A258651 -id:A258651 - cp's OEIS frontend

A258650 Tenth arithmetic derivative of n.

0, 0, 0, 0, 4, 0, 0, 0, 8592, 0, 0, 0, 20096, 0, 0, 3424, 70464, 0, 0, 0, 16304, 0, 0, 0, 32624, 0, 1520, 27, 70464, 0, 0, 0, 235072, 0, 0, 8592, 47872, 0, 0, 20096, 24640, 0, 0, 0, 65264, 8592, 0, 0, 130544, 0, 3424, 8144, 47872, 0, 57996, 20096, 198656, 0, 0

Offset: 0

Alois P. Heinz, Jun 06 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=10 of A258651.

Cf. A003415.

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= n-> A(n, 10):
seq(a(n), n=0..70);

from sympy import factorint
def A258650(n):
    for _ in range(10):
        if n <= 1: return 0
        n = sum((n*e//p for p,e in factorint(n).items()))
    return n # Chai Wah Wu, Nov 03 2022

a(n) = A003415^10(n).

A328383 a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number which is either a divisor or multiple of n, but not both at the same time. If no such number can ever be reached, a(n) is 0 (when either n is of the form p^p, or if the iteration would never stop). When the number reached is a divisor of n, a(n) is -1 * iteration count.

Original entry on oeis.org

-1, -1, 0, -1, -2, -1, 2, -3, -2, -1, 9, -1, -4, 23, 1, -1, -4, -1, 5, -2, -2, -1, 2, -3, 24, 0, 18, -1, -2, -1, 6, -5, -2, 85, 7, -1, -4, 21, 10, -1, -2, -1, 35, 53, -4, -1, 2, -5, 44, 18, 34, -1, 2, 21, 4, -3, -2, -1, 16, -1, -6, 21, 1, -5, -2, -1, 7, 85, -2, -1, 4, -1, 23, 55, 5, -4, -2, -1, 4, 9, -2, -1, 42, -3, 42

Offset: 2

Antti Karttunen, Oct 15 2019

The absolute value of a(n) tells how many columns right from the leftmost column in array A258651 one needs to go at row n, before one (again) finds either a divisor or a multiple of n, with 0's reserved for cases like 4 and 27 where the same value continues forever. If one finds a divisor before a multiple, then the value of a(n) will be negative, otherwise it will be positive.

Question: What is the value of a(91) ?

			For n = 6, its arithmetic derivative A003415(6) = 5 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(5) = 1 is its divisor, thus a(6) = -2.
For n = 8, its arithmetic derivative A003415(8) = 12 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(12) = 16 is its multiple, thus a(8) = +2.
Numbers reached for n=2..28 (with positions of the form p^p are filled with the same p^p): 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 8592, 1, 1, 410267320320, 32, 1, 1, 1, 240, 7, 1, 1, 48, 1, 410267320320, 27, 9541095424. For example, we have a(12) = 9 and the 9th arithmetic derivative of 12 is A003415^(9)(12) = 8592 = 716*12.

Antti Karttunen, Table of n, a(n) for n = 2..90

Cf. A003415, A258651, A328235, A328236.
Cf. A051674 (indices of zeros provided for all n >= 2 either a divisor or multiple can be found).
Cf. A256750, A328248, A328384 for similar counts.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328383(n) = { my(u=A003415(n),k=1); if(u==n,return(0)); while((n%u) && (u%n), k++; u = A003415(u)); if(u%n,-k,k); };

a(A000040(n)) = -1.

a(A051674(n)) = 0.

A328384 If n is of the form p^p, a(n) = 0, otherwise a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number different from n which is either a prime, or whose degree (A051903) differs from the degree of n. If the degree of the final number is <= that of n, then a(n) = -1 * iteration count.

Original entry on oeis.org

-1, -1, -1, 0, -1, -1, -1, -1, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -2, -1, -1, -1, -1, 2, 0, 1, -1, -1, -1, -1, 2, -1, 1, 3, -1, -3, 1, -1, -1, -1, -1, 1, -1, 1, -1, 3, -1, -2, 1, 1, -1, 1, 1, -1, -2, -1, -1, 2, -1, 3, -1, 2, 1, -1, -1, 1, 3, -1, -1, -1, -1, 2, -1, 1, 1, -1, -1, 5, -1, -1, -1, 2, -2, 1, 1, -1, -1, -1, 1, 1, -2, 1

Offset: 1

Antti Karttunen, Oct 15 2019

sign

The records -1, 0, 1, 2, 3, 5, 8, 10, 11, 13, ... occur at n = 1, 4, 12, 26, 36, 80, 108, 4887, 18688, 22384, ...

			For n = 21 = 3*7, A051903(21) = 1, A003415(21) = 10 = 2*5, is of the same degree as A051903(10) = 1, but then A003415(10) = 7, which is a prime, having degree <= of the starting value (as we have A051903(7) <= A051903(21)), thus a(21) = -1 * 2 = -2.
For n = 33 = 3*11, A051903(33) = 1, A003415(33) = 14 = 2*7, is of the same degree, but on the second iteration, A003415(14) = 9 = 3^2, with A051903(9) = 2, which is larger than the initial degree, thus a(33) = +2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A051903, A051674, A157037, A258651, A328252, A328310, A328320.
Cf. A328385 (the number found in the iteration).
Cf. A256750, A328248, A328383 for similar counts.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((n<=1),n-1,vecmax(factor(n)[, 2]));
A328384(n) = { my(d=A051903(n), u=A003415(n), k=1); if(u==n,return(0)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, k++; n = u; u = A003415(u)); if(A051903(u)<=d,-k,k); };

a(1) = -1 as 0 is here considered having a smaller degree than 1.
a(p) = -1 for all primes.
a(A051674(n)) = 0.
a(A157037(n)) = -1.
a(A328252(n)) = -1.
a(A328320(n)) = -1.

A185232 n-th arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 1520, 0, 0, 0, 235072, 0, 0, 705280, 278539264, 0, 0, 0, 226593936, 0, 0, 0, 295266178368, 0, 24143851798528, 27, 10557680820452065280, 0, 0, 0, 2821525007683005301391360, 0, 0, 2821525007683005301391360, 43942858408664114852524638339072

Offset: 0

José María Grau Ribas, Jan 24 2012

nonn

a(n) is zero for all prime n.

Alois P. Heinz, Table of n, a(n) for n = 0..80

Cf. A003415.

Main diagonal of A258651.

dn[0]=0; dn[1]=0; dn[n_] := Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[Nest[dn,n,n], {n,50}]

from sympy import factorint
def A185232(n):
    for _ in range(n):
        if n <= 1: return 0
        n = sum((n*e//p for p,e in factorint(n).items()))
    return n # Chai Wah Wu, Nov 03 2022

A369638 The n-th arithmetic derivative of 2^7.

Original entry on oeis.org

128, 448, 1408, 5056, 15232, 56384, 169216, 677120, 2902784, 12008192, 49750016, 323383296, 2048106496, 13312700416, 79908855808, 520115646464, 3120693882880, 19851642843136, 121321447460864, 754017659596800, 5330629212856320, 37592273278603264, 230970707522453504, 1736195064174952448, 12244744136813133824

Offset: 0

Antti Karttunen, Feb 04 2024

nonn

Iterates of the map k -> k', starting from k=128, where k' stands for the arithmetic derivative, A003415.

Antti Karttunen, Table of n, a(n) for n = 0..89

Row 128 of A258651.
Cf. A003415, A369652 [= A328114(a(n))].
Cf. also A001787, A129150.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369638(n) = if(!n,128,A003415(A369638(n-1)));

up_to = 89;
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369638list(up_to) = { my(v=vector(up_to)); v[1] = 128;
for(i=2,up_to,print1(i,", "); v[i] = A003415(v[i-1])); (v); };
v369638 = A369638list(1+up_to);
A369638(n) = v369638[1+n];

a(0) = 128, and for n > 0, a(n) = A003415(a(n-1)).

A258652 Sum of the k-th arithmetic derivative of n-k for k=0..n.

Original entry on oeis.org

0, 1, 2, 4, 5, 9, 11, 16, 14, 25, 36, 59, 99, 209, 419, 860, 1730, 3862, 9464, 21868, 74371, 244648, 727345, 3098351, 13469007, 56269849, 281642632, 1406177909, 9597415332, 58891421656, 411673964638, 3406742649805, 24202753250241, 176482943622608

Offset: 0

Alois P. Heinz, Jun 06 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Arithmetic derivative

Antidiagonal sums of A258651.

Cf. A003415.

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= proc(n) option remember; add(A(h, n-h), h=0..n) end:
seq(a(n), n=0..40);

d[n_ /; n>1] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[_] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
a[n_] := a[n] = Sum[A[h, n-h], {h, 0, n}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = Sum_{k=0..n} A258651(n-k,k).

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A258650 Tenth arithmetic derivative of n.

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A185232 n-th arithmetic derivative of n.

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A369638 The n-th arithmetic derivative of 2^7.

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A258652 Sum of the k-th arithmetic derivative of n-k for k=0..n.

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