A054723 -id:A054723 - cp's OEIS frontend

A243890 Primes of the form 2n^2+38n+17.

101, 149, 257, 317, 449, 521, 677, 761, 941, 1697, 1949, 2081, 2357, 2801, 2957, 3449, 3797, 4349, 4937, 6221, 6449, 6917, 7649, 7901, 8681, 9221, 9497, 10061, 10937, 12161, 13121, 13781, 15149, 16217, 17321, 18077, 18461, 20441, 20849, 25601, 26981, 27449

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.
Conjecture: except 521, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 327 is a square. - Vincenzo Librandi, Jun 29 2016

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

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..200] | IsPrime(a) where a is 2*n^2+38*n+17];

Select[Table[2 n^2 + 38 n + 17, {n, 800}], PrimeQ]

A243891 Primes of the form 2n^2 + 62n + 29.

Original entry on oeis.org

233, 389, 653, 953, 1061, 1289, 1409, 2069, 2213, 4253, 4649, 5273, 6869, 8933, 9209, 10061, 10949, 13829, 15569, 16661, 17033, 17789, 24413, 26693, 28109, 32573, 35729, 36269, 37361, 42473, 44249, 46061, 48533, 51713, 52361, 55661, 56333, 57689, 59753

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.
Conjecture: except 4253, 2^a(n) - 1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 903 is a square. - Vincenzo Librandi, Jun 29 2016

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

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..200] | IsPrime(a) where a is 2*n^2+62*n+29];

Select[Table[2 n^2 + 62 n + 29, {n, 200}], PrimeQ]

A243957 Primes of the form 2n^2+66n+31.

Original entry on oeis.org

499, 787, 1471, 1867, 2767, 3271, 4999, 5647, 8599, 13099, 14107, 16231, 19687, 22171, 24799, 33547, 40099, 43591, 52951, 63211, 65371, 67567, 79087, 88951, 94099, 99391, 104827, 107599, 116131, 119047, 124987, 131071, 153499, 160231, 167107, 177691, 192307

Offset: 1

Vincenzo Librandi, Jun 16 2014

Primes of the form 36*A056115(k)+31.

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.

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

Cf. A056115, A142110 (supersequence).

Cf. similar sequences listed in A243888.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2+66*n+31];

Select[Table[2 n^2 + 66 n + 31, {n, 800}], PrimeQ]

A243958 Primes of the form 2n^2+86n+41.

Original entry on oeis.org

317, 521, 857, 977, 1229, 1361, 1637, 2081, 2237, 2729, 3257, 3821, 4217, 4421, 5501, 6197, 8501, 9341, 9629, 12401, 13397, 14081, 15137, 15497, 16229, 18521, 18917, 20129, 21377, 22229, 23537, 23981, 26261, 26729, 29129, 31121, 32141, 35837, 36929, 39161

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.

Conjecture: except 521, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.

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

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..300] | IsPrime(a) where a is 2*n^2+86*n+41];

Select[Table[2 n^2 + 86 n + 41, {n, 800}], PrimeQ]

A275938 Numbers m such that d(m) is prime while sigma(m) is not prime (where d(m) = A000005(m) and sigma(m) = A000203(m)).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Altug Alkan, Aug 12 2016

From Robert Israel, Aug 12 2016: (Start)
d(m) is prime iff m = p^k where p is prime and k+1 is prime.
For such m, sigma(m) = 1 + p + ... + p^k = (p*m-1)/(p-1).
The sequence contains 2^(q-1) for q in A054723,
3^(q-1) for q prime but not in A028491,
5^(q-1) for q prime but not in A004061,
7^(q-1) for q prime but not in A004063, etc.
In particular, it contains all odd primes. (End)

			49 is a term because A000005(49) = 3 is prime while sigma(49) = 57 is not.

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

Cf. A000005, A000203, A004061, A004063, A009087, A023194, A028491.

Cf. A000040, A286095.

N:= 1000: # to get all terms <= N
P:= select(isprime, {2,seq(p,p=3..N,2)}):
fp:= proc(p) local q,res;
  q:= 2;
  res:= NULL;
  while p^(q-1) <= N do
     if not isprime((p^q-1)/(p-1)) then res:= res, p^(q-1) fi;
     q:= nextprime(q);
  od;
  res;
end proc:
sort(convert(map(fp, P),list)); # Robert Israel, Aug 12 2016

lista(nn) = for(n=1, nn, if(isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")));

UNION of A000040 and A286095 (except for the term 2). - Bill McEachen, Jul 16 2024

A344515 Primes p such that 2^p-1 has exactly 3 distinct prime factors.

Original entry on oeis.org

29, 43, 47, 53, 71, 73, 79, 179, 193, 211, 257, 277, 283, 311, 331, 349, 353, 389, 409, 443, 467, 499, 563, 577, 599, 613, 631, 643, 647, 683, 709, 751, 769, 829, 919, 941, 1039, 1103, 1117, 1123, 1171, 1193

Offset: 1

Amiram Eldar, May 21 2021

The corresponding Mersenne numbers are in A135977.
a(43) >= 1237.
The following primes are also terms of this sequence: 1301, 1303, 1327, 1459, 1531, 1559, 1907, 2311, 2383, 2887, 3041, 3547, 3833, 4127, 4507, 4871, 6883, 7673, 8233.

			29 is a term since 2^29-1 = 536870911 = 233 * 1103 * 2089 has exactly 3 distinct prime factors.

Subsequence of A054723.

Cf. A000225, A065341, A135975, A135976, A135977, A135978.

Select[Range[200], PrimeQ[#] && PrimeNu[2^# - 1] == 3 &]

2^a(n) - 1 = A135977(n).

A379590 a(n) is the number of prime divisors d of n such that 2^d - 1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 3, 1, 1, 1, 2, 2, 2, 0, 2, 2, 2, 0, 3, 0, 1, 2, 1, 0, 2, 1, 2, 2, 2, 0, 2, 1, 2, 2, 1, 0, 3, 1, 2, 2, 1, 2, 2, 0, 2, 1, 3, 0, 2, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 3, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 2, 0, 2, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Dec 26 2024

nonn

Number of divisors of n that belong to A000043.

Cf. A000005, A000043, A054723.

[#[d: d in Divisors(n) | IsPrime(2^d-1)]: n in [1..100]];

a[n_] := DivisorSum[n, 1 &, PrimeQ[2^# - 1] &]; Array[a, 100] (* Amiram Eldar, Dec 27 2024 *)

a(n) = sumdiv(n, d, isprime(d) && ispseudoprime(2^d-1)); \\ Michel Marcus, Dec 28 2024

A135386 Mersenne composites A065341 with 4 or more prime factors.

Original entry on oeis.org

10384593717069655257060992658440191, 2854495385411919762116571938898990272765493247, 182687704666362864775460604089535377456991567871

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..102

Subsequence of A065341.

Cf. A000225, A054723, A134852, A135975, A135976, A135977, A135979.

A135386 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) > 3) then
   RETURN (i)
fi: end: seq(A135386(n), n=1..37); # Jani Melik, Feb 09 2011

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d >3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

A186884 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k - 2^p for some integer p >= 0 and 2^p <= b.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 71, 127, 173, 199, 233, 251, 257, 379, 491, 613, 881, 2047, 2633, 2659, 3373, 3457, 5501, 5683, 8191, 11497, 13249, 15823, 16879, 18839, 22669, 24763, 25037, 26893, 30139, 45337, 48473, 56671, 58921, 65537, 70687, 74531, 74597, 77023, 79669, 87211, 92237, 102407, 131071, 133493, 181421, 184511, 237379, 250583, 254491, 281381

Offset: 1

Alzhekeyev Ascar M, Feb 28 2011

nonn

This sequence contains A186645 as a subsequence (corresponding to p=0).
All composites in this sequence are 2-pseudoprimes, A001567. This sequence contains all terms of A054723. Another composite term is 4294967297 = 2^32 + 1, which does not belong to A054723. In other words, all known composite terms have the form (2^x + 1) or (2^x - 1). Are there composites not of this form?
This sequence contains all the primes of the forms (2^x + 1) and (2^x - 1), i.e., subsequences A092506 and A000668.

Edited by Max Alekseyev, Mar 14 2011

a(25) and a(26) interchanged by Georg Fischer, Jul 08 2022

A241973 Prime exponents of composite Mersenne numbers in the order of the magnitude of the smallest prime factor.

Original entry on oeis.org

11, 23, 83, 37, 29, 131, 179, 191, 43, 73, 239, 251, 359, 419, 431, 443, 491, 659, 683, 233, 719, 743, 911, 1019, 1031, 1103, 47, 397, 1223, 79, 461, 1439, 1451, 1499, 1511, 1559, 1583, 557, 113, 577, 601, 1811, 1931, 2003, 2039, 2063, 761, 2339, 2351, 2399

Offset: 1

J. Lowell, May 03 2014

nonn

Terms are the same as A054723, but in a different order.
If p is a prime and 2^p-1 is composite, each prime factor of 2^p-1 will be of the form kp+1 for some integer k. Thus, the smallest prime factor of 2^p-1 cannot be smaller than p.
The corresponding smallest prime factors are: 23, 47, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, ....

			83 comes before 37 because 167 (the smallest prime factor of 2^83-1) < 223 (the smallest prime factor of 2^37-1).

Cf. A054723, A136030.

lista() = {vi = readvec("b054723.txt"); vm = vector(#vi, i, 2^vi[i]-1); p = 2; nbf = 0; while ( nbf != #vm, i = 1; while (!(i>#vm) && (!vm[i] || (vm[i] % p)), i++); if (i <= #vm, print1(vi[i], ", "); vm[i] = 0; nbf ++;); p = nextprime(p+1););} \\ Michel Marcus, May 14 2014

More terms from Michel Marcus, May 14 2014

cp's OEIS Frontend

A243890 Primes of the form 2*n^2+38*n+17.

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Magma

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A243891 Primes of the form 2*n^2 + 62*n + 29.

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Magma

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A243957 Primes of the form 2*n^2+66*n+31.

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Magma

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A243958 Primes of the form 2*n^2+86*n+41.

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Magma

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A275938 Numbers m such that d(m) is prime while sigma(m) is not prime (where d(m) = A000005(m) and sigma(m) = A000203(m)).

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Maple

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A344515 Primes p such that 2^p-1 has exactly 3 distinct prime factors.

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Mathematica

Formula

A379590 a(n) is the number of prime divisors d of n such that 2^d - 1 is prime.

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Magma

Mathematica

PARI

A135386 Mersenne composites A065341 with 4 or more prime factors.

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A243890 Primes of the form 2n^2+38n+17.

A243891 Primes of the form 2n^2 + 62n + 29.

A243957 Primes of the form 2n^2+66n+31.

A243958 Primes of the form 2n^2+86n+41.