A104589 -id:A104589 - cp's OEIS frontend

A355958 Positions where primes occur in A104589.

2, 3, 4, 8, 12, 18, 19, 35, 39, 47, 53, 65, 75, 81, 84, 102, 109, 127, 131, 261, 265, 273, 283, 305, 323, 329, 350, 450, 456, 462, 492, 522, 562, 660, 664, 784, 904, 922, 1081, 1213, 1258, 1406, 1443, 1467, 1549, 1655, 1772, 1774, 1786, 1810, 1901, 1919, 2024, 2144, 2161

Offset: 1

Michel Marcus, Jul 21 2022

nonn

			The 1st prime in A104589 is 2, and its position is 2, so a(1) = 2.
The 2nd prime in A104589 is 5, and its position is 3, so a(2) = 3.
The 3rd prime in A104589 is 13, and its position is 4, so a(3) = 4.

Cf. A104589, A355967.

s[1] = 1; s[n_] := s[n] = s[n - 1] + Sum[If[CompositeQ[s[k]], 0, s[k]], {k, 1, n - 1}]; Select[Range[2000], PrimeQ[s[#]] &] (* Amiram Eldar, Jul 21 2022 *)

lista(nn) = my(last=1, s = 1, list = List()); for (n=2, nn, last += s; if (isprime(last), s += last; listput(list, n));); Vec(list);

A355967 Indices of the primes that occur in A104589.

Original entry on oeis.org

1, 3, 6, 25, 104, 634, 1236, 22613, 103409, 929044, 6298419, 80396036, 843325558, 5843115733, 24428345613, 515211289906, 4021634909249, 77930896716918, 387592118891917, 53467625139656294, 258820291307490689, 2600667638804262010, 29899374277934530878

Offset: 1

Michel Marcus, Jul 21 2022

nonn

			The primes in A104589 are 2, 5, 13, 97, ... with prime indices 1, 3, 6, 25, ...

Cf. A000040, A000720, A104589, A355958.

s[1] = 1; s[n_] := s[n] = s[n - 1] + Sum[If[CompositeQ[s[k]], 0, s[k]], {k, 1, n - 1}]; PrimePi[Select[s /@ Range[200], PrimeQ[#] &]] (* Amiram Eldar, Jul 21 2022 *)

lista(nn) = my(last=1, s = 1, list = List()); for (n=2, nn, last += s; if (isprime(last), s += last; listput(list, primepi(last)));); Vec(list);

from sympy import isprime, primepi
from itertools import islice
def A355967_gen(): # generator of terms
    a, b = 1, 1
    while True:
        a += b
        if isprime(a):
            b += a
            yield primepi(a)
A355967_list = list(islice(A355967_gen(),14)) # Chai Wah Wu, Jun 03 2024

a(14)-a(20) from Amiram Eldar, Jul 21 2022

a(21)-a(23) from Chai Wah Wu, Jun 03 2024

A355697 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + g - 1 if a(n-1) is prime, otherwise a(n) = a(n-1) + g + 1, where g = a(n-1) - a(n-2).

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 13, 16, 20, 25, 31, 36, 42, 49, 57, 66, 76, 87, 99, 112, 126, 141, 157, 172, 188, 205, 223, 240, 258, 277, 295, 314, 334, 355, 377, 400, 424, 449, 473, 498, 524, 551, 579, 608, 638, 669, 701, 732, 764, 797, 829, 860, 892, 925, 959, 994, 1030, 1067, 1105, 1144, 1184

Offset: 0

John Tyler Rascoe, Jul 19 2022

			0                              =  0
0 + 1                          =  1
0 + 1 + 2                      =  3 (prime)
0 + 1 + 2 + 1                  =  4
0 + 1 + 2 + 1 + 2              =  6
0 + 1 + 2 + 1 + 2 + 3          =  9
0 + 1 + 2 + 1 + 2 + 3 + 4      = 13 (prime)
0 + 1 + 2 + 1 + 2 + 3 + 4 + 3  = 16

John Tyler Rascoe, Illustration of first 15 terms

Cf. A000217, A104589, A116533, A332410, A356445.

function a = A355697(max_n)
    a = 0; m = 0;
    for n = 1:max_n
        if isprime(a(n))
            m = m - 1;
        else
            m = m + 1;
        end
        a(n+1) = a(n) + m;
    end
end % Thomas Scheuerle, Jul 22 2022

a[0] = 0; a[1] = 1; a[n_] := a[n] = 2*a[n - 1] - a[n - 2] - If[PrimeQ[a[n - 1]], 1, -1]; Array[a, 50, 0] (* Amiram Eldar, Jul 23 2022 *)

lista(nn) = my(va = vector(nn)); va[1] = 0; va[2] = 1; for (n=3, nn, if (isprime(va[n-1]), va[n] = 2*va[n-1]-va[n-2]-1, va[n] = 2*va[n-1]-va[n-2]+1);); va; \\ Michel Marcus, Aug 23 2022

import sympy
A355697 = A = [0,1]
for n in range(1,100):
      if sympy.isprime(A355697[-1]):
            y = A[-1] - A[-2] - 1
      else:
            y = A[-1] - A[-2] + 1
      A.append(A[-1] + y)

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A355958 Positions where primes occur in A104589.

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A355967 Indices of the primes that occur in A104589.

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A355697 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + g - 1 if a(n-1) is prime, otherwise a(n) = a(n-1) + g + 1, where g = a(n-1) - a(n-2).

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