Displaying 1-10 of 19 results found.
Numbers with 37 divisors.
+10
19
68719476736, 150094635296999121, 14551915228366851806640625, 2651730845859653471779023381601, 30912680532870672635673352936887453361, 12646218552730347184269489080961456410641
COMMENTS
36th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139573.
Numbers with 43 divisors.
+10
18
4398046511104, 109418989131512359209, 227373675443232059478759765625, 311973482284542371301330321821976049, 54763699237492901685126120802225273763666521, 61040881526285814362156628321386486455989674569
COMMENTS
42nd powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139575.
Numbers with 41 divisors.
+10
17
1099511627776, 12157665459056928801, 9094947017729282379150390625, 6366805760909027985741435139224001, 452592555681759518058893560348969204658401
COMMENTS
40th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572.
Numbers with 47 divisors.
+10
17
70368744177664, 8862938119652501095929, 142108547152020037174224853515625, 749048330965186233494494102694564493649, 801795320536133573571931534665380233173841533961
COMMENTS
46th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574.
Numbers with 53 divisors.
+10
12
4503599627370496, 6461081889226673298932241, 2220446049250313080847263336181640625, 88124787089723195184393736687912818113311201, 1420429319844313329730664601483335671261683881745483121, 8415003868347247618489696679505181495471801448798649088081
COMMENTS
52nd powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575.
Numbers with 59 divisors.
+10
11
288230376151711744, 4710128697246244834921603689, 34694469519536141888238489627838134765625, 10367793076318844190248738727596255138212949486449
COMMENTS
Also, 58th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533.
Numbers with 61 divisors.
+10
11
1152921504606846976, 42391158275216203514294433201, 867361737988403547205962240695953369140625, 508021860739623365322188197652216501772434524836001
COMMENTS
Also, 60th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062.
Numbers with 30 divisors.
+10
8
720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2592, 2736, 2800, 3312, 3564, 3888, 3920, 4050, 4176, 4212, 4400, 4464, 4608, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396
COMMENTS
Numbers of the form p^29 (subset of A122970), p*q^2*r^4 ( A179669), p^4*q^5 ( A179702), p^2*q^9 (like 4608) or p*q^14, where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010
MATHEMATICA
Select[Range[10000], DivisorSigma[0, #]==30&] (* Harvey P. Dale, Feb 18 2011 *)
PROG
(PARI) list(lim)=
{
my(f=(v, s)->concat(v, listsig(lim, s, 1)));
Set(fold(f, [[], [29], [5, 4], [9, 2], [14, 1], [4, 2, 1]]));
}
listsig(lim, sig, coprime)=
{
my(e=sig[1]);
if(#sig<2,
if(#sig==0 || sig[1]==0, return(if(lim<1, [], [1])));
my(P=primes([2, sqrtnint(lim\1, e)]));
if(coprime==1, return(if(e>1, apply(p->p^e, P), P)));
P=select(p->gcd(p, coprime)==1, P);
if(e>1, P=apply(p->p^e, P));
return(P);
);
my(v=List(), ss=sig[2..#sig], t=leastOfSig(ss));
forprime(p=2, sqrtnint(lim\t, e),
if(coprime%p,
my(u=listsig(lim\p^e, ss, coprime*p));
for(i=1, #u, listput(v, p^e*u[i]));
)
);
Vec(v);
Numbers with 101 divisors.
+10
8
1267650600228229401496703205376, 515377520732011331036461129765621272702107522001, 7888609052210118054117285652827862296732064351090230047702789306640625, 3234476509624757991344647769100216810857203198904625400933895331391691459636928060001
COMMENTS
Also, 100th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
EXAMPLE
a(1) = 2^100, a(2) = 3^100, a(3) = 5^100, a(4) = 7^100.
CROSSREFS
Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085.
Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.
+10
8
1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293
COMMENTS
If n = p - 1 where p is prime, then row n lists the numbers with p divisors.
The partial sums of column k give the column k of A319076.
FORMULA
T(n,k) = A000040(k)^n, n >= 0, k >= 1.
EXAMPLE
The corner of the square array is as follows:
A000040 2, 3, 5, 7, 11, 13, 17, ...
A001248 4, 9, 25, 49, 121, 169, 289, ...
A030078 8, 27, 125, 343, 1331, 2197, 4913, ...
A030514 16, 81, 625, 2401, 14641, 28561, 83521, ...
A050997 32, 243, 3125, 16807, 161051, 371293, 1419857, ...
A030516 64, 729, 15625, 117649, 1771561, 4826809, 24137569, ...
A092759 128, 2187, 78125, 823543, 19487171, 62748517, 410338673, ...
A179645 256, 6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...
PROG
(PARI) T(n, k) = prime(k)^n;
CROSSREFS
Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031. Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100). Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
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