A066100 -id:A066100 - cp's OEIS frontend

A063783 Numbers k such that the sum of the cubes of divisors of k is a prime.

4, 9, 121, 36481, 72361, 146689, 259081, 654481, 683929, 786769, 1985281, 2036329, 3193369, 3636649, 3798601, 4583881, 5031049, 5470921, 5555449, 6135529, 6713281, 7284601, 7778521, 16589329, 20403289, 21557449, 22915369, 26739241, 27426169, 30261001, 30591961

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Solutions to sigma_3(x) = prime.

			All these terms are squares of primes {2, 3, 11, 191, 269, 383, 509, 809, 827, 887, 1409, 1427, 1787, 1907, 1949, 2141, 2243, 2339, 2357, 2477, 2591, 2699, 2789, ...}, so their sigma_3(p^2) = p^6 + p^3 + 1 has polynomial of degree 6.
sigma_3(9) = 1 + 27 + 729 = 757 is a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001158, A023194, A066100.

Select[Prime[Range[500]]^2, PrimeQ@ DivisorSigma[3, #] &] (* Michael De Vlieger, Jul 16 2017 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(p^6 + p^3 + 1), write("b063783.txt", n++, " ", p^2); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

a(n) = A066100(n)^2. - Amiram Eldar, Aug 16 2024

A063784 Primes that are the sum of cubes of divisors of some integer.

Original entry on oeis.org

73, 757, 1772893, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 7824668707707203971, 8443914727229480773, 32564717507686012813, 48095468363380957093, 54811417636756749151

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Primes of the form p^6 + p^3 + 1 where p is a prime. - Amiram Eldar, Aug 16 2024

			sigma_3(9) = 1 + 27 + 729 = 757, a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001158, A023194, A060883, A063783, A066100.

Select[Table[p^6 + p^3 + 1, {p, Prime[Range[500]]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(q=p^6 + p^3 + 1), write("b063784.txt", n++, " ", q); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

Primes of form p = sigma_3(k).
From Amiram Eldar, Aug 16 2024: (Start)
a(n) = A001158(A063783(n)).
a(n) = A060883(A066100(n)). (End)

A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

Original entry on oeis.org

2, 3, 2, 5, 3, 2, 17, 11, 11, 191, 41, 191, 263, 311, 4457, 59, 269, 557, 557, 5867, 3803, 71, 383, 761, 659, 7001, 13859, 1889, 89, 509, 797, 887, 7019, 22961, 16829, 17, 101, 809, 863, 1607, 7541, 31223, 62549, 69677, 113921, 131, 827, 977, 2309, 8609, 44351, 67103, 102647, 176459, 24071

Offset: 1

Martin Becker, May 19 2021

T(n,k)^(3^k), for all n >= 1, k >= 0, arranged by increasing values, is A342690. It is conjectured that all columns are infinite. If 3^k was replaced by k in the definition, all additional columns would be empty, as x^(2*k) + x^k + 1 is reducible if k has prime factors other than 3. For checking the property, Pocklington-Lehmer type primality tests seem particularly effective, as n-1 always has a large smooth factor p^(3^k), cf. the paper of Brillhart, Lehmer and Selfridge (1975), Theorem 5.

This array describes the essence of A342690 and A342691 in much more terse form. T(1, 8) = 113921 matches the 33177-digit value q = 113921^3^8 in A342690 and the 66353-digit prime q^2+q+1 in A342691.

			Array begins:
===============================================================
n\k |   0    1    2    3     4     5      6      7      8     9
----+----------------------------------------------------------
  1 |   2    2    2  191  4457  3803   1889     17 113921 24071
  2 |   3    3   11  311  5867 13859  16829  69677 176459 ...
  3 |   5   11  263  557  7001 22961  62549 102647 ...
  4 |  17  191  557  659  7019 31223  67103 164963 ...
  5 |  41  269  761  887  7541 44351 181931 170669 ...
  6 |  59  383  797 1607  8609 45737 188333 207923 ...
  7 |  71  509  863 2309  8627 61751 205433 235679 ...
  8 |  89  809  977 2621 21773 63377 210407 342833 ...
  9 | 101  827 1091 2687 22871 79481 219761 459209 ...

J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+-1, Math. Compl. 29 (1975) 620-647.

The first column T(n,0) is A053182(n). The second column T(n,1) is A066100(n).

Cf. A342690, A342691.

N=5; K=2; m=matrix(N, K+1); for(k=0, K, i=0; forprime(p=2, , q=p^3^k;if(isprime(q^2+q+1, 1), i+=1; m[i,k+1]=p; if(i==N, break)))); m

A195332 Numbers such that the sum of the cube of the odd divisors is prime.

Original entry on oeis.org

9, 18, 36, 72, 121, 144, 242, 288, 484, 576, 968, 1152, 1936, 2304, 3872, 4608, 7744, 9216, 15488, 18432, 30976, 36481, 36864, 61952, 72361, 72962, 73728, 123904, 144722, 145924, 146689, 147456, 247808, 259081, 289444, 291848, 293378, 294912

Offset: 1

Michel Lagneau, Sep 15 2011

nonn

a(n) is of the form m^2 or 2*m^2.
See the comments in A195268 (numbers such that the sum of the odd divisors is prime).
It is interesting to observe that the intersection of this sequence with A195268 gives {9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 146689, 147456, 293378, 294912,...} and contains the sequence A005010(n) (numbers of the form 9*2^n), but is not equal to this sequence. For example, up to n = 400000, the numbers 146689 and 293378 are not divisible by 9.

			The divisors of 18 are  { 1, 2, 3, 6, 9, 18}, and the sum of the cube of the odd divisors 1^3 + 3^3 + 9^3 =757 is prime. Hence 18 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005010, A066100 (sqrt of odd numbers here), A195268.

with(numtheory):for n from 1 to 400000 do:x:=divisors(n):n1:=nops(x):s:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s:=s+x[m]^3:fi:od:if type(s,prime)=true  then printf(`%d, `,n): else fi:od:

  Module[{c=Range[800]^2,m},m=Sort[Join[c,2c]];Select[m,PrimeQ[Total[ Select[ Divisors[#],OddQ]^3]]&]](* Harvey P. Dale, Jul 31 2012 *)

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A063783 Numbers k such that the sum of the cubes of divisors of k is a prime.

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A063784 Primes that are the sum of cubes of divisors of some integer.

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A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

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A195332 Numbers such that the sum of the cube of the odd divisors is prime.

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