2905, 3745, 17747, 16975, 43075, 107150, 370705, 686, 3871, 5145, 29953, 115395, 1001035, 4475975, 11285575, 5145, 115591, 176645, 234955, 40817, 156065, 156065, 1193297

Offset: 1

Antti Karttunen, Feb 11 2022

At point n=104776, where a(104776) = 6121569170076203821789253759640129542895524171255601586612637263670135

and A351255(104776) = 144537549602172859330715888995919357193998109417395984504745753750, the ratio a(n)/A351255(n) obtains another record (~ 42352.7947), which motivates a conjecture that it is not bound from above. See also A351079.

			From A351255(27) = 2625 it takes 12 iterations of map x -> A003415(x) to reach zero: 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0. The largest term on this path is 2825, therefore a(27) = 2825.

Antti Karttunen, Table of n, a(n) for n = 1..12878 (computed for all 17-smooth terms of A351255)
Antti Karttunen, 105368 initial terms, without indices (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)

Cf. A003415, A276086, A351079, A351255, A351257, A351259.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A351079(n) = { my(m=n); while(n>1, n = A003415checked(n); m = max(m,n)); if(n,m); };
for(n=0, 2^9, u=A276086(n); m = A351079(u); if(m>0,print1(m, ", ")));

a(n) = A351079(A351255(n)).

A351256 Maximal exponent in the prime factorization of A351255(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 3, 4, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

The largest digit in the primorial base representation of A328116(n).

The scatter plot gives some sense of how it is harder to eventually reach zero by iterating A003415, when starting from a number with large exponent(s) in its prime factorization. See also A099308.

			A328116(27) = 50, and A049345(50) = "1310", where the largest primorial base digit is 3, therefore a(27) = 3. Equally, A351255(27) = 2625 = 3 * 5^3 * 7, thus A051903(2625) = 3 and a(27) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..105368 (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)

Cf. A003415, A049345, A051903, A099308, A276086, A328114, A328116, A351255, A351257, A351258.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(A051903(u), ", ")));

a(n) = A328114(A328116(n)) = A051903(A351255(n)).

For all n, a(n) < A351257(n). [See A351258 for the differences].

A351259 First noncomposite number reached when iterating the map x -> x', when starting from x = A351255(n). Here x' is the arithmetic derivative of x, A003415.

Original entry on oeis.org

1, 2, 3, 5, 5, 7, 5, 7, 31, 7, 41, 71, 191, 2711, 7, 5, 7, 41, 103, 59, 71, 271, 71, 1031, 2887, 439, 5, 5, 7, 631, 251, 401, 3491, 1031, 1319, 17747, 9733, 1931, 16319, 77351, 131, 5, 419, 7079, 22343, 971, 5981, 6861581, 419, 18731, 11903, 33937, 7079, 15287, 15287, 6143, 6944111, 1415651, 11, 13, 5, 61, 103, 401, 631

Offset: 1

Antti Karttunen, Feb 11 2022

For the initial 105367 19-smooth terms of A351255, the last 7 occurs here at a(54796), with A351255(54796) = 289993286583 = 3^2 * 7 * 11 * 13^2 * 19^5, and the last 5 occurs here at a(65777), with A351255(65777) = 391899820830375516750 = 2 * 3^2 * 5^3 * 7^3 * 13^3 * 17^3 * 19^6, already a moderately high starting value, in whose vicinity most ending primes for successful iterations are much larger. This observation motivates a conjecture: Even from large numbers with high exponents in their prime factorization it is sometimes possible to reach a small prime. Compare to the conjecture 8 in Ufnarovski & Åhlander paper.

			From A351255(27) = 2625 it takes 12 iterations of the map x -> A003415(x) to reach zero: 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0. Two steps before the final zero is the first and only prime on the path, 5, therefore a(27) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..12878 (computed for all 17-smooth terms of A351255)
Antti Karttunen, 105368 initial terms, without indices (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A000040, A003415, A276086, A327975, A327977, A351078, A351255, A351257, A351261.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };
for(n=0, 2^9, u=A276086(n); p = A351078(u); if(p>0,print1(p, ", ")));

a(n) = A351078(A351255(n)).

a(1) = 1, and for n > 1, a(n) = A003415^[A351257(n)-2](A351255(n)). [This means: take the (A351257(n)-2)-th arithmetic derivative of A351255(n)].

A351258 a(n) = A099307(A351255(n)) - A051903(A351255(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 1, 4, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 9, 4, 5, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 9, 2, 2, 2, 5, 6, 2, 2, 2, 3, 2, 2, 2, 2, 6, 2, 6, 1, 2, 5, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 8, 5, 2, 2, 6, 2, 2, 9, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 3, 7, 3, 3, 3, 4, 2, 2, 3, 2, 8

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

All terms are > 0 because from any k > 0, one certainly cannot reach 1 in less than A051903(k) iterations of the map x -> A003415(x).

One of the records occur at a(20457) = 38. The corresponding term of A351255 is A351255(20457) = A276086(A328116(20457)) = A276086(688352) = 442600020398400142264711707660915237 = 3 * 7^6 * 11^10 * 13^11 * 17^5 * 19. When starting iterating from this value with A003415, it first goes relatively smoothly in 11 steps to the first squarefree number encountered, 6201461846617177861789236821121654153, but after that, it still meanders for the additional 37 iterations (visiting mostly squarefree numbers, but also six numbers with max. exponent = 2, and one number with max. exponent = 3), before finally reaching zero.

Antti Karttunen, Table of n, a(n) for n = 1..105368 (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)

Cf. A003415, A051903, A099307, A351255, A351256, A351257.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(c-A051903(u), ", ")));

a(n) = A351257(n) - A351256(n) = A099307(A351255(n)) - A051903(A351255(n)).

cp's OEIS Frontend

A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A351261 a(n) is the largest term encountered on the path to 0 when iterating the map x -> x', and starting from x = A351255(n). Here x' means the arithmetic derivative of x, A003415.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351256 Maximal exponent in the prime factorization of A351255(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351259 First noncomposite number reached when iterating the map x -> x', when starting from x = A351255(n). Here x' is the arithmetic derivative of x, A003415.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351258 a(n) = A099307(A351255(n)) - A051903(A351255(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula