A006037 -id:A006037 - cp's OEIS frontend

A335008 Weird numbers (A006037) with more divisors than any smaller weird number.

70, 836, 4030, 45356, 1713592, 15126992, 29465852, 1550860550

Offset: 1

Amiram Eldar, May 20 2020

The corresponding numbers of divisors are 8, 12, 16, 24, 32, 40, 48, 96, ...

			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.

Cf. A000005, A006037, A330244.

A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 17 2019

nonn

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), ...

			512468 is in the sequence since both 512468 and 512470 are weird numbers.

Amiram Eldar, Table of n, a(n) for n = 1..22426 (terms below 10^10).

Cf. A006037, A125109, A231086 (supersequence), A231964.

A329190 Weird admirable numbers: numbers that are both weird (A006037) and admirable (A111592).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Admirable numbers that are not pseudoperfect (A005835).

Differs from A258250 at n >= 13.

Amiram Eldar, Table of n, a(n) for n = 1..66 (terms below 10^11)

Intersection of A006037 and A111592.

Cf. A005835, A258250.

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[#] &]

A371950 Weird numbers (A006037) whose sum of aliquot divisors is also a weird number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 14 2024

nonn

Terms k of A006037 such that A001065(k) is also a term of A006037.

			97930 is a term because it is a weird number, and A001065(97930) = sigma(97930) - 97930 = 103670 is also a weird number.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Weird number.

Subsequence of A006037 and A125639.

Cf. A000203 (sigma), A001065, A371951, A371952.

With[{weirds = Import["https://oeis.org/a006037/b006037.txt", "Table"][[;; , 2]]}, Select[weirds, (s = DivisorSigma[1, #] - #) <= Last[weirds] && MemberQ[weirds, s] &]]

A138850 Weird numbers (A006037) not divisible by 5.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Apr 02 2008

nonn

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)

Cf. A006037, A100696.

t=0; A138850=vector(50,i,until( (t+=2)%5 && is_A006037(t),);t) \\ [Updated by M. F. Hasler, Jan 07 2014]

A328562 Unitary weird numbers (A064114) that are not weird numbers (A006037).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 19 2019

nonn

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A006037, A013929, A064114, A292705, A328563.

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]
(* after M. F. Hasler's pari code at A006037 *)

A330244 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).

Original entry on oeis.org

70, 10430, 1554070, 5681270, 6365870

Offset: 1

Amiram Eldar, Dec 06 2019

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.

			The abundancy indices of the terms are sigma(a(n))/a(n) = 2.0571... < 2.0709... < 2.0710... < 2.0716... < 2.0716...

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.

Stan Benkoski, Are All Weird Numbers Even?, Problem E2308, The American Mathematical Monthly, Vol. 79, No. 7 (1972), p. 774.
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.

Cf. A000203, A006037.

A275487 Number of weird numbers (A006037) less than 2^n.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Jul 30 2016

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.

Cf. A006037.

print1(s=0);for(k=0,30,print1(","s+=sum(n=1<A006037(2*n))))

a(25)-a(33) from Amiram Eldar, Mar 17 2019

a(34)-a(36) from Amiram Eldar, May 26 2023

A337740 Weird numbers (A006037) with an even sum of divisors that are not Zumkeller numbers (A083207).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 17 2020

nonn

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.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A006037 and A171641.

Cf. A000203, A083207.

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]

A371951 a(n) is the least number whose aliquot sequence begins with exactly n weird numbers (A006037), or -1 if no such numbers exists.

Original entry on oeis.org

1, 70, 97930, 597730, 77420770, 459940810, 11835050710

Offset: 0

Amiram Eldar, Apr 14 2024

a(7) > 10^11, if it exists.

			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

Wikipedia, Aliquot sequence.
Wikipedia, Weird number.

Cf. A001065, A006037, A371950.

cp's OEIS Frontend

A335008 Weird numbers (A006037) with more divisors than any smaller weird number.

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A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

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A329190 Weird admirable numbers: numbers that are both weird (A006037) and admirable (A111592).

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A371950 Weird numbers (A006037) whose sum of aliquot divisors is also a weird number.

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A138850 Weird numbers (A006037) not divisible by 5.

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PARI

A328562 Unitary weird numbers (A064114) that are not weird numbers (A006037).

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A330244 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).

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A275487 Number of weird numbers (A006037) less than 2^n.

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PARI

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A337740 Weird numbers (A006037) with an even sum of divisors that are not Zumkeller numbers (A083207).

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