Displaying 31-40 of 55 results found.
Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.
+10
3
10, 11, 12, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 165, 181, 201, 202, 221, 222, 261, 262, 282, 302, 303, 322, 323, 342, 343, 363, 403, 404, 423, 424, 444, 463, 483, 504, 505, 525, 545, 564, 584, 585, 605, 606, 645, 646, 666, 686, 706
COMMENTS
These numbers are related to the taxicab number 1729. This is why they might be called "additive Hardy-Ramanujan numbers".
EXAMPLE
For k = 11 the sum of the digits is 2 and the multiplier is 5: 2 * 5 = 10 and 10 + 01 = 11.
For k = 747 the sum of the digits is 18 and the multiplier is 7: 18 * 7 = 126 and 126 + 621 = 747.
MATHEMATICA
Block[{k, d, j}, Reap[Do[k = 1; d = Total@ IntegerDigits[i]; While[Nor[k > i, Set[j, # + IntegerReverse@ #] == i &[d k]], k++]; If[j == i, Sow[{i, k}]], {i, 720}]][[-1, 1, All, 1]] ] (* Michael De Vlieger, Jan 28 2020 *)
a(n) is the smallest number which can be represented as the sum of two distinct positive n-th powers in exactly n ways, or -1 if no such number exists.
+10
3
EXAMPLE
a(3) = 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.
Number of sums i^3 + j^3 that occur more than once for 1<=i<=j<=n.
+10
2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 7, 7, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 13, 13, 14, 15, 16, 16, 16, 17, 17, 19, 19, 19, 19, 20, 20, 20, 21, 23, 24, 24, 24, 25, 25, 25, 25
PROG
(PARI) a(n) = my(v=vector(2*n^3, i, 0)); for(i=1, n, for(j=i, n, v[i^3+j^3]+=1)); sum(i=1, #v, v[i]>1); \\ Seiichi Manyama, May 14 2024 (Ruby)
def A(n)
h = {}
(1..n).each{|i|
(i..n).each{|j|
k = i * i * i + j * j * j
if h.has_key?(k)
h[k] += 1
else
h[k] = 1
end
}
}
h.to_a.select{|i| i[1] > 1}.size
end
(1..n).map{|i| A(i)}
end
Smallest number that is the difference between two positive cubes in n ways.
+10
2
7, 721, 3367, 4118877, 1412774811, 424910390480793
COMMENTS
a(7) <= 15490327057569000, a(8) <= 123922616460552000. - Giovanni Resta, Mar 19 2020
EXAMPLE
Pairs (x, y) such that x^3 - y^3 = a(1), ..., a(6):
7 = (2, 1);
721 = (16, 15), (9, 2);
3367 = (34, 33), (16, 9), (15, 2)l
4118877 = (162, 51), (165, 72), (178, 115), (678, 675);
1412774811 = (1134, 357), (1155, 504), (1246, 805), (2115, 2004), (4746, 4725);
424910390480793 = (596001, 595602), (317982, 316575), (141705, 134268), (83482, 53935), (77385, 33768), (75978, 23919).
Primes which are the sum of three distinct positive cubes of prime numbers in two or more distinct ways.
+10
2
185527, 451837, 591751, 1265471, 1266929, 1618973, 1626227, 1664713, 2586277, 2754683, 2765519, 2805193, 3422303, 3740309, 3748499, 4154779, 5336479, 5483953, 5557987, 6130151, 6586091, 7231013, 7361801, 7726571, 8205553
EXAMPLE
185527 = 47^3+43^3+13^3=53^3+31^3+19^3.
MATHEMATICA
lst={}; Do[Do[Do[If[PrimeQ[p=Prime[a]^3+Prime[b]^3+Prime[c]^3], AppendTo[lst, p]], {c, b-1, 1, -1}], {b, a-1, 1, -1}], {a, 88}]; lst1=Sort@lst; lst={}; Do[If[lst1[[n]]==lst1[[n+1]], AppendTo[lst, lst1[[n]]]], {n, Length[lst1]-1}]; lst
Select[Tally[Select[Total/@Subsets[Prime[Range[50]]^3, {3}], PrimeQ]], #[[2]]> 1&] [[All, 1]]//Sort (* Harvey P. Dale, Sep 26 2020 *)
Numbers expressible as the sum of four nonnegative fourth-powers in four different ways.
+10
2
236674, 260658, 282018, 300834, 334818, 478338, 637794, 650034, 650658, 671778, 708483, 708834, 729938, 789378, 811538, 816578, 832274, 849954, 941859, 989043, 1042083, 1045539, 1099203, 1099458, 1102258, 1179378, 1243074, 1257954, 1283874, 1323234, 1334979
COMMENTS
A natural extension of the two-sets-of-two-cubes taxi-cab numbers ( A001235). a(4) is the first number which contains distinct fourth-powers in all four sets of four, and is therefore also A146756(4).
LINKS
Christian N. K. Anderson, Decomposition of the first 1000 terms into four sets of four fourth powers.
EXAMPLE
a(1) = 236674 = 1^4+2^4+7^4+22^4 = 3^4+6^4+18^4+19^4 = 7^4+14^4+16^4+19^4 = 8^4+16^4+17^4+17^4.
Taxi-cab numbers n such that n-1 and n+1 are both prime.
+10
2
32832, 513000, 2101248, 8647128, 43570872, 46661832, 152275032, 166383000, 175959000, 351981000, 543449088, 610991208, 809557632, 970168752, 1710972648, 2250265752, 2262814272, 2560837032, 3222013032, 3308144112, 3582836712, 4505949000, 4543936488, 4674301632, 4868489178
COMMENTS
Taxi-cab numbers that are in A014574. There are two versions of "taxicab numbers" that are A001235 and A011541. This sequence focuses on the version A001235. First six terms are 2^6*3^3*19, 2^3*3^3*5^3*19, 2^12*3^3*19, 2^3*3^3*7^2*19*43, 2^3*3^6*31*241, 2^3*3^8*7*127.
This sequence contains many terms that are divisible by 6^3 = 216. But there are also terms that are not divisible by 6^3. For example, 166383*10^3 and 351981*10^3 are terms that are not divisible by 216.
EXAMPLE
Taxi-cab number 32832 is a term because 32831 and 32833 are twin primes.
PROG
(PARI) T=thueinit(x^3+1, 1);
isA001235(n)=my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1
p=2; forprime(q=3, 1e9, if(q-p==2 && isA001235(p+1), print1(p+1", ")); p=q) \\ Charles R Greathouse IV, May 09 2016
Taxicab numbers that are sandwiched between squarefree numbers.
+10
2
4104, 32832, 39312, 110808, 171288, 262656, 314496, 373464, 513000, 886464, 1016496, 1075032, 1195112, 1331064, 1370304, 1407672, 1609272, 1728216, 1734264, 1774656, 2101248, 2515968, 2864288, 2987712, 2991816, 3511872, 3512808, 3551112, 4104000, 4342914, 4467528, 4511808, 4607064
COMMENTS
All terms are even numbers.
LINKS
Christian Boyer, Les nombres Taxicabs, in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
EXAMPLE
4104 = 2^3 * 3^3 * 19 (between 4103 = 11 * 373 and 4105 = 5 * 821).
32832 = 2^6 * 3^3 *19 (between 32831 and 32833, which are twin primes).
39312 = 2^4 * 3^3 * 7 * 13 (between 39311 = 19 * 2069 and 39313, which is prime).
MATHEMATICA
Select[Range[300000], And @@ SquareFreeQ /@ (# + {-1, 1}) && Length[PowersRepresentations[#, 2, 3]] > 1 &] (* Amiram Eldar, Mar 29 2024 *)
a(n) is the smallest number which can be represented as the sum of two nonzero square pyramidal numbers in exactly n ways, or -1 if no such number exists.
+10
2
EXAMPLE
a(2) = 60 = 5 + 55 = 30 + 30.
a(n) is the smallest number which can be represented as the sum of two nonzero pentagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.
+10
2
EXAMPLE
a(2) = 1471 = 1 + 1470 = 288 + 1183.
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