A023394 -id:A023394 - cp's OEIS frontend

A229856 Primes of the form 384*k + 257.

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041, 40193, 40577, 40961, 41729, 43649, 44417

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Every Fermat number greater than 257 has a prime factor of the form 384*k + 257, k > 0.

Wikipedia, Fermat number

Subsequence of A107181 (primes of the form 8x^2+9y^2).

Cf. A000215, A023394, A094358, A229853, A229855, A229856.

[384*n+257 : n in [0..115] | IsPrime(384*n+257)];

Select[Table[384*n + 257, {n, 0, 115}], PrimeQ]

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

A229854 Primes of the form 384*k + 1.

Original entry on oeis.org

769, 1153, 2689, 3457, 4993, 6529, 7297, 7681, 9601, 10369, 10753, 12289, 13441, 14593, 15361, 18049, 18433, 20353, 21121, 22273, 23041, 26113, 26497, 26881, 29569, 31489, 31873, 32257, 33409, 36097, 37633, 39937, 43777, 45697, 49537, 49921, 52609, 53377

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Not every composite Fermat number has a prime factor of the form 384*k + 1.

Wikipedia, Fermat number

Subsequence of A002476 (primes of form 6m + 1).

Cf. A000215, A023394, A094358, A229850, A229853, A229855, A229856.

[384*n+1 : n in [1..139] | IsPrime(384*n+1)];

Select[Table[384*n + 1, {n, 139}], PrimeQ]

A242017 Smallest prime factor of composites in the sequence A000051(n) = 2^n+1.

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 3, 17, 3, 5, 3, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 274177, 3, 5, 3

Offset: 1

Felix Fröhlich, Aug 11 2014

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1786

Cf. A000051, A023394, A242016.

FactorInteger[#][[1,1]]&/@Select[(2^Range[70]+1),CompositeQ] (* Harvey P. Dale, Feb 17 2017 *)

for(n=1, 1e2, if(!ispseudoprime(2^n+1), p=factor(2^n+1)[1, 1]; print1(p, ", ")))

A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

Original entry on oeis.org

274177, 319489, 6700417, 825753601, 1214251009, 6487031809, 646730219521, 6597069766657, 25409026523137, 31065037602817, 46179488366593, 151413703311361, 231292694251438081, 1529992420282859521, 2170072644496392193, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Feb 07 2022

nonn

Subsequence of A014752.

			a(1) = 503^2 + 27*28^2 = 274177 is a prime factor of 2^(2^6) + 1;
a(2) = 383^2 + 27*80^2 = 319489 is a prime factor of 2^(2^11) + 1;
a(3) = 887^2 + 27*468^2 = 6700417 is a prime factor of 2^(2^5) + 1;
a(4) = 27017^2 + 27*1884^2 = 825753601 is a prime factor of 2^(2^16) + 1;
a(5) = 2561^2 + 27*6688^2 = 1214251009 is a prime factor of 2^(2^15) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A002476, A014752, A023394, A204620, A229850, A229853.

isok(p) = if(p%6==1 && isprime(p), my(z=znorder(Mod(2, p))); z>>valuation(z, 2)==1, return(0));

A002476 INTERSECT A023394.

A228845 Least m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 7, 8, 9, 11, 11, 12, 13, 14

Offset: 0

Arkadiusz Wesolowski, Sep 05 2013

a(n) >= n + 2 for n >= 2.

a(n) = A228846(n) if 2^(2^n) + 1 is prime or semiprime.

Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A019434, A023394, A046052, A228846.

A360652 Primes of the form x^2 + 432*y^2.

Original entry on oeis.org

433, 457, 601, 1657, 1753, 1777, 1801, 2017, 2089, 2113, 2281, 2689, 2833, 2953, 3457, 3889, 4057, 4129, 4153, 4177, 4513, 4657, 4729, 5113, 5209, 5449, 5569, 5737, 5953, 6217, 6361, 6673, 6961, 7057, 7321, 7369, 7537, 7753, 7873, 8353, 8377, 8713, 8761, 8929

Offset: 1

Arkadiusz Wesolowski, Feb 15 2023

nonn

Supersequence of A351332. Thus every prime congruent to 1 mod 3 that divides a Fermat number is in this sequence.

Every Fermat number that is a semiprime has a prime of this form as a factor.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A014752, A023394, A229850, A351332.

[p: p in PrimesUpTo(8929) | NormEquation(432, p) eq true];

select(p->my(m=Mod(2, p)^(p\12)); p>11 && (m==1||m==p-1), primes(1110))

A367648 Primes p such that the multiplicative order of 3 modulo p is a power of 3.

Original entry on oeis.org

2, 13, 109, 433, 757, 3889, 8209, 17497, 52489, 58321, 70957, 1190701, 1705861, 2598157, 6627097, 13463173, 57395629, 23245229341, 79320757897, 1069604540569, 1631815099669, 5774114968057, 8635817966221, 23765922477217, 43781455818469, 307283335691329

Offset: 1

Jianing Song, Nov 25 2023

Prime factors of numbers of the form 3^3^i - 1: p divides 3^3^i - 1 if and only if the multiplicative order of 3 modulo p is a power of 3 not exceeding 3^i.

			13 is a term since the multiplicative order of 3 modulo 13 is 3 = 3^1, which means that 13 is a factor of 3^3^1 - 1.
109 is a term since the multiplicative order of 3 modulo 109 is 27 = 3^3, which means that 109 is a factor of 3^3^3 - 1.

Subsequence of A367265.

Cf. A023394 (ord(2,p) being a power of 2, prime factors of numbers of the form 2^2^i - 1 (or of the form 2^2^i + 1)), A367649 (ord(3,p) being 2 times a power of 3, prime factors of numbers of the form 3^3^i + 1).

isA367648(n) = isprime(n) && (n!=3) && isprimepower(3*znorder(Mod(3,n)))

a(18)-a(19) from Michel Marcus, Nov 27 2023
a(20)-a(25) from Max Alekseyev, Jul 22 2024
a(26) from Jinyuan Wang, Jan 29 2025

A367649 Primes p such that the multiplicative order of 3 modulo p is 2 times a power of 3.

Original entry on oeis.org

7, 19, 37, 163, 487, 1297, 1459, 2917, 19441, 19927, 39367, 59779, 131221, 208657, 224209, 572023, 2051893, 5062663, 8503057, 19131877, 44457337, 86093443, 113863969, 133923133, 258280327, 565571323, 600830137, 859270843, 1319934691, 4161183031, 5366491219, 5879415781

Offset: 1

Jianing Song, Nov 25 2023

nonn

Odd prime factors of numbers of the form 3^3^i + 1: for odd primes p, p divides 3^3^i + 1 if and only if the multiplicative order of 3 modulo p is 2 times a power of 3 not exceeding 3^i.

			37 is a term since the multiplicative order of 3 modulo 37 is 18 = 2*3^2, which means that 37 is a factor of 3^3^2 + 1.
163 is a term since the multiplicative order of 3 modulo 163 is 162 = 2*3^4, which means that 163 is a factor of 3^3^4 + 1.

Subsequence of A367266.

Cf. A023394 (ord(2,p) being a power of 2, prime factors of numbers of the form 2^2^i - 1 (or of the form 2^2^i + 1)), A367648 (ord(3,p) being a power of 3, prime factors of numbers of the form 3^3^i - 1).

isA367649(n) = my(d); isprime(n) && (n!=3) && ((d=znorder(Mod(3,n)))%2==0) && isprimepower(3*d/2)

a(28)-a(31) from Chai Wah Wu, Nov 26 2023

a(32) from Jinyuan Wang, Jan 29 2025

A373580 Numbers k that divide 2^(2^k) - 2^k + 1.

Original entry on oeis.org

1, 3, 5, 17, 257, 641, 1605, 4369, 32113, 65537, 94945, 114689, 159441, 164737, 274177, 319489, 974849, 2424833, 3862465, 6700417, 13631489, 13906833, 16843009, 26017793, 42009217, 45592577, 63766529, 70463489, 167772161, 175747457, 825753601, 1214251009, 1227890731, 1251711641

Offset: 1

Thomas Ordowski, Jun 10 2024

nonn

Numbers k that divide A119564(k).
A term is prime if and only if it is in A023394.
If k is in A307843, then it is a term of this sequence.
The terms that are not divisors of Fermat numbers are 1605, 4369, 32113, 94945, ... (they are all composite). Are there infinitely many of them?
Note that 2^(2^k) - 2^k + 1 = (2^(2^k) - 1) - (2^k - 2).

Cf. A023394 (primes in this sequence), A119564, A307843 (subsequence).

q[k_] := Mod[PowerMod[2, 2^k, k] - PowerMod[2, k, k] + 1, k] == 0; Select[Range[1, 10^5, 2], q] (* Amiram Eldar, Jun 10 2024 *)

isok(k) = Mod(Mod(2, k)^(2^k) - Mod(2,k)^k + 1, k) == 0; \\ Michel Marcus, Jun 12 2024

More terms from Amiram Eldar, Jun 10 2024

cp's OEIS Frontend

A229856 Primes of the form 384*k + 257.

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A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

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A229854 Primes of the form 384*k + 1.

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A242017 Smallest prime factor of composites in the sequence A000051(n) = 2^n+1.

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A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

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A228845 Least m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

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A360652 Primes of the form x^2 + 432*y^2.

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A367648 Primes p such that the multiplicative order of 3 modulo p is a power of 3.

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A367649 Primes p such that the multiplicative order of 3 modulo p is 2 times a power of 3.

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