A178490 -id:A178490 - cp's OEIS frontend

A292447 Primes p such that sigma((p + 1) / 2) is a prime q.

3, 7, 17, 31, 127, 577, 3361, 4801, 6961, 8191, 31249, 131071, 171697, 524287, 982801, 1062881, 1104097, 1367857, 1407841, 1468897, 2705137, 3770257, 6822817, 7785457, 10941841, 14183137, 15557041, 18495361, 20749681, 25304497, 36278161, 38878561, 44575681

Offset: 1

Jaroslav Krizek, Sep 16 2017

A companion sequence of A249902.
Mersenne primes p = 2^k - 1 (A000668) are terms: sigma((p + 1) / 2) = sigma((2^k - 1 + 1) / 2) = sigma(2^(k - 1)) = 2^k - 1.
A subsequence of A178490. - Altug Alkan, Oct 02 2017

			17 is a term because sigma((17 + 1) / 2) = sigma(9) = 13 (prime).

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

Cf. A000203, A000668, A178490, A249902, A292448.

[n: n in [1..10^8] | IsPrime(n) and IsPrime(SumOfDivisors((n+1) div 2))];

Select[Prime@ Range[10^6], PrimeQ@ DivisorSigma[1, (# + 1)/2] &] (* Michael De Vlieger, Sep 16 2017 *)

lista(nn) = forprime(p=3, nn, if(isprime(sigma((p+1)/2)), print1(p, ", "))); \\ Altug Alkan, Oct 02 2017

a(n) = 2*A249902(n) - 1. - Altug Alkan, Oct 02 2017

A178491 Primes of the form 2*p^k-1, where p is prime and k > 1.

Original entry on oeis.org

7, 17, 31, 53, 97, 127, 241, 337, 577, 1249, 3361, 3697, 4373, 4801, 6961, 8191, 10657, 13121, 23761, 25537, 31249, 32257, 33613, 37537, 49297, 59581, 64081, 65521, 77617, 79201, 89041, 126001, 131071, 138337, 153457, 159013, 171697, 193441

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Oct 09 2010

nonn

Includes the Mersenne primes > 3 (A000668) and primes of the form 2p^2-1 (A092057) as subsequences. Its union with A005383 gives A178490.

			a(1) = 7 = 2*2^2-1 and a(2) = 17 = 2*3^2-1 are also in A092057, and a(3) = 31 = 2*2^4-1 = A000668(3), but a(4) = 53 = 2*3^3-1 is in neither of these subsequences.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000668, A005383, A092057, A178490.

N:= 10^6: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..floor(sqrt((N+1)/2)),2)]):
R:= NULL:
for p in P do
  for k from 2 do
    v:= 2*p^k-1;
    if v > N then break fi;
    if isprime(v) then R:= R,v fi;
od od:
sort([R]); # Robert Israel, Feb 20 2024

Select[Prime[Range[20000]],!PrimeQ[(#+1)/2]&&Length[FactorInteger[(#+1)/2]]==1&]

is_A178491(n) = isprime(n) & ispower((n+1)/2,,&n) & isprime(n)

A370452 Prime powers of the form 2*p^k-1, where p is prime and k >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 17, 25, 31, 37, 49, 53, 61, 73, 81, 97, 121, 127, 157, 193, 241, 277, 313, 337, 361, 397, 421, 457, 541, 577, 613, 625, 661, 673, 733, 757, 841, 877, 997, 1093, 1153, 1201, 1213, 1237, 1249, 1321, 1381, 1453, 1621, 1657, 1681, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2401, 2473, 2557, 2593, 2797, 2857

Offset: 1

Keith J. Bauer, Feb 18 2024

Also, sizes of finite fields such that halving the size of their unit group is also the unit group of a field. - Keith J. Bauer, Jun 20 2024

Original motivation for this sequence: let k be a term of this sequence. Then consider the finite field of k elements, denoted by F_k. Adjoin the hyperbolic unit j^2 = 1 to F_k to form a ring whose elements are of the form a + bj for a, b in F_k. Let M be the multiplication monoid of F_k[j] and let ~ be the equivalence relation on the elements of M defined by a + bj ~ b + aj (with no further unnecessary equivalences). Then M/~ is isomorphic to the multiplication monoid of the ring F_k x F_(k+1)/2 and therefore there exists a ring with M/~ as its multiplication. For prime powers k not in this sequence, no such ring will exist. See the link for a proof of this fact.

			3 = 2*2^1 - 1 = 3^1;
5 = 2*3^1 - 1 = 5^1;
7 = 2*2^2 - 1 = 7^1;
9 = 2*5^1 - 1 = 3^2.

Keith J. Bauer, The monoid (F_n[j], *)/(a + bj ~ b + aj) is isomorphic to (F_n X F_(n+1)/2, *), when such fields exist.

Cf. A178490, A246655 (prime powers).

filter:= n -> nops(numtheory:-factorset(n))=1 and nops(numtheory:-factorset((n+1)/2))=1:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Feb 20 2024

Select[Range[3000], PrimePowerQ[#] && PrimePowerQ[(# + 1)/2] &] (* Amiram Eldar, Feb 19 2024 *)

isok(q) = isprimepower(q) && (q%2) && isprimepower((q+1)/2); \\ Michel Marcus, Jun 14 2024

A380101 Numbers k such that omega(k-th triangular number) = 2, where omega = A001221.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 17, 18, 22, 25, 26, 31, 37, 46, 49, 53, 58, 61, 73, 81, 82, 97, 106, 121, 127, 157, 162, 166, 178, 193, 226, 241, 242, 250, 256, 262, 277, 313, 337, 346, 358, 361, 382, 397, 421, 457, 466, 478, 486, 502, 541, 562, 577, 586, 613

Offset: 1

Juri-Stepan Gerasimov, Jan 12 2025

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A077065 and of A178490.

Cf. A000217, A001221, A119663.

[k: k in [1..400] | #PrimeDivisors(k*(k+1) div 2) eq 2];

filter:= proc(n) local W1, n1, W2; uses numtheory;
     if n::odd then nops(factorset(n)) = 1 and nops(factorset((n+1)/2)) = 1
     else nops(factorset(n/2)) = 1 and nops(factorset(n+1)) = 1
     fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 12 2025

Select[Range[600], PrimeNu[#*(#+1)/2] == 2 &] (* Amiram Eldar, Jan 12 2025 *)

isok(k) = omega(k*(k+1)/2) == 2; \\ Michel Marcus, Jan 14 2025

cp's OEIS Frontend

A292447 Primes p such that sigma((p + 1) / 2) is a prime q.

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A178491 Primes of the form 2*p^k-1, where p is prime and k > 1.

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A370452 Prime powers of the form 2*p^k-1, where p is prime and k >= 1.

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Maple

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PARI

A380101 Numbers k such that omega(k-th triangular number) = 2, where omega = A001221.

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Magma

Maple

Mathematica

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