A189710 -id:A189710 - cp's OEIS frontend

A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 161, 163, 167, 173, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is a prime.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial numbers

Cf. A002110, A003415, A024451 (arith. deriv. of primorials), A068346, A276086, A327859, A328233.
Union of A000040 and A327978 (gives the composite terms).
Differs from A189710 for the first time by having term a(39) = 161, which is not included in A189710, while A189710(44) = 185 is the first term in latter that is not included here.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327859(n) = A276086(A003415(n));
isA328232(n) = isprime(A327859(n));

A189639 Numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

Original entry on oeis.org

161, 209, 221, 1935, 4265, 15941, 22217, 24041, 25637, 30377, 38117, 39077, 48617, 49097, 55877, 68441, 73817, 76457, 80357, 88457, 95237, 98117, 99941, 105641, 110057, 115397, 122537, 130217, 131141, 136517, 143237, 147941, 148697, 152357, 154457, 159077

Offset: 1

Giorgio Balzarotti, Apr 24 2011

The arithmetic derivative of a(n) is a Giuga's number A007850 (solution of n' = n+1).

			161' = 30, 161'' = 30' = 31 ==> 161'' = 161'+1 so 161 is a term.

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)

Cf. A003415, A007850, A054377, A132632, A189639.

import Data.List (elemIndices)
a189710 n = a189710_list !! (n-1)
a189710_list = elemIndices 0 $
   zipWith (-) (map a003415 a003415_list) (map pred a003415_list)
-- Reinhard Zumkeller, May 09 2011

/* using Michael B. Porter's code from A003415: */
A003415(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))} /* arithmetic derivative */
for(n=1,10^6,d1=A003415(n);d2=A003415(d1);if(d2==d1+1,print1(n,", "))); /* show terms */
/* Joerg Arndt, Apr 25 2011 */

A188145 Solutions of the equation n" - n' - n = 0, where n' and n" are the first and second arithmetic derivatives (see A003415).

Original entry on oeis.org

0, 20, 135, 164, 1107, 15625, 43692, 128125, 188228, 294921, 1270539, 4117715, 33765263, 34134375, 147053125, 8995560189, 19348535652, 38753462951

Offset: 1

Paolo P. Lava, Mar 22 2011

Solutions of the similar equation n”-n’+n=0 are 30, 858, 1722, etc., apparently Giuga numbers whose derivative is a prime number. In fact the equation can be rewritten as n'=n+n" and if n"=1 it is the conjecture in A007850.
a(16) > 2*10^9. - Donovan Johnson, Apr 30 2011
a(19) > 10^11. - Giovanni Resta, Jun 04 2016

			n=20, n’=24, n”=44 -> 44-24-20=0;  n=135, n’=162, n”=297 -> 297-162-135=0

Cf. A003415, A007850, A165558, A165561, A165562, A189639, A189710.

import Data.List (zipWith3, elemIndices)
a188145 n = a188145_list !! (n-1)
a188145_list = elemIndices 0 $ zipWith3 (\x y z -> x - y - z)
   (map a003415 a003415_list) a003415_list [0..]
-- Reinhard Zumkeller, May 10 2011

readlib(ifactors):
Der:= proc(n)
local a,b,i,p,pfs;
for i from 0 to n do
  if i<=1 then a:=0;
  else pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs) ;
  fi;
  if a<=1 then b:=0;
  else pfs:=ifactors(a)[2]; b:=a*add(op(2,p)/op(1,p),p=pfs) ;
  fi;
  if b-a=i then lprint(i,a,b); fi;
od
end:
Der(10000000);

a(13)-a(15) from Donovan Johnson, Apr 30 2011

Corrected a(9) and a(16)-a(18) from Giovanni Resta, Jun 04 2016

A189803 Composite numbers n such that n'' = n'-1 where n' and n'' are the first and the second arithmetic derivative of n.

Original entry on oeis.org

9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909, 103073, 126509, 143009, 165773, 181793, 184973, 191309, 228653, 231713, 246893, 291233, 311309, 316973, 319793, 329357, 353009, 358433, 374513, 398093, 405809, 431009, 460193, 467309

Offset: 1

Giorgio Balzarotti, Apr 27 2011

nonn

The sequence A189710 (n"=n'-1) includes all prime numbers because p'=1 and p" = 0. Composite numbers are not very frequent.

Are all terms semiprimes? These terms appear to be p*q such that p+q is a term in A054377, which has solutions to the equation n' = n-1. - T. D. Noe, Apr 27 2011

			9' = 6, 9''= 6'= 5, 9" = 9'- 1 -> 9 is in the sequence.

Cf. A002808, A003415, A054377, A068346, A189710.

ader(n) = my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1])); \\ A003415
isok(k) = if (!isprime(k), my(d=ader(k)); ader(d) == d - 1); \\ Michel Marcus, Mar 13 2023

cp's OEIS Frontend

A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A189639 Numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

A188145 Solutions of the equation n" - n' - n = 0, where n' and n" are the first and second arithmetic derivatives (see A003415).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Maple

Extensions

A189803 Composite numbers n such that n'' = n'-1 where n' and n'' are the first and the second arithmetic derivative of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI