Displaying 1-10 of 25 results found.
Numbers k such that the k-th partition number A000041(k) is prime.
+10
50
2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, 188, 212, 216, 302, 366, 417, 440, 491, 498, 525, 546, 658, 735, 753, 825, 841, 863, 1085, 1086, 1296, 1477, 1578, 1586, 1621, 1793, 2051, 2136, 2493, 2502, 2508, 2568, 2633, 2727, 2732, 2871, 2912, 3027, 3098, 3168, 3342, 3542, 3641, 4118
PROG
(PARI) for(n=0, 10^5, my(p=numbpart(n)); if(isprime(p), print1(n, ", "))); \\ Joerg Arndt, May 09 2013 (Python)
from sympy import isprime, npartitions
print([n for n in range(1, 5001) if isprime(npartitions(n))]) # Indranil Ghosh, Apr 10 2017
CROSSREFS
Cf. A000041, A035359, A049575, A051143, A111036, A111045, A114165, A111389, A113499, A114166, A114167, A114168, A114169, A114170, A115214.
EXTENSIONS
b-file extended by Max Alekseyev, Jul 07 2009, Jun 14 2011, Jan 08 2012, May 19 2014
Number of ways to write n = k + m with k > 0 and m > 2 such that p(k + phi(m)/2) is prime, where p(.) is the partition function ( A000041) and phi(.) is Euler's totient function.
+10
17
0, 0, 0, 1, 2, 3, 4, 5, 5, 4, 4, 4, 2, 2, 3, 5, 4, 2, 4, 2, 3, 2, 3, 2, 3, 1, 0, 3, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 3, 3, 3, 1, 0, 4, 2, 4, 1, 1, 2, 2, 3, 2, 2, 0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 4, 2, 1, 0, 1, 3, 1, 0, 2, 4, 3, 1, 6, 2, 2, 1, 2, 4, 3, 1, 2, 6, 2, 3, 2, 2, 2, 2, 3, 3
COMMENTS
Conjecture: a(n) > 0 if n > 3 is not among 27, 34, 50, 61, 74, 78, 115, 120, 123, 127.
This implies that there are infinitely many primes in the range of the partition function p(n).
EXAMPLE
a(26) = 1 since 26 = 2 + 24 with p(2 + phi(24)/2) = p(6) = 11 prime.
a(54) = 1 since 54 = 27 + 27 with p(27 + phi(27)/2) = p(36) = 17977 prime.
a(73) = 1 since 73 = 1 + 72 with p(1 + phi(72)/2) = p(36) = 17977 prime.
a(110) = 1 since 110 = 65 + 45 with p(65 + phi(45)/2) = p(77) = 10619863 prime.
a(150) = 1 since 150 = 123 + 27 with p(123 + phi(27)/2) = p(132) = 6620830889 prime.
a(170) = 1 since 170 = 167 + 3 with p(167 + phi(3)/2) = p(168) = 228204732751 prime.
MATHEMATICA
f[n_, k_]:=PartitionsP[k+EulerPhi[n-k]/2]
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
CROSSREFS
Cf. A000010, A000040, A000041, A049575, A232504, A233307, A233346, A233918, A234200, A234246, A234309, A234337, A234344, A234347, A234359, A234360, A234451, A234475
Primes p with P(p-1) also prime, where P(.) is the partition function ( A000041).
+10
12
3, 5, 7, 37, 367, 499, 547, 659, 1087, 1297, 1579, 2137, 2503, 3169, 3343, 4457, 4663, 5003, 7459, 9293, 16249, 23203, 34667, 39971, 41381, 56383, 61751, 62987, 72661, 77213, 79697, 98893, 101771, 127081, 136193, 188843, 193811, 259627, 267187, 282913, 315467, 320563, 345923, 354833, 459029, 482837, 496477, 548039, 641419, 647189
COMMENTS
By the conjecture in A234567, this sequence should have infinitely many terms. It seems that a(n+1) < a(n) + a(n-1) for all n > 5. The b-file lists all terms not exceeding the 500000th prime 7368787. Note that P(a(113)-1) is a prime having 2999 decimal digits.
See also A234572 for primes of the form P(p-1) with p prime.
EXAMPLE
a(1) = 3 since P(2-1) = 1 is not prime, but P(3-1) = 2 is prime.
a(2) = 5 since P(5-1) = 5 is prime.
a(3) = 7 since P(7-1) = 11 is prime.
MATHEMATICA
n=0; Do[If[PrimeQ[PartitionsP[Prime[k]-1]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 10^6}]
CROSSREFS
Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234572, A234615, A234644
Number of distinct partition numbers dividing n.
+10
9
1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 1, 3, 4, 2, 1, 3, 1, 3, 3, 4, 1, 3, 2, 2, 2, 3, 1, 6, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 5, 1, 4, 4, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 2, 2, 1, 5, 2, 2, 2, 4, 1, 6
FORMULA
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)
EXAMPLE
For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.
For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.
MATHEMATICA
p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* Giovanni Resta, Jan 22 2014 *)
Nonprimes whose proper divisors are partition numbers.
+10
7
4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 303, 505, 707, 1111, 10201, 35954, 53931, 89885, 125839, 197747, 1815677, 21239726, 31859589, 53099315, 74339041, 116818493, 323172529, 1072606163, 13241661778, 19862492667, 33104154445, 46345816223, 72829139779
EXAMPLE
10 is in the sequence because 10 is a nonprime number and the proper divisors of 10 are 1, 2, 5, which are also partition numbers.
MAPLE
isA000041 := proc(n)
local k, P;
for k from 1 do
P := combinat[numbpart](k) ;
if P > n then
return false;
elif P = n then
return true ;
end if;
end do:
end proc:
isA236108 := proc(n)
local pdvs, d ;
if n =1 or isprime(n) then
return false;
end if;
pdvs := numtheory[divisors](n) minus {n} ;
for d in pdvs do
if not isA000041(d) then
return false;
end if;
end do:
return true;
end proc:
for n from 1 to 300000 do
if isA236108(n) then
printf("%d, ", n) ;
end if;
MATHEMATICA
partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
Select[Range[2, 10000],
If[! PrimeQ[#],
ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* Julien Kluge, Dec 03 2016 *)
Primes of the form P(p-1), where p is a prime and P(.) is the partition function ( A000041).
+10
6
2, 5, 11, 17977, 790738119649411319, 2058791472042884901563, 27833079238879849385687, 8121368081058512888507057, 675004412390512738195023734124239, 1398703012615213588677365804960180341, 16193798232344933888778097136641377589301, 204931453786129197483756438132982529754356479553, 3019564607799532159016586951616642980389816614848623, 22757918197082858017617136646280039394687006502870793231847, 1078734573992480956821414895441907729656949308800686938161281
COMMENTS
Though the primes in this sequence are very rare, by the conjecture in A234567 there should be infinitely many such primes. See A234569 for a list of known primes p with P(p-1) also prime.
EXAMPLE
a(1) = 2 since 2 = P(3-1) with 2 and 3 both prime.
a(2) = 5 since 5 = P(5-1) with 5 prime.
a(3) = 11 since 11 = P(7-1) with 7 and 11 both prime.
MATHEMATICA
Table[PartitionsP[p[n]-1], {n, 1, 15}]
Numbers whose divisors are partition numbers.
+10
6
1, 2, 3, 5, 7, 11, 15, 22, 77, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939
COMMENTS
By definition all terms are partition numbers.
All members of A049575 are in this sequence. Conjecture: the only composite numbers in this sequence are 15, 22, and 77. - Jon E. Schoenfield, Feb 05 2014
EXAMPLE
15 is in the sequence because the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.
Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.
+10
4
2, 3, 4, 6, 9, 13, 18, 26, 35, 48, 65, 87, 115, 152, 199, 258, 333, 427, 545, 692, 875, 1102, 1381, 1725, 2145, 2659, 3285, 4046, 4967, 6080, 7423, 9037, 10974, 13293, 16065, 19370, 23304, 27977, 33519, 40080, 47833, 56981, 67757, 80431, 95316
COMMENTS
This conjecture is false, for correct approximation see the formula below. - Vaclav Kotesovec, Apr 03 2017
REFERENCES
John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 95.
FORMULA
a(n) = round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))). - Alonso del Arte, May 21 2011 a(n) - A000041(n) ~ (1/Pi + Pi/72) * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (9 + Pi^2/48)*Pi/((72 + Pi^2)*sqrt(6*n))). - Vaclav Kotesovec, Apr 03 2017
PROG
(UBASIC) input N:print round(#e^(pi(1)*sqrt(2*N/3))/(4*N*sqrt(3)))
Numbers k such that the k-th prime is a partition number.
+10
4
1, 2, 3, 4, 5, 26, 2061, 702993, 307058572, 3350187739, 9088200428, 43794115173, 51932790219, 378210209388, 521301342188, 297064987225918, 19677201507658441, 437852535314831447, 1673669998972800207, 29252504332047744188, 42842701894337201916
EXTENSIONS
a(16)-a(21) using Kim Walisch's primecount, from Amiram Eldar, Jul 26 2019
15, 22, 77, 1255, 2012558, 2679689, 9289091, 18004327, 38887673, 56634173, 72533807, 82010177, 104651419, 2056148051, 2552338241, 20390982757, 27517052599, 118159068427, 749474411781, 5134205287973, 18028182516671
COMMENTS
Enoch Haga asks if this is a finite sequence. The larger these numbers get, the more opportunity for more factors.
EXAMPLE
E.g., the 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.
MATHEMATICA
Select[PartitionsP[Range[0, 450]], PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 19 2016 *)
PROG
(PARI) { n=0; for (m=1, 10^9, p=numbpart(m); if (bigomega(p) == 2, write("b065728.txt", n++, " ", p); if (n==100, return)) ) } \\ Harry J. Smith, Oct 28 2009
Search completed in 0.013 seconds
|