Displaying 1-10 of 69 results found.
Weird numbers ( A006037) with more divisors than any smaller weird number.
+20
5
70, 836, 4030, 45356, 1713592, 15126992, 29465852, 1550860550
COMMENTS
The corresponding numbers of divisors are 8, 12, 16, 24, 32, 40, 48, 96, ...
EXAMPLE
The first 3 weird numbers, 70, 836 and 4030, have an increasing number of divisors, 8, 12 and 16. The least weird number with more than 16 divisors is the 94th weird number, 45356, which has 24 divisors.
Lesser member of twin weird numbers: weird numbers n ( A006037) such that n+2 is also weird.
+20
4
512468, 540890, 688028, 1390268, 1565828, 1741388, 2268068, 3525410, 3848108, 4374788, 6481508, 6657068, 7534868, 7885988, 7914410, 8089970, 8838968, 9143330, 9290468, 10021130, 10343828, 10898930, 12654530, 12801668, 12872510, 13152788, 13181210, 14234570
COMMENTS
Number of terms below 10^k for k = 6, 7, ... 10: 19, 231, 2111, 22426.
The first occurrences of 2 consecutive pairs of twin weirds are (21607670, 21607672, 21608090, 21608092), (873951608, 873951610, 873951890, 873951892), ...
EXAMPLE
512468 is in the sequence since both 512468 and 512470 are weird numbers.
Weird admirable numbers: numbers that are both weird ( A006037) and admirable ( A111592).
+20
4
70, 836, 4030, 5830, 7192, 7912, 10792, 17272, 45356, 83312, 91388, 113072, 243892, 254012, 388076, 786208, 1713592, 4145216, 4199030, 4632896, 9928792, 11547352, 13086016, 15126992, 17999992, 29465852, 29581424, 34869056, 74899952, 89283592, 95327216, 120888092
COMMENTS
Admirable numbers that are not pseudoperfect ( A005835).
MATHEMATICA
admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; weirdQ[n_] := Module[{d = Most[Divisors[n]]}, If[Total[d] <= n, False, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]]; Select[Range[10000], admQ[#] && weirdQ[#] &]
Weird numbers ( A006037) whose sum of aliquot divisors is also a weird number.
+20
3
97930, 132230, 146930, 191030, 205730, 215530, 244930, 259630, 279230, 308630, 362530, 411530, 440930, 524230, 529130, 583030, 597730, 602630, 632030, 646730, 705530, 730030, 808430, 891730, 921130, 955430, 970130, 1014230, 1024030, 1028930, 1102430, 1215130, 1435630
EXAMPLE
97930 is a term because it is a weird number, and A001065(97930) = sigma(97930) - 97930 = 103670 is also a weird number.
MATHEMATICA
With[{weirds = Import["https://oeis.org/a006037/b006037.txt", "Table"][[;; , 2]]}, Select[weirds, (s = DivisorSigma[1, #] - #) <= Last[weirds] && MemberQ[weirds, s] &]]
Weird numbers ( A006037) not divisible by 5.
+20
2
836, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 222952, 243892, 254012, 338572, 343876, 351956, 388076, 407132, 410476, 465652, 470668, 475684, 477356, 482372, 490732, 495748, 500764, 502436, 507452, 512468, 515812
COMMENTS
Up to 200000, there are only 11 weird numbers not divisible by 5.
Since no A006037(k) < 10^17 is odd, at least up to there, "divisible by 5" is equivalent to "ending in 0" (in base 10). It appears that 4*11*19*p is an element of this sequence for p=1 and all primes p>547. Moreover, these seem to comprise most of the terms of this sequence.
Up to n=500, the only indices for which a(n) is not of this form are n=2,...,16, 18, 34, 38, 43, 64, 83, 148, 158, 236, 266, 296, 310.
5390, 11830, 17010, 20230, 25270, 37030, 51030, 58870, 67270, 93170, 95830, 117670, 129430, 153090, 153790, 154630, 196630, 243670, 260470, 314230, 343910, 352870, 373030, 436870, 459270, 480130, 482230, 554470, 658630, 714070, 742630, 801430, 831670, 851690, 893830
COMMENTS
All the terms are nonsquarefree, since unitary weird numbers that are squarefree are necessarily also weird.
Nonsquarefree unitary weird numbers that are also weird numbers are listed in A328563.
MATHEMATICA
weirdQ[n_, d_, s1_, m1_] := weirdQ[n, d, s1, m1] = Module[{s = s1, m = m1}, If[m == 0, False, While[d[[m]] > n, s -= d[[m]]; m--]; d[[m]] < n && If[s > n, weirdQ[n - d[[m]], d, s - d[[m]], m - 1] && weirdQ[n, d, s - d[[m]], m - 1], s < n && m < Length[d] - 1]]];
wQ[n_] := Module[{d = Divisors[n]}, s = Total@d - n; m = Length[d] - 1; weirdQ[n, d, s, m]];
uQ[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] == 1 &]}, s = Total@d - n; m = Length[d] - 1; weirdQ[n, d, s, m]];
aQ[n_] := uQ[n] && ! wQ[n]; Select[Range[10^6], aQ]
Weird numbers m ( A006037) such that sigma(m)/m > sigma(k)/k for all weird numbers k < m, where sigma(m) is the sum of divisors of m ( A000203).
+20
2
70, 10430, 1554070, 5681270, 6365870
COMMENTS
Benkoski and Erdős asked whether sigma(n)/n can be arbitrarily large for weird number n. Erdős offered $25 for the solution of this question.
No more terms below 10^10.
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.
LINKS
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Mathematics of Computation, Vol. 28, No. 126 (1974), pp. 617-623, alternative link, corrigendum, ibid., Vol. 29, No. 130 (1975), p. 673. Paul Erdős, Problems and results on combinatorial number theory III, in: M. B. Nathanson (ed.), Number Theory Day, Proceedings of the Conference Held at Rockefeller University, New York 1976, Lecture Notes in Mathematics, Vol 626, Springer, Berlin, Heidelberg, 1977, pp. 43-72. See page 47. Paul Erdős, Some problems I presented or planned to present in my short talk, in: B. C. Berndt, H. G. Diamond, and A. J. Hildebrand (eds.), Analytic Number Theory, Volume 1, Proceedings of a Conference in Honor of Heini Halberstam, Progress in Mathematics, Vol. 138, Birkhäuser Boston, 1996, pp. 333-335.
EXAMPLE
The abundancy indices of the terms are sigma(a(n))/a(n) = 2.0571... < 2.0709... < 2.0710... < 2.0716... < 2.0716...
Number of weird numbers ( A006037) less than 2^n.
+20
1
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 6, 25, 66, 134, 266, 505, 952, 1850, 3515, 6643, 12920, 26893, 55786, 114992, 234719, 473299, 946406, 1884728, 3754316, 7467998, 14845968, 29628007, 59231668, 118528637
COMMENTS
It is known that the density of weird numbers is positive, but smaller than 0.0101. The three smallest weird numbers are 70, 836 and 4030, therefore a(n) = 0 for n < 7; a(n) = 1 for 7 <= n < 10, and a(n) = 2 for 10 <= n < 12.
PROG
(PARI) print1(s=0); for(k=0, 30, print1(", "s+=sum(n=1<<k, 2<<k, is_ A006037(2*n))))
Weird numbers ( A006037) with an even sum of divisors that are not Zumkeller numbers ( A083207).
+20
1
73616, 682592, 2081824, 3963968, 4960448, 5440192, 6621632, 8000704, 8134208, 12979264, 31297472, 33736064, 43955584, 55691392, 58433152, 58904704, 160074368, 254533504, 263654656, 266828032, 267369728, 272240768, 352668416, 353383168, 357542656, 431462656, 530110208
COMMENTS
Non-deficient numbers ( A023196) with an even sum of divisors ( A000203) that are neither pseudoperfect numbers ( A005835) nor Zumkeller numbers ( A083207). Equivalently, numbers k such that sigma(k) >= 2*k and sigma(k) == 0 (mod 2), such that no subset of the aliquot divisors of k sums to k or to sigma(k)/2.
EXAMPLE
73616 is a term since sigma(73616) = 147312 is even and larger than 2 * 73616 = 147232. No subset of the aliquot divisors of 73616 sums to 73616 or to sigma(73616)/2 = 73656.
MATHEMATICA
seqQ[n_] := Module[{d = Divisors[n], sum, c, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, c = CoefficientList[Product[1 + x^i, {i, d}], x]; c[[1 + 2*n]] == 0 && c[[1 + sum/2]] == 0]]; Select[Range[10^6], seqQ]
a(n) is the least number whose aliquot sequence begins with exactly n weird numbers ( A006037), or -1 if no such numbers exists.
+20
1
1, 70, 97930, 597730, 77420770, 459940810, 11835050710
COMMENTS
a(7) > 10^11, if it exists.
EXAMPLE
The iterations of A001065 over the terms a(1)..a(6): n | a(n) | Iterations
--+-------------+--------------------------------------------------------------
1 | 70 | 70
2 | 97930 | 97930 -> 103670
3 | 597730 | 597730 -> 632030 -> 668290
4 | 77420770 | 77420770 -> 82246430 -> 86946370 -> 92477630
5 | 459940810 | 459940810 -> 487175990 -> 515884810 -> 546184310 -> 582130570
6 | 11835050710 | 11835050710 -> 12515648810 -> 13235404630 -> 13991713610
| | -> 14797250230 -> 15649107530
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