Douglas E. Iannucci - cp's OEIS frontend

A275986 Positive integers of the form x*10^k + y which also equal x^2 + y^2 (x, y and k being positive integers).

101, 1233, 8833, 10001, 10100, 990100, 1000001, 5882353, 94122353, 99009901, 100000001, 100010000, 1765038125, 2584043776, 7416043776, 8235038125, 9901009901, 10000000001, 48600220401, 116788321168, 123288328768, 601300773101, 876712328768, 883212321168, 990100990100, 999900010000, 1000000000001, 1000001000000

Offset: 1

Douglas E. Iannucci, Aug 15 2016

nonn

The condition x^2 + y^2 = x*10^k + y is equivalent to (2x-10^k)^2 + (2y-1)^2 = 10^2k + 1, so to find these sequence elements it is necessary to write 10^2k + 1 as the sum of two squares.
The number of elements in this sequence corresponding to a fixed k is tau(10^2k + 1) - 1, where tau counts the (positive) divisors of a natural number. For all k, 10^2k + 1 is itself a member of the sequence corresponding to k, and is the only one such if it is prime. The elements themselves are arranged according to magnitude, indexed here by n. There is some disruption of the order of the terms versus the corresponding exponent k. For example, the twelfth member of the sequence, 100010000, corresponds to k=6, yet the thirteenth, 1765038125, corresponds to the smaller k=5.
Contains 10^(2*i) + 10^(4*i) and 10^(6*i) - 10^(4*i) + 10^(2*i) for each i >= 1 (corresponding to k = 3*i). - Robert Israel, Mar 30 2017

			a(1) = 101 corresponds to k = 1, x = 10, and y = 1.
a(2) = 1233 corresponds to k = 2, x = 12, y = 33.

Steven Charlton, Square sum concatenation - Number theory challenge
A. van der Poorten, K. Thomsen, and M. Wiebe, A curious cubic identity and self-similar sums of squares, The Mathematical Intelligencer, v.29(2), pp. 69-73, June 2007.

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]

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]

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

A258884 Primitive weird numbers, pwn, of the form 2^kpqrs with k > 0 and where p < q < r < s are odd primes.

Original entry on oeis.org

4199030, 29465852, 120888092, 259858324, 260378492, 410832532, 775397948, 785187524, 903217276, 989226964, 1609445332, 2358115084, 3254323124, 3381352084, 3381872252, 3781448788, 3782267372, 5056717796, 5065605532, 5066413508, 9210347984, 9772585048, 9776437996, 9843119884

Offset: 1

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

nonn

This differs from the sequence of primitive weird numbers with 5 (or 4 odd) distinct prime factors from a(54) on, the 54th number of that form being 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203. - M. F. Hasler, Jul 08 2016

			a(1) = 4199030 = 2*5*11*59*647.
a(2) = 29465852 = 2^2*13*23*71*347.
a(3) = 120888092 = 2^2*13*23*61*1657.

M. F. Hasler, Table of n, a(n) for n = 1..60

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

select(t -> factor(t)[,2][^1]~ == [1,1,1,1], A002975) \\ Assuming that A002975 is defined as vector holding enough terms of that sequence. - M. F. Hasler, Jul 08 2016

Definition edited (to require p > 2) by M. F. Hasler, Jul 08 2016

Incorrect Mathematica code (allowed 114141404156) removed by M. F. Hasler, Jun 29 2016

A258883 Primitive weird numbers (PWN) of the form 2^kpq*r with k > 0 and where p < q < r are odd primes.

Original entry on oeis.org

4030, 5830, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 1713592, 8812312, 9928792, 11339816, 11547352, 15126992, 17999992, 29581424, 38546576, 74899952, 85389368, 89283592, 95327216, 141659096, 146764264, 162079768, 173482552, 569494624, 632874016

Offset: 1

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

nonn

The condition k > 0 is not really a limitation since a product of three odd primes cannot be weird. -- Numbers of the form 2^k*p^2*q having only two distinct odd prime divisors, e.g., A258401(45) = 2319548096 = 2^6 * 137^2 * 1931 or A258401(143) = 232374697216 = 2^8 * 797^2 * 1429, are neither in A258882 nor in the present sequence as it is currently defined, although they are in the set of weird numbers 2^k*p*q*r with odd primes p,q,r. (PWN with nonsquarefree odd part are listed in A273815.) - M. F. Hasler, Jul 18 2016, amended Nov 09 2017

It appears that there are (2, 7, 12, 18, 41, ...) terms with k = valuation(a(n),2) = 1, 2, 3, etc. The smallest and largest such are (4030, 45356, 1713592, 15126992, 569494624, 5353519168, 96743686016, 1009572479744, ...) resp. (5830, 388076, 173482552, 6587973136, 297512429728, ...). - M. F. Hasler, Nov 09 2017

			a(1) = 4030 = 2*5*13*31.
a(2) = 5830 = 2*5*11*53.
a(3) = 45356 = 2^2*17*23*29.

M. F. Hasler, Table of n, a(n) for n = 1..121 (Using A002975(1..1073) calculated by _Robert G. Wilson v_.)

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

(* copy the terms from A002975, assign them to 'lst' and then *) Select[ lst, PrimeNu@# == 4 &] (* WARNING: this code selects PWN with 3 distinct odd prime factors but does not exclude that they occur with multiplicity > 1, which is forbidden by definition of this sequence. - M. F. Hasler, Jul 12 2016 *)

select(w->factor(w)[,2][^1]~==[1,1,1], A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 12 2016

A258882 Primitive weird numbers of the form 2^kpq with k > 0 and where p < q are odd primes.

Original entry on oeis.org

70, 836, 7192, 7912, 9272, 10792, 17272, 73616, 83312, 113072, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1901728, 2081824, 2189024, 3963968, 4128448, 4145216, 4486208, 4559552, 4632896, 4960448, 5440192, 5568448, 6460864, 6621632, 7354304, 7470272, 8000704, 8134208

Offset: 1

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

nonn

The number of terms < 10^n: 0, 1, 2, 5, 9, 15, 35, 61, 114, 204, 380, 696, 1703, 3548, 6726, 13137, ....
If 2^k*p*q is a weird number, it is necessarily primitive, and 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
No odd weird numbers are known and any even weird number must have at least 3 distinct prime factors, since all numbers of the form 2^k*p^m are deficient or pseudoperfect or perfect (iff m = 1 and p = 2^(k+1)-1 is a Mersenne prime). Sequence A258333 lists the number of terms in this sequence for given k. - M. F. Hasler, Jul 11 2016
Kravitz has shown that 2^k*p*q is a primitive weird number when the primes p and q satisfy p = (2^(k+1)*q-q-1)/(q+1-2^(k+1)). Many terms in this sequence are of this form, e.g., a(n) with n = 1, 2, 3, 4, 6, 7, 9, 10, 15, 23, 26, 38, 45, 75, 94, 144, 157, 187, 287, 327, 368, 370, 459, 607, 657, 658, .... Sequences A242025, A242998, ... are related to the special case where q is a Mersenne prime (A000668). - M. F. Hasler, Jul 12 2016
Weird numbers of the form 2^k*p*q are always primitive, so this condition could be omitted in the definition of this sequence. - M. F. Hasler, Jul 13 2016
About 35 years after Kravitz's work, the topic of weird numbers has regained interest after a CWU press release about students who used Kravitz's formula to find a large PWN of this form. See A242025 and A320875. - M. F. Hasler, Nov 20 2018

			a(1) = A002975(1) = 70 = 2*5*7.
a(2) = A002975(2) = 836 = 2^2*11*19.
A002975(3) = 4030 = 2*5*13*31 is not in this sequence since it is not of the required form.
The same is true for A002975(4) = 5830.
a(3) = A002975(5) = 7192 = 2^3*29*31, etc.
A002975(179) = 2319548096 = 2^6 * 137^2 * 1931 is the first term of A002975 with only two odd prime divisors, but not of the required form. - _M. F. Hasler_, Nov 20 2018

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9 (1976), 82-85 (1977). Zbl 0365.10003

Douglas E. Iannucci and Robert G. Wilson v, Table of n, a(n) for n = 1..15384, updated Dec 06 2015; corrected by _M. F. Hasler_, Jul 16 2016
R. Bagula et al., A very big weird number, Number Theory group on LinkedIn (web.archive.org snapshot; page no longer available). Dec. 2013
Central Washington University, CWU Math Students Break World Record for Largest Weird Number [alternate article]
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, pp 508-514.
Eric Weisstein's World of Mathematics, Weird Number.
Wikipedia, Weird number

Cf. A002975, A258401 (PWN not of this form), A258374, A258375, A258883, A258884, A258885.
Cf. A242025, A242993, A242998, A242999, A243003 (related to the subsequence with q = (2^k*p-p-1)/(p+1-2^k) and p a Mersenne prime in A000668).
Cf. A320875 (more general case of Karavitz' formula).

(* copy the terms from A002975, assign them equal to 'lst' and then *) fQ[n_] := Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; PrimeOmega@ m == 2]; Select[lst, fQ]

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

A258882 union A258401 is A002975.

Edited by M. F. Hasler, Jul 11 2016, Nov 20 2018

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

A258333 Number of (primitive) weird numbers of the form 2^npq, with odd primes p < q.

Original entry on oeis.org

1, 1, 5, 3, 10, 23, 29, 53, 115, 210, 394, 683, 1389, 3118, 6507, 9120

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, May 27 2015

Sequence taken from page 3 of "On primitive weird numbers of the form 2^k*p*q".
The (primitive) weird numbers considered here are listed in A258882, a proper subset of A002975.
If 2^k*p*q is weird, then 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
This being the case the number of possible pwn of the form 2^n*p*q with p unique is: 1, 2, 4, 7, 12, 23, 43, 75, 137, 255, 463, 872, 1612, 3030, 5708, ....
However, p is usually not unique, e.g., for k=3, p=19 we have two pwn (with q=61 and q=71), and for k=5, p=71 yields two pwn (for q=523 and q=541) and p=67 yields three pwn (for q=887, 971 and 1021). I conjecture that there is an increasing number of pwn with, e.g., p=nextprime(2^(k+1)). Also, if 2^k p q and 2^k p' q are both weird, then usually 2^k p" q is weird for all p" between p and p'. There is one exception [p, p', q] = [2713, 2729, 8191] for k=10, five exceptions [6197, 6203, 12049], [6113, 6131, 12289], [6113, 6131, 12301], [6121, 6133, 12323], [5441, 5449, 16411] for k=11, and seven exceptions for k=12. These exceptions occur when q/p is close to an integer, (p, q) ~ (3/4, 3/2)*2^(k+2) or (2/3, 2)*2^(k+2). - M. F. Hasler, Jul 16 2016

			The only primitive weird number of the form 2*p*q is 70 so a(1) = 1;
The only primitive weird number of the form 2^2*p*q is 836 so a(2) = 1;
There are 5 primitive weird numbers of the form 2^3*p*q and they are 5704, 7912, 9272, 10792 & 17272; so a(3) = 5; etc.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Cf. A002975, A258882.

A258333(n)={ local(s=0,p,M=2^(n+1)-1,qn,T(P=p-1)=is_A006037(qn*p=precprime(P)) && s+=1); forprime(q=2*M,M*(M+1), qn=q<M, T() || T() || break)); s} \\ Not very efficient, for illustrative purpose only. - M. F. Hasler, Jul 18 2016

a(15) from Robert G. Wilson v, Jun 14 2015

a(16) from Robert G. Wilson v, Dec 06 2015

A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

Original entry on oeis.org

1, 12, 24, 72, 168, 240, 336, 360, 504, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 15120, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800

Offset: 1

Douglas E. Iannucci, Oct 22 2008

nonn

Sequence A206027 has the number of solutions.

			sigma(m)=1 has only one solution: m=1.
sigma(m)=12 has two solutions, m=6 and m=11; 12 is the smallest number with more than one such solutions.
sigma(m)=24 has three solutions, m=14,m=15 and m=23; 24 is the smallest number with more than two such solutions.
sigma(m)=72 has five solutions, m=30, m=46, m=51, m=55 and m=71; 72 is the smallest number with more than three such solutions.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000203 (sum of divisors of n), A054973 (number of numbers whose divisors sum to n), A007368 (smallest k such that sigma(x) = k has exactly n solutions).
Cf. A206027.
Cf. Untouchable numbers (A005114), sigma-untouchable numbers (A007369) and highly touchable numbers (A238895).

t = DivisorSigma[1, Range[10^6]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)

{m=3650000; v=vectorsmall(m); for(n=1, m, s=sigma(n); if(s<=m, v[s]++)); g=0; j=1; while(j<=m, if(v[j]<=g, j++, g=v[j]; print1(j, ",")))} \\ Klaus Brockhaus, Oct 27 2008

Extended beyond a(15) by Klaus Brockhaus, Oct 27 2008

cp's OEIS Frontend

User: Douglas E. Iannucci

A275986 Positive integers of the form x*10^k + y which also equal x^2 + y^2 (x, y and k being positive integers).

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

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A265726 Primitive weird numbers whose abundance is a record.

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A258884 Primitive weird numbers, pwn, of the form 2^k*p*q*r*s with k > 0 and where p < q < r < s are odd primes.

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A258883 Primitive weird numbers (PWN) of the form 2^k*p*q*r with k > 0 and where p < q < r are odd primes.

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A258882 Primitive weird numbers of the form 2^k*p*q with k > 0 and where p < q are odd primes.

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

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A258884 Primitive weird numbers, pwn, of the form 2^kpqrs with k > 0 and where p < q < r < s are odd primes.

A258883 Primitive weird numbers (PWN) of the form 2^kpq*r with k > 0 and where p < q < r are odd primes.

A258882 Primitive weird numbers of the form 2^kpq with k > 0 and where p < q are odd primes.

A258333 Number of (primitive) weird numbers of the form 2^npq, with odd primes p < q.