A189441 -id:A189441 - cp's OEIS frontend

A157037 Numbers with prime arithmetic derivative A003415.

6, 10, 22, 30, 34, 42, 58, 66, 70, 78, 82, 105, 114, 118, 130, 142, 154, 165, 174, 182, 202, 214, 222, 231, 238, 246, 255, 273, 274, 282, 285, 286, 298, 310, 318, 345, 357, 358, 366, 370, 382, 385, 390, 394, 399, 418, 430, 434, 442, 454, 455, 465, 474, 478

Offset: 1

Reinhard Zumkeller, Feb 22 2009

nonn

Equivalently, solutions to n'' = 1, since n' = 1 iff n is prime. Twice the lesser of the twin primes, 2*A001359 = A108605, are a subsequence. - M. F. Hasler, Apr 07 2015

All terms are squarefree, because if there would be a prime p whose square p^2 would divide n, then A003415(n) = (A003415(p^2) * (n/p^2)) + (p^2 * A003415(n/p^2)) = p*[(2 * (n/p^2)) + (p * A003415(n/p^2))], which certainly is not a prime. - Antti Karttunen, Oct 10 2019

			A003415(42) = A003415(2*3*7) = 2*3+3*7+7*2 = 41 = A000040(13), therefore 42 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Reinhard Zumkeller)

Cf. A003415, A010051, A038554, A192082, A192189, A192190,  A327978, A328233, A328240, A328384, A328385.
Cf. A189441 (primes produced by these numbers), A241859.
Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime).
Cf. A108605, A256673 (subsequences).
Subsequence of following sequences: A005117, A099308, A235991, A328234 (A328393), A328244, A328321.

a157037 n = a157037_list !! (n-1)
a157037_list = filter ((== 1) . a010051' . a003415) [1..]
-- Reinhard Zumkeller, Apr 08 2015

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]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* T. D. Noe, Mar 07 2013 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA157037(n) = isprime(A003415(n)); \\ Antti Karttunen, Oct 19 2019

from itertools import count, islice
from sympy import isprime, factorint
def A157037_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), count(2))
A157037_list = list(islice(A157037_gen(),20)) # Chai Wah Wu, Jun 23 2022

A010051(A003415(a(n))) = 1; A068346(a(n)) = 1; A099306(a(n)) = 0.

A003415(a(n)) = A328385(a(n)) = A241859(n); A327969(a(n)) = 3. - Antti Karttunen, Oct 19 2019

A189483 Primes that are not the arithmetic derivative (A003415) of any integer.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 37, 47, 53, 67, 79, 83, 89, 97, 107, 127, 137, 149, 157, 163, 173, 179, 197, 223, 233, 257, 277, 307, 317, 337, 353, 367, 373, 379, 389, 397, 409, 443, 449, 457, 499, 509, 547, 569, 577, 593, 613, 659, 673, 677, 683, 709, 733, 757, 769

Offset: 1

T. D. Noe, Apr 22 2011

nonn

These are the prime terms of A098700. Complement of A189441 in the primes.

If we consider rational numbers we can find anti-derivatives for these numbers: for instance 11385/4 = 2846+1/4 is an anti-derivative of 3. - Paolo P. Lava, Aug 02 2012

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A098700, A099302.

Primes p such that A099302(p) = 0.

A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

Original entry on oeis.org

0, 1, 2, 3, 0, 5, 5, 7, 0, 5, 7, 11, 0, 13, 5, 0, 0, 17, 7, 19, 0, 7, 13, 23, 0, 7, 0, 0, 0, 29, 31, 31, 0, 5, 19, 0, 0, 37, 7, 0, 0, 41, 41, 43, 0, 0, 7, 47, 0, 5, 0, 0, 0, 53, 0, 0, 0, 13, 31, 59, 0, 61, 5, 0, 0, 7, 61, 67, 0, 0, 59, 71, 0, 73, 0, 0, 0, 7, 71, 79, 0, 0, 43, 83, 0, 13, 0, 0, 0, 89, 0, 0, 0, 19, 5

Offset: 0

Antti Karttunen, Feb 11 2022

Primes of A189483 occur only once, on the corresponding indices, while A189441 may also occur in other positions.

There are interesting white "filament-like regions" in the scatter plot.

			For n = 15, if we iterate with A003415, we get a path 15 -> 8 -> 12 -> 16 -> 32 -> 80 -> 176 -> 368 -> ..., where the terms just keep on growing without ever reaching a prime or 1, therefore a(15) = 0.
For n = 18, its path down to zero, when iterating A003415 is: 18 -> 21 -> 10 -> 7 -> 1 -> 0, and the first noncomposite term on the path is prime 7, therefore a(18) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A189441, A189483, A351079, A351259 [= a(A351255(n))].
Cf. A099309 (positions of zeros after the initial one at a(0)=0), A328115 (positions of 5's), A328117 (positions of 7's).
Cf. also A327968.

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));
A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };

For all n, a(4*n) = a(27*n) = a((p^p)*n) = a(A099309(n)) = 0.

a(p) = p for all primes p.

A369250 Primes for which there is at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

71, 103, 131, 151, 167, 191, 199, 211, 239, 251, 263, 271, 311, 331, 359, 383, 419, 431, 439, 467, 479, 487, 491, 503, 563, 587, 599, 607, 631, 647, 691, 719, 727, 739, 743, 751, 811, 823, 839, 859, 863, 887, 911, 919, 971, 983, 991, 1019, 1031, 1051, 1063, 1091, 1103, 1151, 1163, 1187, 1223, 1231, 1279, 1283, 1291

Offset: 1

Antti Karttunen, Jan 22 2024

nonn

All such primes are by necessity of the form 4m+3 (in A002145). See A369249 for those 4m+3 primes that do not have such a representation.

Also by necessity, in these cases the primes in the sum (p*q + p*r + q*r) must all be distinct, that is, we actually need p < q < r, otherwise the sum would not be a prime.

			71 is present as 71 = (3*5) + (3*7) + (5*7) = A003415(105).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Primes with such a representation vs. 4m+3 primes without such a representation (Plot2 comparison of the densities)

Primes in A369251.
Setwise difference A002145 \ A369249.
Subsequence of A189441.
Cf. A003415, A369054, A369056.

isA369250(n) = (isprime(n) && (A369054(n)>0)); \\ Needs also program from A369054.

A241859 1st Arithmetic derivative of numbers with prime arithmetic derivative (A157037).

Original entry on oeis.org

5, 7, 13, 31, 19, 41, 31, 61, 59, 71, 43, 71, 101, 61, 101, 73, 113, 103, 151, 131, 103, 109, 191, 131, 167, 211, 151, 151, 139, 241, 167, 191, 151, 227, 271, 199, 191, 181, 311, 269, 193, 167, 433, 199, 211, 269, 311, 293, 281, 229, 191, 263, 401, 241, 251

Offset: 1

Freimut Marschner, Apr 30 2014

nonn

The arithmetic derivative of numbers with prime arithmetic derivative (A157037) leads to a sequence of a selection of prime numbers. It is possible for several numbers to produce the same primes (see A189441). The next arithmetic derivative is A000012, the simplest sequence of positive numbers: the all 1's sequence, A000012=(A157037)’’.

For the arithmetic derivative of n see A003415. - Wolfdieter Lang, May 10 2014

			a(10) = (A157037(10))'= (78)' =  (2*3*13)' = 2*3+3*13+2*13 = 71,
a(12) = (A157037(12))'= (105)' = (3*5*7)' = 3*5+5*7+3*7 = 71.

Freimut Marschner, Table of n, a(n) for n = 1..1000

Cf. A157037, A003415, A189441.

a(n) = (A157037(n))’.

a(n) = A003415(A157037(n)).

A351096 Semiprimes that are the arithmetic derivative (A003415) of some number.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 51, 55, 58, 62, 69, 74, 77, 82, 85, 86, 87, 91, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 146, 155, 158, 159, 161, 166, 169, 178, 183, 185, 187, 194, 201, 202, 203, 206, 213, 214, 215, 218, 221, 226, 235, 247, 249, 253

Offset: 1

Antti Karttunen, Feb 03 2022

nonn

Complement of A351095 in A001358. Subsequence of A239433.

Cf. A003415, A189441.

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA351096(n) = if(2!=bigomega(n), 0, for(k=1,A002620(n),if(A003415(k)==n,return(1))); (0));

cp's OEIS Frontend

A157037 Numbers with prime arithmetic derivative A003415.

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A189483 Primes that are not the arithmetic derivative (A003415) of any integer.

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A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

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A369250 Primes for which there is at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A241859 1st Arithmetic derivative of numbers with prime arithmetic derivative (A157037).

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A351096 Semiprimes that are the arithmetic derivative (A003415) of some number.

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A369250 Primes for which there is at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.