A192189 -id:A192189 - cp's OEIS frontend

A192190 Degree of A192189(n) (polynomial-like numbers) in term of number derivatives: number of A003415-iterations up to reaching 0.

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

Offset: 1

Vladimir Shevelev, Jun 25 2011

nonn

a(n) = 1 iff A192189(n) is prime.

Cf. A003415, A038554, A192082, A192189.

A157037 Numbers with prime arithmetic derivative A003415.

Original entry on oeis.org

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

A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

Original entry on oeis.org

9, 21, 25, 57, 85, 93, 121, 126, 145, 161, 185, 201, 206, 209, 221, 237, 242, 253, 265, 289, 305, 315, 326, 333, 341, 365, 369, 377, 381, 413, 417, 437, 453, 458, 490, 495, 497, 517, 537, 542, 545, 565, 566, 575, 578, 597, 605, 633, 637, 638, 649, 666, 685

Offset: 1

Vladimir Shevelev, Jun 25 2011

nonn

The fourth A003415-iteration of a(n) is the first to be 0.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (the first 1000 terms from T. D. Noe)

Cf. A003415, A038554, A068346, A192082, A192189, A192190, A327969, A328240.
Cf. A157037, A328239 (the first and third derivative is prime).
Subsequence of following sequences: A328234, A328244, A328246.

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[1000], dn[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]));
isA192192(n) = isprime(A003415(A003415(n))); \\ Antti Karttunen, Oct 19 2019

For all n, A327969(a(n)) <= 4. - Antti Karttunen, Oct 19 2019

More terms from Olivier Gérard, Jul 04 2011

New primary definition added to the name by Antti Karttunen, Oct 19 2019

A193262 Number of representations of 2p_n as sum of two primes p,q such that pq-2 is prime (p_n is the n-th prime).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 0, 3, 0, 2, 3, 4, 2, 1, 3, 4, 2, 0, 4, 2, 5, 2, 2, 5, 2, 2, 5, 2, 4, 1, 0, 1, 2, 0, 8, 3, 0, 2, 2, 5, 3, 0, 1, 5, 7, 1, 3, 1, 2, 4, 5, 5, 1, 0, 3, 2, 4, 3, 4, 2, 3, 3, 1, 3, 2, 0, 8, 3, 4, 3, 0, 9, 1, 6, 0, 2, 5, 2, 2, 9, 1, 5, 4, 3, 1, 7, 5, 2, 4, 2, 1

Offset: 1

Vladimir Shevelev, Aug 04 2011

Sequence arising in connection with conjecture in comment to A192189.

Conjecture: There exists n_0, such that, for n>n_0, a(n)>0.

			a(4)=2 since 2*p(4) = 14 = 3+11 = 7+7, and 3*11-2 = 31, 7*7-2 = 47 are prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A045917.

a:= proc(n) local t, s, p, q;
      t:= 2*ithprime(n);
      s:= 0;
      p:= 2;
      do q:= t-p;
         if qAlois P. Heinz, Aug 04 2011

a[n_] := Module[{t = 2 Prime[n], s = 0, p = 2, q}, While[True, q = t - p; If[q < p, Break[]]; If[PrimeQ[q] && PrimeQ[p q - 2], s++]; p = NextPrime[p]]; s];
Array[a, 100] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

A193262(n,c=0)={ n=2*prime(n); forprime(p=1,n/2,isprime(n-p) || next; isprime(p*(n-p)-2) & c++);c}  \\ M. F. Hasler, Aug 06 2011

A192931 Numbers n for which A193262(n)=0.

Original entry on oeis.org

11, 13, 22, 35, 38, 41, 46, 58, 70, 75, 79, 98, 109, 125, 158, 171, 193, 237, 257, 286, 305, 462, 738

Offset: 1

Vladimir Shevelev, Aug 05 2011

There are no more terms <= 10000.
Conjecture: The sequence is finite.
There are no more terms <= 100000. - Michel Marcus, Dec 18 2018

Cf. A192189, A193262.

cp's OEIS Frontend

A192190 Degree of A192189(n) (polynomial-like numbers) in term of number derivatives: number of A003415-iterations up to reaching 0.

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A157037 Numbers with prime arithmetic derivative A003415.

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A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

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A193262 Number of representations of 2*p_n as sum of two primes p,q such that p*q-2 is prime (p_n is the n-th prime).

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A192931 Numbers n for which A193262(n)=0.

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A193262 Number of representations of 2p_n as sum of two primes p,q such that pq-2 is prime (p_n is the n-th prime).