A139306
Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).
Original entry on oeis.org
8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272
Offset: 1
a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.
Cf.
A000079,
A000396,
A019279,
A061645,
A061652,
A133033,
A135652,
A135653,
A135654,
A135655,
A139286,
A139294,
A139307.
Original entry on oeis.org
9, 125, 161051, 410338673, 925103102315013629321, 1271991467017507741703714391419, 49593099428404263766544428188098203, 165163983801975082169196428118414326197216835208154294976154161023
Offset: 1
A126043
Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 3.
Original entry on oeis.org
2, 0, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2
Offset: 1
-
Array[Mod[MersennePrimeExponent@ #, 3] &, 45] (* Michael De Vlieger, Apr 07 2018 *)
-
forprime(p=1, 1e3, if(isprime(2^p-1), print1(p%3, ", "))) \\ Felix Fröhlich, Aug 12 2014
A126059
Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 19.
Original entry on oeis.org
2, 3, 5, 7, 13, 17, 0, 12, 4, 13, 12, 13, 8, 18, 6, 18, 1, 6, 16, 15, 18, 4, 3, 6, 3, 10, 18, 2, 18, 18, 4, 12, 6, 6, 2, 4, 16, 11, 2, 4, 6, 7, 6, 13, 1, 11, 13, 8
Offset: 1
A330819
Numbers of the form M_p^2(M^p+2)^2, where M_p is a Mersenne prime (A000668) and p is a Mersenne exponent (A000043). Also the first element of the spectral basis of A330817.
Original entry on oeis.org
225, 3969, 1046529, 268402689, 4503599493152769, 295147905144993087489, 75557863725364567605249, 21267647932558653957237540927630737409, 28269553036454149273332760011886696242605918383730576346715242738439159809
Offset: 1
If p=2, then M_2=3, and a(1) = 3^2(3+2)^2 = 15^2 = 225.
-
A330819:=[]:
for www to 1 do
for i from 1 to 31 do
#ithprime(31)=127
p:=ithprime(i);
q:=2^p-1;
if isprime(q) then x:=2^(2*p+1)*q^2; A330819:=[op(A330819),x]; fi;
od;
od;
A330819;
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(m = 2^MersennePrimeExponent[Range[9]] - 1)^2 * (m + 2)^2 (* Amiram Eldar, Jan 03 2020 *)
A332211
Lexicographically earliest permutation of primes such that a(n) = 2^n - 1 when n is one of the Mersenne prime exponents (in A000043).
Original entry on oeis.org
2, 3, 7, 5, 31, 11, 127, 13, 17, 19, 23, 29, 8191, 37, 41, 43, 131071, 47, 524287, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 2147483647, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 2305843009213693951, 269, 271, 277, 281, 283, 293, 307, 311, 313
Offset: 1
For p in A000043: 2, 3, 5, 7, 13, 17, 19, ..., a(p) = (2^p)-1, thus a(2) = 2^2 - 1 = 3, a(3) = 7, a(5) = 31, a(7) = 127, a(13) = 8191, a(17) = 131071, etc., with the rest of positions filled by the least unused prime:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
2, 3, 7, 5, 31, 11, 127, 13, 17, 19, 23, 29, 8191, 37, 41, 43, 131071, ...
-
up_to = 127;
A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1,up_to, if(isprime(q=((2^n)-1)), v[n] = q, while(mapisdefined(xs,prime(i)), i++); v[n] = prime(i)); mapput(xs,v[n],n)); (v); };
v332211 = A332211list(up_to);
A332211(n) = v332211[n];
\\ For faster computing of larger values, use precomputed values of A000043:
v000043 = [2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217];
up_to = v000043[#v000043];
A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1,up_to, if(vecsearch(v000043,n), q = (2^n)-1, while(mapisdefined(xs,prime(i)), i++); q = prime(i)); v[n] = q; mapput(xs,q,n)); (v); };
A145041
Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 31 mod 6!.
Original entry on oeis.org
5, 17, 89, 521, 4253, 9689, 9941, 11213, 19937, 21701, 859433, 1398269, 2976221, 3021377, 6972593, 32582657, 43112609, 57885161
Offset: 1
Cf.
A000043,
A000668,
A124477,
A139484,
A145038,
A112633,
A145041,
A145042,
A145044,
A145045,
A145046.
-
p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 720] == 32 &] (* Amiram Eldar, Oct 19 2024 *)
A145042
Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 127 mod 6!.
Original entry on oeis.org
7, 19, 31, 127, 607, 1279, 2203, 4423, 110503, 216091, 1257787, 20996011, 24036583
Offset: 1
Cf.
A000043,
A000668,
A124477,
A139484,
A145038,
A112633,
A145041,
A145042,
A145044,
A145045,
A145046.
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p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 127, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
A253850
Mersenne exponents (A000043) that are the sum of the divisors (A000203) of some n.
Original entry on oeis.org
3, 7, 13, 31, 127
Offset: 1
Mersenne exponent 7 is in the sequence because sigma(4) = 7.
Mersenne exponent 31 is in the sequence because there are two numbers n (16 and 25) with sigma(n) = 31.
A324200
a(n) = 2^(A000043(n)-1) * ((2^A059305(n)) - 1), where A059305 gives the prime index of the n-th Mersenne prime, while A000043 gives its exponent.
Original entry on oeis.org
6, 60, 32752, 137438953408
Offset: 1
Showing 1-10 of 693 results.
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