A165562 -id:A165562 - cp's OEIS frontend

A165561 Primes that are the sum of an integer n and its arithmetic derivative.

3, 11, 17, 23, 31, 41, 47, 53, 59, 61, 71, 79, 83, 89, 107, 113, 127, 131, 149, 151, 167, 179, 191, 193, 197, 227, 239, 251, 263, 269, 271, 293, 311, 313, 347, 359, 383, 401, 419, 431, 439, 443, 449, 457, 479, 491, 503, 521, 523, 587, 593, 599, 607, 617, 631

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Sep 25 2009

Some primes are the sum of an integer and its derivative in more than one way (e.g., 23, 71, 191 (not a complete listing within the range shown)). Just calculating this sequence from A165562 gives a list that is not sorted in ascending order and contains duplicate items. However, since in the range from 1 to 10000 only the number 1 and the primes have arithmetic derivatives that are less than their square roots, I feel confident that the list given above is not missing some term that corresponds to a large value in A165562. In other words, for a term to be missing from the list above, its corresponding value in A165562 would have to be less than 625. - Alonso del Arte, Oct 30 2009

			71 is in the list because: n=46 -> n'=25 -> n+n'=71; n=51 -> n'=20 -> n+n'=71; n=55 -> n'=16 -> n+n'=71.

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

Cf. A003415, A165562

isA165561 := proc(n)
    if isprime(n) then
        for i from 1 to n do
            if n = A129283(i) then
                return true ;
            end if;
        end do:
        false ;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA165561(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 04 2022

(*First run the program given in A165562*) SetAttributes[a, Listable]; A165561 = Union[A165562 + a[A165562]]

{p in A000040: p in A129283}. - R. J. Mathar, Feb 04 2022

Terms verified by Alonso del Arte, Oct 30 2009

A229269 Numbers k for which k - k' is prime, k' being the arithmetic derivative of k.

Original entry on oeis.org

3, 9, 10, 14, 15, 21, 26, 33, 35, 38, 39, 50, 51, 62, 65, 66, 69, 70, 77, 78, 86, 91, 93, 95, 102, 110, 111, 114, 122, 123, 129, 130, 133, 138, 146, 154, 159, 161, 170, 174, 190, 201, 203, 206, 209, 213, 215, 217, 218, 221, 222, 230, 238, 249, 258, 278, 282, 287

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			15 is in the list because 15’ = 8 and 15 - 8 = 7 that is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A003415, A165561, A165562, A229270-A229272.

with(numtheory); P:=proc(q) local a,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); if isprime(n-a) then print(n); fi; od; end: P(10^5);

from sympy import isprime, factorint
A229269 = [n for n in range(1,10**4) if isprime(n-sum([int(n*e/p) for p,e in factorint(n).items()]))] # Chai Wah Wu, Aug 21 2014

A229270 Numbers k for which k' - k is prime, k' being the arithmetic derivative of k.

Original entry on oeis.org

18, 210, 315, 330, 390, 462, 510, 546, 690, 726, 798, 870, 930, 966, 1110, 1218, 1230, 1290, 1302, 1554, 1590, 1770, 2010, 2130, 2190, 2310, 2370, 2490, 2730, 2910, 3030, 3210, 3270, 3570, 3810, 4110, 4290, 4470, 4530, 4830, 4890, 5010, 5430, 5790, 5910, 5970

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			315 is in the list because 315’ = 318 and 318 - 315 = 3 that is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A003415, A165561, A165562, A229269, A229271, A229272.

with(numtheory); P:=proc(q) local a,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); if isprime(a-n) then print(n); fi; od; end: P(10^5);

from sympy import isprime, factorint
A229270 = [n for n in range(1,10**5) if isprime(sum([int(n*e/p) for p,e in factorint(n).items()])-n)] # Chai Wah Wu, Aug 21 2014

A229272 Numbers n for which n' + n and n' - n are both prime, n' being the arithmetic derivative of n.

Original entry on oeis.org

210, 330, 390, 690, 798, 966, 1110, 1230, 2190, 2310, 2730, 3270, 4110, 4530, 4890, 5430, 6090, 6270, 6810, 6990, 7230, 7890, 8310, 8490, 9030, 9210, 9282, 10470, 10590, 10770, 12090, 12210, 12270, 12570, 12810, 12930, 13110, 13830, 14070, 17070, 17094, 17310

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

Intersection of A165561 and A229270.

Paolo P. Lava, Table of n, a(n) for n = 1..300

Cf. A003415, A165561, A165562, A229269-A229271.

with(numtheory); P:=proc(q) local a,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if isprime(a+n) and isprime(a-n) then print(n); fi;
od; end: P(10^5);

from sympy import isprime, factorint
A229272 = []
for n in range(1, 10**5):
    np = sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
    if isprime(np+n) and isprime(np-n):
        A229272.append(n)
# Chai Wah Wu, Aug 21 2014

A229271 Numbers k for which k + k' and k - k' are both prime, k' being the arithmetic derivative of k.

Original entry on oeis.org

10, 14, 15, 21, 26, 33, 35, 38, 51, 65, 66, 78, 86, 93, 102, 110, 111, 123, 161, 201, 203, 206, 209, 215, 221, 230, 258, 278, 282, 321, 371, 374, 395, 398, 402, 413, 438, 470, 471, 485, 530, 533, 543, 545, 551, 590, 626, 671, 678, 698, 723, 755, 779, 803, 815

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

Intersection of A165561 and A229269.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A003415, A165561, A165562, A229269, A229270, A229272.

with(numtheory); P:=proc(q) local a,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if isprime(n+a) and isprime(n-a) then print(n); fi;
od; end: P(10^5);

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

A210935 Numbers n for which n*n'/(n+n') is an integer, where n' is the arithmetic derivative of n.

Original entry on oeis.org

1, 4, 64, 80, 108, 270, 351, 432, 729, 768, 864, 2916, 5184, 5832, 6250, 6912, 12096, 13500, 16384, 25600, 32832, 34992, 37500, 39366, 43200, 46656, 50000, 73008, 74304, 81648, 84375, 110592, 131250, 138240, 143748, 153664, 172800, 176418, 200000, 225000

Offset: 1

Paolo P. Lava, May 11 2012

nonn

Only eleven odd numbers in the first 150 terms: a(1)=1, a(7)=351, a(9)=729, a(31)=84375, a(76)=2470629, a(78)=2709375, a(87)=4159375, a(89)=4348377, a(115)=13286025, a(126)=22235661, a(128)=25059375.

			n=729; n'=1458; n*n'/(n+n')=486.

Paolo P. Lava and Donovan Johnson, Table of n, a(n) for n = 1..500 (first 150 terms from Paolo P. Lava)

Cf. A003415, A165562.

with(numtheory);
A210935:= proc(n)
local a,i,p,pfs;
for i from 1 to n do
    pfs:=ifactors(i)[2];  a:=i*add(op(2,p)/op(1,p),p=pfs) ;
    if a*i/(a+i)=trunc(a*i/(a+i)) then print(i); fi;
od; end:
A210935(100000000);

cp's OEIS Frontend

A165561 Primes that are the sum of an integer n and its arithmetic derivative.

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A229269 Numbers k for which k - k' is prime, k' being the arithmetic derivative of k.

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A229270 Numbers k for which k' - k is prime, k' being the arithmetic derivative of k.

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A229272 Numbers n for which n' + n and n' - n are both prime, n' being the arithmetic derivative of n.

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A229271 Numbers k for which k + k' and k - k' are both prime, k' being the arithmetic derivative of k.

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

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A210935 Numbers n for which n*n'/(n+n') is an integer, where n' is the arithmetic derivative of n.

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Maple