A258250 -id:A258250 - cp's OEIS frontend

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

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[#] &]

A265727 Least primitive weird number, pwn, (A002975) which is divisible by the n-th prime (A000040).

Original entry on oeis.org

70, 70, 836, 4030, 17272, 836, 7912, 7192, 4030, 113072, 83312, 7912, 8812312, 5830, 4199030, 9272, 91388, 10792, 23941578736, 786208, 682592, 569494624, 555616, 539744, 15126992, 73616, 519712

Offset: 3

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

As of Dec 14 2015, there is no known pwn which is divisible by 3. Therefore the offset denotes the third prime number, 5.

			a(6) is 4030 since it is the first pwn to be divisible by the sixth prime number, 13. 4030 = 13 * 310.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Linked In, Number Theory, A very big weird number
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pp. 508-514.
Wikipedia, Weird number.

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265726, A265728.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := Select[lst, Mod[#, Prime@ n] == 0 &][[1]]; Array[f, 27, 3]

A354282 Weird numbers k such that k+1 is the sum of a subset of the aliquot divisors of k.

Original entry on oeis.org

70, 836, 4030, 5830, 7192, 7912, 10792, 17272, 45356, 83312, 91388, 113072, 222952, 243892, 254012, 388076, 410476, 786208, 1713592, 4145216, 4199030, 4632896, 6911512, 7257530, 7354304, 7607530, 9928792, 10402490, 10580624, 11339816, 11547352, 12052390, 13086016

Offset: 1

Amiram Eldar, May 22 2022

nonn

First differs from A258250 and A329190 at n=13.

There are 17270452 weird numbers below 10^10 and only 94 are in this sequence.

			70 is a term since it is a weird number, its aliquot divisors are {1, 2, 5, 7, 10, 14, 35} and 71 = 5 + 7 + 10 + 14 + 35.

Amiram Eldar, Table of n, a(n) for n = 1..94

Subsequence of A006037.
A354283 is a subsequence.
Cf. A005835, A258250, A329190, A354281.

q[n_] := Module[{d = Most @ Divisors[n], x, s, c}, If[Plus @@ d <= n, False, s = Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n + 1}]; c = SeriesCoefficient[s, #] & /@ (n + {0, 1}); c[[1]] == 0 && c[[2]] > 0]]; Select[Range[10000], q]

is(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n, d, s) && is_A005835(n+1, d, s)}; \\ using is_A005835() by M. F. Hasler at A005835

A265726 Primitive weird numbers whose abundance is a record.

Original entry on oeis.org

70, 836, 7192, 9272, 73616, 243892, 338572, 1188256, 1901728, 3963968, 28279232, 36228736, 91322752, 141659096, 263144192, 351295232, 664373504, 2113834496, 5522263024, 6933503488, 19179527168, 22755515392, 31574500724, 98620009472, 135895635968

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

Although the abundance A(n) = sigma(n) - 2n is increasing, the (relative) abundancy sigma(n)/n is decreasing, except at indices {3, 6, 8, 15, 16, 19, 24 ...}. No term has larger abundancy than 2 + 2/35, that of a(1). - M. F. Hasler, Nov 14 2018

			a(1) = 70 since it is the first term in A002975; its abundance is 4.
a(2) is 836 since its abundance, 8, exceeds that of a(1); 4.
a(3) is 7192 = A002975(5) since its abundance, 16, exceeds that of a(2) and that of A002975(1..4).

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pgs 508-514.
Wikipedia, Weird number

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265727, A265728.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := DivisorSigma[1, n] - 2n; k = 1; lsu = {}; mx = 0; While[k < 647, ds = f@ lst[[k]]; If[ds > mx, mx = ds; AppendTo[lsu, lst[[k]]]]; k++]; lsu

A265728 Least primitive weird number, pwn, (A002975) whose abundance is divisible by the n-th prime (A000040), or 0 if no such pwn exists.

Original entry on oeis.org

70, 232374697216, 73616, 9272, 243892, 343876, 4128448, 519712, 1901728, 338572, 5568448, 6621632, 272240768, 4960448, 7470272, 1673087984, 146279296, 5440192, 91322752, 8134208, 35442304, 286717696, 54962343424, 110232704, 6460864, 2812606976, 44473216, 141659096, 33736064, 58668928, 9537494528, 37499776, 292335872, 795730688, 530110208, 18657360896, 16995175424, 664373504, 266311424, 23049995264, 15152370176, 17124699136, 64015565312, 52059008

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

			a(1) = 70 since it is the least pwn whose abundance, 4, is divisible by the first prime, 2.
a(2) = 0 since there is no known odd pwn and if there were, there is no reason why the abundance would be == 0 (mod 3).
a(3) = 73616 since it is the first pwn whose abundance, 80, is divisible by the third prime, 5.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Linked In, Number Theory, A very big weird number
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pp. 508-514.
Wikipedia, Weird number

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265726, A265727.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := Select[lst, Mod[ DivisorSigma[1, #] - 2#, Prime@ n] == 0 &][[1]]; Array[f, 30]

A319735 Primitive weird numbers (pwn; A002975) congruent to 2 mod 4.

Original entry on oeis.org

70, 4030, 5830, 4199030, 1550860550, 66072609790

Offset: 1

M. F. Hasler and Robert G. Wilson v, Sep 26 2018

Primitive weird numbers divisible by 2 but not by 4.

10805836895078390 = 2 * 5 * 11 * 89 * 167 * 829 * 7972687 is a term.

			a(1) is 70 = 2 * 5 * 7 with abundance of 4;
a(2) is 4030 = 2 * 5 * 13 * 31 with abundance of 4;
a(3) is 5830 = 2 * 5 * 11 * 53 with abundance of 4;
a(4) is 4199030 = 2 * 5 * 11 * 59 * 647 with abundance of 20;
a(5) is 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 with abundance of 20;
a(6) is 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 with abundance of 4; etc.
From _M. F. Hasler_, Nov 28 2018: (Start)
The larger terms are in other sequences related to PWN with many prime factors. We have the following relations:
   a(3) = 70 = A258882(1) = A258374(3) = A258250(1) = A002975(1).
   a(3) = 4030 = A258883(1) = A258374(4) = A258401(1) = A258250(3) = A002975(3).
   a(3) = 5830 = A258883(2) = A258401(2) = A258250(4) = A002975(4).
   a(4) = 4199030 = A258884(1) = A258374(5) = A258401(11) = A265727(15).
   a(5) = 1550860550 = A258885(1) = A273815(1) = A258374(6).
   a(6) = 66072609790 = A258885(3) = A273815(3). (End)

Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton. Primitive weird numbers having more than three distinct prime factors. Rivista di Matematica della Università degli studi di Parma, 2016, 7(1), pp. 153-163. (hal-01684543)

G. Amato, M. Hasler, G. Melfi and M. Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.
Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972) 774.
S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.
Eric Weisstein's World of Mathematics, Weird Number.
Wikipedia, Weird number

Cf. A002975, A258375, A258401, A258882, A258883, A258884, A258885.

(* import the b-file in A002975 and assign it to lst *);
Select[lst, IntegerExponent[#, 2] == 1 &]

cp's OEIS Frontend

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

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A265727 Least primitive weird number, pwn, (A002975) which is divisible by the n-th prime (A000040).

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A354282 Weird numbers k such that k+1 is the sum of a subset of the aliquot divisors of k.

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A265726 Primitive weird numbers whose abundance is a record.

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A265728 Least primitive weird number, pwn, (A002975) whose abundance is divisible by the n-th prime (A000040), or 0 if no such pwn exists.

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A319735 Primitive weird numbers (pwn; A002975) congruent to 2 mod 4.

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