A240693 -id:A240693 - cp's OEIS frontend

A259417 Even powers of the odd primes listed in increasing order.

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649

Offset: 1

Hartmut F. W. Hoft, Jun 26 2015

nonn

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:
A001248(p) = p^2,  A030514(p) = p^4,  A030516(p) = p^6,
A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,
A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,
A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,
A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,
A183062(p) = p^58, A183085(p) = p^60.
See also the link to the OEIS Wiki.
The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.
The odd numbers in A023194 are a subsequence of this sequence.

			a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473
OEIS Wiki, Index entries for number of divisors.

a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000},Select[Union[Flatten[Table[Prime[Range[2,Floor[ Sqrt[ upto]]]]^n,{n,0,Log[2,upto],2}]]],#<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

Original entry on oeis.org

12207031, 2141993519227, 178250690949465223, 2346320474383711003267, 398341412240537151131351, 79545183674814239059370551, 494424256962371823779424877, 8271964541879648991904246901, 32142180034067960734115528951, 91264002187709396686868598317

Offset: 1

Hartmut F. W. Hoft, May 05 2017

nonn

			Prime number 12207031 = Sum_{i=0..10} 5^i is the first in the sequence since 23 divides 88573 = Sum_{i=0..10} 3^i as well as 2047 = Sum_{i=0..10} 2^i.

Subsequence of A060885, A162861 and A193574.

Cf. A162862, A198244, A240693.

a286301[n_] := Select[Map[(Prime[#]^11-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a286301[150] (* data *)

A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 117, 120, 132, 140, 182, 306, 380, 496, 552, 672, 775, 870, 992, 1080, 1287, 1406, 1428, 1680, 1722, 1892, 2016, 2184, 2256, 2480, 2793, 2862, 3276, 3540, 3640, 3782, 3960, 4060, 4556, 4560, 4650, 5112, 5382, 5402, 5460, 6120, 6320, 6552

Offset: 1

Marius A. Burtea, Dec 06 2020

nonn

All terms are nonprimes.
The sequence includes all numbers of the form p*(p + 1) with p prime. Indeed: sigma(p*(p + 1)) = sigma(p)*sigma(p + 1) = (p + 1)*sigma(p + 1). So A036690 is a subsequence. Thus, the sequence is infinite.
Let k >= 1. If p and q = 1 + p + ... + p^(2*k) are prime numbers, then m = p^(2*k)*q is a term. Indeed, sigma(m) = sigma(p^(2*k)*q) = sigma(p^(2*k))*sigma(q) = q*sigma(q).
p is in: A053182 (k = 1), A065509 (k = 2), A163268 (k = 3), and A240693 (k = 5).
For k = 4 there are no prime p because 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 + p^7 + p^8 = (p^6 + p^3 + 1)*(p^2 + p + 1).
If m = 2^(p - 1)*(2^p - 1), p >= 1, (see A006516), then sigma(m) = sigma(2^(p - 1)*(2^p - 1)) = sigma(2^(p - 1))*sigma(2^p - 1) = (2^p - 1)*sigma(2^p - 1), so m is a term.
Thus, A006516(n) and A000396(n), for n >= 1, are terms.

			sigma(6) = 12 = 3*4 = 3*sigma(3), so 6 is a term.
sigma(12) = 28 = 4*7 = 4*sigma(4), so 12 is a term.
sigma(30) = 72 = 6*12 = 6*sigma(6), so 30 is a term.
sigma(56) = 120 = 8*15 = 8*sigma(8), so 56 is a term.
sigma(117) = 182 = 13*14 = 13*sigma(13), so 117 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000203, A000396, A006516, A036690, A053182, A064987, A065509, A163268, A327165, A327599.

Magma
```
s:=func; [n:n in [1..6600]|s(n)];
```

q[n_] := Module[{d = Divisors[n], s}, s = Plus @@ d; AnyTrue[d, #*DivisorSigma[1, #] == s &]]; Select[Range[7000], q] (* Amiram Eldar, Dec 06 2020 *)

isok(k) = my(sk=sigma(k)); fordiv(k, d, if (d*sigma(d) == sk, return(1))); \\ Michel Marcus, Dec 06 2020

A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 20, 21, 30, 60, 86, 172, 195, 212, 224, 258, 268, 272, 319, 339, 355, 365, 366, 390, 398, 414, 480, 504, 534, 539, 543, 567, 592, 626, 654, 735, 756, 766, 770, 778, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1194, 1210, 1239, 1243, 1264, 1309, 1311

Offset: 1

Bernard Schott, May 16 2019

nonn

A240693 Union {this sequence} = A162862.

The corresponding prime numbers, (11111111111)_k, are Brazilian primes and belong to A085104 and A285017 (except 11).

			(11111111111)_20 = (20^11 - 1)/19 = 10778947368421 is prime, thus 20 is a term.

Cf. A162861, A162862, A240693, A286301.
Intersection of A064108 and A285017.
Similar to A182253 for k^2+k+1, A286094 for k^4+k^3+k^2+k+1, A288939 for k^6+k^5+k^4+k^3+k^2+k+1.

[1] cat [n:n in [2..1500]|not IsPrime(n) and IsPrime(Floor((n^11-1)/(n-1)))]; // Marius A. Burtea, May 16 2019

filter:= n -> not isprime(n) and isprime((n^11-1)/(n-1)) : select(filter, [$2..5000]);

Select[Range@ 1320, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 10]]] &] (* Michael De Vlieger, Jun 09 2019 *)

isok(n) = !isprime(n) && isprime(polcyclo(11, n)); \\ Michel Marcus, May 19 2019

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A259417 Even powers of the odd primes listed in increasing order.

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A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

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A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

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A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

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