A002975 -id:A002975 - cp's OEIS frontend

A258401 Primitive weird numbers (A002975) of the form 2^kpq*x with k >= 0 and odd p, q, x >= 3.

4030, 5830, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 1713592, 4199030, 8812312, 9928792, 11339816, 11547352, 15126992, 17999992, 29465852, 29581424, 38546576, 74899952, 85389368, 89283592, 95327216, 120888092, 141659096, 146764264, 162079768, 173482552

Offset: 1

Robert G. Wilson v, May 28 2015

nonn

The complement of A258882 in A002975, i.e., primitive weird numbers not of the form 2^k*p*q with primes p, q. Equivalently, subsequence of A002975 for numbers with at least 3 odd prime factors, counting multiplicity. (No weird number is of the form 2^k*p^m.) Note that, e.g., a(40) = 2^6 * 137^2 * 1931 and a(143) = 2^8 * 797^2 * 1429 have only 3 distinct prime factors.
Primitive weird numbers of the excluded set (of the form 2^k*p*q, cf. A258882) are well studied and comparably easier to produce, see the Douglas E. Iannucci link; therefore this sequence is noteworthy and harder to produce.
More rare are the primitive weird numbers in which there is an odd prime squared factor, for example:
a(40) = A002975(156) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211,
a(45) = A002975(179) = 2319548096 = 2^6 * 137^2 * 1931,
a(117) = A002975(483) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223,
a(123) = A002975(508) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203,
a(143) = A002975(725) = 232374697216 = 2^8 * 797^2 * 1429.
These PWN with an odd square factor are now listed as A273815. - M. F. Hasler, Jul 10 2016

Robert G. Wilson v, Table of n, a(n) for n = 1..186 (52 terms were from M. F. Hasler)
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Cf. A002975, A258401, A258374, A258375, A258882, A258883, A258884, A258885.

(* copy the terms from A002975, assign them equal to 'pwn' and then *) fQ[n_] :=
Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; Total[Last@# & /@ FactorInteger@ m] > 2]; Select[pwn, fQ] (* Robert G. Wilson v, May 28 2015 and modified Mar 30 2017 *)

select(t->factor(t)[,2][^1]<>[1,1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 11 2016

Edited and definition corrected by M. F. Hasler, Jul 10 2016

A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

Original entry on oeis.org

1550860550, 44257207676, 66072609790

Offset: 1

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

a(4) <= 5976833582079328 = 2^5*181*197*353*431*34429 and a(5) <= 48083019473926272314825065088 = 2^7*257*97213*97973*100957*1520132521 that is certainly in this sequence. - Giuseppe Melfi, Oct 26 2015

a(4) <= 125258675788784 = 2^4 * 47 * 149 * 353 * 1307 * 2423. - M. F. Hasler, Jul 12 2016

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A273815(1). (Abundance = 20)
a(2) = 44257207676 = 2^2 * 11 * 37 * 59 * 523 * 881. (Abundance = 8, cf. A088833)
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A273815(3). (Abundance = 4, cf. A088832)

G. Amato, M. F. 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.
G. Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, 147 (2015) 508-514; Lemma 2.

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

(* copy the terms from A002975, assign them to 'lst' and then *)
Select[ lst, PrimeNu@# == 6 &]

select(w->omega(w)==6, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 12 2016

One more term added and definition corrected by Giuseppe Melfi, Nov 02 2015

A258250 Primitive weird numbers (pwn) (A002975) whose abundance (A033880) is a power of 2 (A000079).

Original entry on oeis.org

70, 836, 4030, 5830, 7192, 7912, 10792, 17272, 45356, 83312, 91388, 113072, 254012, 388076, 786208, 1713592, 4145216, 4559552, 4632896, 9928792, 11547352, 13086016, 15126992, 17999992, 29465852, 29581424, 34869056, 37111168, 38546576, 74899952, 89283592, 95327216

Offset: 1

Robert G. Wilson v, Jun 19 2015

nonn

Number of terms < 10^n: 0, 1, 2, 6, 11, 15, 20, 32, 38, 48, 65, ..., .
Of the total of 499 terms < 10^11 which are pwn, only about 13% have an abundance which are powers of two.
Least term whose abundance has an exponent, e, of two > 1: 70, 836, 7192, 83312, 786208, 4145216, 98196134272, 4559552, 37111168, 22889716736, 141145802752, ?13?, 3307637248, ?15?, 154153326592, ..., .
Least term which has k prime factors, not counting multiplicity > 2: 70, 4030, 29465852, 44257207676, ..., .
Least term which has k prime factors, counting multiplicity > 2: 70, 836, 7192, 83312, 786208, 4145216, 37111168, 270788864, 2529837568, 22889716736, 141145802752, ..., .

			70 is in the sequence since sigma(70) = 144 which yields an abundance of 4 = 2^2.

Robert G. Wilson v, Table of n, a(n) for n = 1..87
Mark Jaybee S. Biano, Bernadette S. Estoque, Lynden Aaron D. Galera, Maria Elena S. Rulloda, Daisy Ann A. Disu, On Weird Numbers, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 157-162.

Cf. A002975, A000079, A033880.

(* copy the terms from A002975 and assign them to lst and then *) f[n_] := DivisorSigma[1, n] - 2n; lst[[#]] & /@ Select[ Range@ 695, IntegerQ@ Log2@ f@ lst[[#]] &]

Corrected by Robert G. Wilson v, Dec 08 2015

A275491 Number of primitive weird numbers (A002975) below 10^n.

Original entry on oeis.org

0, 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN; A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. G. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Partial sums of A275492.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275492, A275493, A275494.

my(s=0); vector(10,n,s+=sum(n=10^n\20+1,10^n\2,is_A002975(n*2)))

vector(10,n,#select(t->t<10^n,A002975)) \\ If A002975 is defined as set, vector, or list with enough terms.

a(12) from Robert G. Wilson v, May 25 2018

a(10) corrected by Amiram Eldar, Sep 02 2023

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]

A275492 Number of primitive weird numbers (A002975) between 10^n and 10^(n+1).

Original entry on oeis.org

0, 1, 1, 5, 6, 11, 24, 37, 67, 124, 223, 382

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN, A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275491, A275493, A275494.

a(n)=sum(n=10^n\2+1,5*10^n,is_A002975(n*2))

Wrong initial term removed and a(11) added by Amiram Eldar, Sep 02 2023

A275493 Number of primitive weird numbers (A002975) below 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 6, 8, 9, 10, 14, 16, 20, 24, 29, 33, 46, 55, 61, 79, 87, 103, 128, 152, 173, 234, 266, 313, 403, 483, 561, 756, 850, 940

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN, A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275494, A275491, A275492.

vector(40,n,#select(t->t<2^n,A002975)) \\ assuming that A002975 is defined as a set, vector or list with enough terms.

a(0) inserted, a(39) corrected, and a(40) added by Amiram Eldar, Sep 02 2023

A275494 Number of primitive weird numbers (A002975) between 2^n and 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 1, 1, 4, 2, 4, 4, 5, 4, 13, 9, 6, 18, 8, 16, 25, 24, 21, 61, 32, 47, 90, 80, 78, 195, 94, 90

Offset: 0

M. F. Hasler, Jul 30 2016

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN, A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

			The first primitive weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792, ..., so there is one between 2^6 and 2^7 = 128, one between 2^9 and 2^10 = 1024, one between 2^11 and 2^12 = 4096, three between 2^12 and 2^13, etc.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Cf. A002975, A006037, A258333, A258882, A275493, A275491, A275492.

PARI
```
a(n)=sum(n=2^n\2+1,2^n,is_A002975(n*2))
```

a(n) = A275493(n+1) - A275493(n).

a(39) from Amiram Eldar, Sep 02 2023

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

A258401 Primitive weird numbers (A002975) of the form 2^k*p*q*x with k >= 0 and odd p, q, x >= 3.

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A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

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A258250 Primitive weird numbers (pwn) (A002975) whose abundance (A033880) is a power of 2 (A000079).

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A275491 Number of primitive weird numbers (A002975) below 10^n.

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

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A275492 Number of primitive weird numbers (A002975) between 10^n and 10^(n+1).

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A275493 Number of primitive weird numbers (A002975) below 2^n.

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A275494 Number of primitive weird numbers (A002975) between 2^n and 2^(n+1).

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A258401 Primitive weird numbers (A002975) of the form 2^kpq*x with k >= 0 and odd p, q, x >= 3.