A083186 -id:A083186 - cp's OEIS frontend

A109790 Numbers k such that A083186(k) is prime.

1, 3, 5, 7, 9, 15, 19, 33, 47, 49, 51, 55, 63, 73, 97, 105, 117, 119, 131, 165, 189, 195, 223, 229, 243, 245, 253, 257, 263, 273, 277, 291, 295, 297, 329, 331, 357, 367, 371, 389, 391, 395, 397, 399, 445, 487, 497, 551, 577, 603, 605, 637, 641, 643, 683, 685

Offset: 1

N. J. A. Sloane, Jun 11 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A006450, A083186, A126144.

Select[Range[2000], PrimeQ[Sum[Prime[Prime[i]], {i, 1, #}]] &] (* Stefan Steinerberger, Jun 23 2007 *)
Flatten[Position[Accumulate[Prime[Prime[Range[800]]]],?PrimeQ]] (* _Harvey P. Dale, Jun 24 2012 *)

More terms from Stefan Steinerberger, Jun 23 2007

A263559 a(n) = A083186(n) mod A007504(n).

Original entry on oeis.org

1, 3, 9, 2, 11, 26, 51, 3, 17, 39, 73, 119, 175, 237, 307, 8, 49, 88, 151, 220, 295, 380, 479, 584, 705, 848, 999, 1158, 1321, 1486, 1687, 51, 139, 241, 355, 477, 611, 763, 919, 1085, 1253, 1435, 1633, 1839, 2055, 2277, 2519, 2813, 3111, 3413, 3719, 4023, 4341, 4683, 5019

Offset: 1

Altug Alkan, Oct 21 2015

nonn

Sequence is interesting because of its graph. a(n)-a(n-1) < 0 at some points such as n=4 and n=8, although usually a(n)-a(n-1) > 0.

			a(1) = 1 because prime(prime(1)) mod prime(1) = 3 mod 2 = 1.

Cf. A007504, A083186, A262744.

Table[Mod[Sum[Prime@ Prime@ k, {k, n}], Sum[Prime@ k, {k, n}]], {n, 55}] (* Michael De Vlieger, Oct 21 2015 *)

vector(100, n, sum(k=1, n, prime(prime(k))) % sum(k=1, n, prime(k)))

A086749 Partial sums of A038580.

Original entry on oeis.org

5, 16, 47, 106, 233, 412, 689, 1020, 1451, 2050, 2759, 3678, 4741, 5894, 7191, 8714, 10501, 12348, 14569, 16950, 19427, 22176, 25177, 28436, 32073, 36016, 40107, 44380, 48777, 53326, 58707, 64330, 70199, 76312, 82973, 89796, 96989, 104596

Offset: 1

Cino Hilliard, Jul 30 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A083186.

Accumulate[Nest[Prime,Range[45],3]] (* Harvey P. Dale, Oct 25 2011 *)

f(n) = y=0; for(x=1,n,y+=prime(prime(prime(x))); print1(y","))

Edited by N. J. A. Sloane, Apr 17 2007

A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

Original entry on oeis.org

0, 3, 5, 9, 11, 8, 17, 27, 25, 14, 31, 14, 41, 20, 16, 81, 59, 28, 67, 20, 22, 34, 83, 32, 121, 44, 125, 26, 109, 19, 127, 243, 36, 62, 28, 34, 157, 70, 46, 38, 179, 25, 191, 40, 36, 86, 211, 86, 289, 124, 64, 50, 241, 128, 42, 44, 72, 112, 277, 25, 283, 130, 42, 729, 52, 39, 331, 68, 88, 31

Offset: 1

Ilya Gutkovskiy, May 09 2018

nonn

			a(12) = 14 because 12 = 2^2*3 and prime(2)^2 + prime(3) = 3^2 + 5 = 14.

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A006450, A008475, A038580, A064988, A083186, A222416, A304037, A304253 (fixed points).

f:= proc(n) local t;
   add(ithprime(t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2024

a[n_] := Plus @@ (Prime[#[[1]]]^#[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 70}]

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,1])^f[k,2]); \\ Michel Marcus, May 09 2018

a(prime(i)^k) = prime(prime(i))^k.
a(A000040(k)) = A006450(k).
a(A006450(k)) = A038580(k).
a(A002110(k)) = A083186(k).

A126144 Primes in A109790.

Original entry on oeis.org

3, 5, 7, 19, 47, 73, 97, 131, 223, 229, 257, 263, 277, 331, 367, 389, 397, 487, 577, 641, 643, 683, 701, 739, 743, 757, 797, 857, 877, 1031, 1291, 1297, 1423, 1451, 1543, 1579, 1637, 1697, 1723, 1823, 1949, 2039, 2081, 2381, 2477, 2539, 2617, 2659, 2689, 2749

Offset: 1

J. M. Bergot, Jun 11 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A083186, A109790.

Select[Select[Range[3000], PrimeQ[Sum[Prime[Prime[i]],{i,1,#}]]&],PrimeQ[ # ] &] (* Stefan Steinerberger, Jun 23 2007 *)

More terms from Stefan Steinerberger, Jun 23 2007

A263541 Numbers k such that k divides the sum of the first k primes with prime indices.

Original entry on oeis.org

1, 2, 4, 6, 40, 43, 705, 789, 1148, 2140, 4276, 5512, 6672, 8754, 38434, 174501, 493578, 598249, 628064, 702774, 1368196, 4584004, 13813057, 36425906, 87964443, 447997476, 1964288296

Offset: 1

Altug Alkan, Oct 20 2015

nonn

There are 8 values of a(n) < 1000 although A045345 has 4 values A045345(n) < 1000. How do these sequences compare asymptotically?

Heuristics suggest that the ratio of the number of terms in each sequence up to x should approach 1 as x increases without bound. In the Cramér model, log a(n) and log A045345(n) are Erlang-distributed with shape n and rate 1. - Charles R Greathouse IV, Oct 20 2015

			1 is in the sequence because prime(prime(1)) = 3 is divisible by 1.
2 is in the sequence because prime(prime(1)) + prime(prime(2)) = 3 + 5 = 8 is divisible by 2.

Cf. A006450, A045345, A083186.

list(lim)=my(v=List(),k,s,t); forprime(p=2,, if(isprime(t++), s+=p; k++; if(s%k==0, listput(v, k)); if(k>=lim, return(Vec(v))))) \\ Charles R Greathouse IV, Oct 20 2015

a(12)-a(26) from Charles R Greathouse IV, Oct 20 2015

a(27) from Charles R Greathouse IV, Oct 21 2015

cp's OEIS Frontend

A109790 Numbers k such that A083186(k) is prime.

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A263559 a(n) = A083186(n) mod A007504(n).

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A086749 Partial sums of A038580.

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A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

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A126144 Primes in A109790.

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A263541 Numbers k such that k divides the sum of the first k primes with prime indices.

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