A048161 -id:A048161 - cp's OEIS frontend

A118939 Primes p such that (p^2+3)/4 is prime.

3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 67, 83, 101, 109, 139, 151, 157, 179, 181, 199, 211, 223, 239, 263, 277, 283, 307, 311, 337, 347, 353, 379, 389, 419, 431, 463, 491, 557, 577, 587, 619, 659, 673, 739, 757, 797, 809, 811, 829, 853, 907, 911, 953, 991, 1051

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2+3. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118940, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+3)/4]&]

A341211 Smallest prime p such that (p^(2^n) + 1)/2 is prime.

Original entry on oeis.org

3, 3, 3, 13, 3, 3, 3, 113, 331, 3631, 827, 3109, 4253, 7487, 71

Offset: 0

Jon E. Schoenfield, Feb 06 2021

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2.
a(12) <= 4253, a(13) <= 7487, a(14) <= 71. - Daniel Suteu, Feb 07 2021
a(13) > 2500 and a(14) = 71. - Jinyuan Wang, Feb 07 2021

			No term is smaller than 3 (since 2 is the only smaller prime, and (2^(2^n) + 1)/2 is not an integer).
(3^(2^0) + 1)/2 = (3^1 + 1)/2 = (3 + 1)/2 = 4/2 = 2 is prime, so a(0)=3.
(3^(2^1) + 1)/2 = (3^2 + 1)/2 = 5 is prime, so a(1)=3.
(3^(2^2) + 1)/2 = (3^4 + 1)/2 = 41 is prime, so a(2)=3.
(3^(2^3) + 1)/2 = (3^8 + 1)/2 = 3281 = 17*193 is not prime, nor is (p^8 + 1)/2 for any other prime < 13, but (13^8 + 1)/2 = 407865361 is prime, so a(3)=13.

Dario Alpern, Integer factorization calculator

Cf. A005383, A048161, A176116, A340480.

Cf. A093625 and A171381 (both for when p=3).

x=3;x=N(x);NOT IsPrime((x^8192+1)/2);N(x)
# Martin Ehrenstein, Feb 08 2021

a(n) = my(p=3); while (!isprime((p^(2^n) + 1)/2), p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021

from sympy import isprime, nextprime
def a(n):
  p, pow2 = 3, 2**n
  while True:
    if isprime((p**pow2 + 1)//2): return p
    p = nextprime(p)
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021

a(11) from Daniel Suteu, Feb 07 2021
a(12) from Jinyuan Wang, Feb 07 2021
a(13)-a(14), using Dario Alpern's integer factorization calculator and prior bounds, from Martin Ehrenstein, Feb 08 2021

A341230 Primes p such that (p^128 + 1)/2 is prime.

Original entry on oeis.org

113, 499, 2081, 2287, 5807, 6151, 7823, 9203, 9629, 11069, 11497, 13463, 16987, 17891, 18049, 19889, 24091, 26981, 27259, 27953, 28319, 28597, 31219, 35899, 39047, 41381, 41603, 43403, 44839, 45343, 49529, 50753, 50857, 55079, 60793, 62219, 66721, 72679, 76771

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^7=128, respectively.

			(3^128 + 1)/2 = 5895092288869291585760436430706259332839105796137920554548481 = 257*275201*138424618868737*3913786281514524929*153849834853910661121, so 3 is not a term.
(113^128 + 1)/2 = 3111793506...0421698561 (a 263-digit number) is prime, so 113 is a term. Since 113 is the smallest prime p such that (p^128 + 1)/2 is prime, it is a(1) and is also A341211(7).

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), (this sequence) (k=7).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

isok(p) = (p>2) && isprime(p) && ispseudoprime((p^128 + 1)/2); \\ Michel Marcus, Feb 07 2021

A118941 Primes p such that (p^2-5)/4 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 53, 61, 71, 79, 83, 89, 97, 101, 107, 109, 113, 131, 137, 167, 173, 179, 193, 229, 241, 251, 263, 269, 277, 281, 283, 307, 311, 317, 349, 353, 373, 383, 419, 431, 439, 461, 463, 467, 563, 571, 577, 593, 607, 613, 619, 647

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2-5. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2-5)/4]&]

A118942 Primes p such that (p^2-13)/12 is prime.

Original entry on oeis.org

7, 13, 17, 19, 23, 31, 37, 41, 53, 67, 71, 73, 89, 103, 107, 113, 131, 139, 157, 163, 181, 199, 211, 233, 239, 257, 269, 283, 307, 311, 337, 359, 373, 379, 401, 419, 463, 487, 491, 499, 509, 521, 577, 593, 607, 617, 631, 647, 653, 683, 701, 733, 761, 769, 787

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 12 divides q^2-13. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118941.

Select[Prime[Range[200]],PrimeQ[(#^2-13)/12]&]

A154428 Primes of the form 50n^2 + 10n + 1.

Original entry on oeis.org

61, 1301, 1861, 2521, 5101, 7321, 8581, 9941, 14621, 16381, 20201, 24421, 26681, 34061, 36721, 51521, 68821, 76441, 97241, 101701, 106261, 110921, 135721, 163021, 168781, 199081, 205441, 218461, 252761, 282001, 304981, 312841, 337021, 353641

Offset: 1

Vincenzo Librandi, Jan 09 2009

Subsequence of A027862 associated with the values of A027861 that are multiples of 5. [R. J. Mathar, Jan 12 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A017281, A048161.

Filtered(List([1..100],n->50*n^2+10*n+1),IsPrime); # Muniru A Asiru, Apr 25 2019

[a: n in [0..100] | IsPrime(a) where a is 50*n^2 + 10*n + 1]; // Vincenzo Librandi, Jul 23 2012

select(isprime,[50*n^2+10*n+1$n=1..100])[]; # Muniru A Asiru, Apr 25 2019

Select[Table[50n^2+10n+1,{n,0,200}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

for (n=0, 100, if (isprime (k=50*n^2+10*n+1), print1 (k, ", "))); \\ Vincenzo Librandi, Jul 23 2012

Replaced 13721 by 135721 - R. J. Mathar, Jan 12 2009

A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 29, 79, 101, 349, 409, 449, 521, 569, 571, 661, 739, 991, 1091, 1129, 1181, 1459, 1489, 1531, 1901, 2269, 2281, 2341, 2351, 2389, 2549, 2659, 2671, 2719, 2729, 2731, 3109, 4049, 4349, 5279, 5431, 5471, 5531, 5591, 5669, 6329, 6359, 6871, 7559, 7741

Offset: 1

Giuseppe Coppoletta, Mar 19 2017

nonn

Primes which correspond to the short leg of an integral right triangle whose hypotenuse is part of a twin prime pair.

Each term p of the sequence must be part of a Pythagorean triple of the form {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponding to {a(n), A284035(n) - 1, A284035(n)}.

			The prime p = 79 is in the sequence because (p^2-3)/2 = 3119 and (p^2+1)/2 = 3121 are twin primes. Remark that {79, 3120, 3121} is a Pythagorean triple.

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

Cf. A284035, A048161, A165635, A284036, A051642.

Select[Prime@ Range[10^3], Function[p, Times @@ Boole@ Map[PrimeQ[(p^2 + #)/2 ] &, {-3, 1}] == 1]] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[1000]],AllTrue[{(#^2-3)/2,(#^2+1)/2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2017 *)

isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017

[p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]

A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.

Original entry on oeis.org

9, 21, 25, 33, 57, 85, 93, 121, 133, 145, 177, 205, 213, 217, 253, 361, 393, 445, 553, 565, 633, 697, 793, 817, 841, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653

Offset: 1

Hartmut F. W. Hoft, Jan 09 2021

nonn

For the squares p^2 in this sequence the area of the central region of the three regions in the symmetric representation of sigma(p^2) is equal to p.

p^2 is a term iff p is in A048161, and this subsequence of p^2 is A263951. - Bernard Schott, Jan 10 2021

			a(1) = 9 = 3*3 is the first number for which SRS(a(1)) consists of three regions ( 5, 3, 5 ).
a(6) = 85 = 5*17, both (1+85)/2 = 43 and (5+17)/2 = 11 are primes, and SRS(a(6)) consists of the 4 regions ( 43, 11, 11, 43 ).

Union of A128283 and A263951.
Subsequence of A046315 (all odd semiprimes).
Cf. A048161, A070552, A128281, A237591, A237593, A249223, A262045, A340482.

dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
a340482[n_] := Select[Range[n], PrimeOmega[#]==2&&dQ[#]&]
a340482[2700]

isok(m) = if ((m % 2) && (bigomega(m)==2), if (issquare(m), isprime((m+1)/2), my(p=factor(m)[1,1], q=factor(m)[2,1]); isprime((p*q+1)/2) && isprime((p+q)/2))); \\ Michel Marcus, Jan 10 2021

A341234 Primes p such that (p^256 + 1)/2 is prime.

Original entry on oeis.org

331, 1783, 2591, 2791, 7127, 8443, 9007, 9859, 10133, 10883, 10889, 11621, 12101, 13183, 15391, 17737, 19309, 19571, 21863, 24043, 24203, 31159, 32717, 33377, 34267, 35023, 35531, 38177, 39929, 42397, 43499, 46867, 49499, 49943, 50087, 51137, 53101, 53377

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, A341230, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^8=256, respectively.

			(3^256 + 1)/2 = 6950422618...4449717761 (a 122-digit number) = 12289 * 8972801 * 891206124520373602817 * (a 90-digit prime), so 3 is not a term.
(331^256 + 1)/2 = 5955749334...7416010241 (a 645-digit number) is prime, so 331 is a term. Since 331 is the smallest prime p such that (p^256 + 1)/2 is prime, it is a(1) and is also A341211(8).

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), A341230 (k=7), (this sequence) (k=8).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

Select[Range[20000], PrimeQ[#] && PrimeQ[(#^256 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

A356743 Numbers k such that k and k+2 both have exactly 6 divisors.

Original entry on oeis.org

18, 50, 242, 243, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3123, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591, 13023, 13075

Offset: 1

Jianing Song, Aug 25 2022

nonn

If an even number has exactly 6 divisors, then it is of the form 32, 4*p or 2*p^2 for an odd prime p. Note that 4*p + 2 = 2*q^2 is impossible since q^2 - 1 is divisible by 24 for prime q >= 5. As a result, if k is an even term, then it is of the form 2*p^2 such that (p^2+1)/2 is a prime (p is in A048161).

			50 is a term since 50 and 52 both have 6 divisors.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A048161.
Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), this sequence (m=6), A356744 (m=8).
Cf. also A049103 (numbers k such that k and k+1 both have exactly 6 divisors).

isA356743(n) = numdiv(n)==6 && numdiv(n+2)==6

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A118939 Primes p such that (p^2+3)/4 is prime.

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A341211 Smallest prime p such that (p^(2^n) + 1)/2 is prime.

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A341230 Primes p such that (p^128 + 1)/2 is prime.

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A118941 Primes p such that (p^2-5)/4 is prime.

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A118942 Primes p such that (p^2-13)/12 is prime.

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A154428 Primes of the form 50n^2 + 10n + 1.

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A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

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A340482 Numbers that are the product of two not necessarily distinct odd primes p*q with the property that (p*q+1)/2 and (p+q)/2 are primes.

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A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.