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A002975
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Primitive weird numbers: weird numbers with no proper weird divisors.
(Formerly M5340)
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40
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70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448
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OFFSET
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1,1
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COMMENTS
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Sidney Kravitz notes that a(21) = 539744; it was misprinted as 539774 in the Benkoski & Erdős article. - Charles R Greathouse IV, Apr 04 2012
It appears that a weird number is primitive iff, divided by its largest prime factor, it is not weird. Is there a simple proof for this? - M. F. Hasler, Aug 20 2014 [The comment below does not answer this question.]
Yes, any primitive weird number, pwn, multiplied by any prime > sigma_1(pwn) is also weird. - Robert G. Wilson v, Jun 09 2015
Number of terms < 10^n: 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881, ..., . - Robert G. Wilson v, Jun 21 2017
The primitive weird number (pwn) 176405960704 is the least term which has as its abundance a pwn. Two other terms are 81152249741312, 14327148694372352. - Robert G. Wilson v, Sep 22 2017
Primitive weird numbers == 2 (mod 4): {70, 4030, 5830, 4199030, 1550860550, 66072609790, ...}. All the terms in A258374 appear so far. - Robert G. Wilson v, Nov 21 2015
Let n be a weird number and d be a divisor of n. If n/d is not weird, then either it is deficient or it is pseudoperfect. But if n/d is pseudoperfect, then multiplying the subset of the divisors of n/d that sums to n/d by d gives a solution for n, contradicting the assumption that n is weird. Therefore, n/d must be deficient. Of all the prime factors of n contributing to sigma(n)/n, the largest prime will contribute the least, and so if n/gpf(n) is deficient, then n/d is deficient for all divisors d of n, and n is a primitive weird number. - Charlie Neder, Oct 08 2018
The second part of the above reasoning is incorrect: gpf(n) may contribute more to sigma(n)/n than a smaller prime factor. For example, for n = 24, we have n/3 deficient, but n/2 abundant; for n = 350, n/7 is deficient but n/5 is abundant. - M. F. Hasler, Jan 25 2020
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B2.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972) 774.
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EXAMPLE
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10430 = A006037(8) is weird but not primitive weird because it has the proper weird divisor 70 = A006037(1).
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MATHEMATICA
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(* first do *) << Combinatorica` (* then *) fQ[n_] := Block[{d = Most@ Divisors@ n, l = 2^(DivisorSigma[0, n] - 1), i = 1}, i = 1; While[i < l && Plus @@ NthSubset[i, d] != n, i++ ]; i == l]; lst = {}; Do[m = n; If[ Mod[n, 6] != 0 && DivisorSigma[1, n] > 2 n && Union[ Mod[ n, Join[lst, {n + 1}]]][[1]] != 0 && fQ@n, AppendTo[lst, n]; Print@n], {n, 2, 42000000, 2}] (* Robert G. Wilson v, Aug 04 2009 *)
(* Input: Range of even numbers --- Output: Primitive weird numbers *)
Block[{$RecursionLimit = Infinity},
subOfSum[ss_, kk_, rr_] :=
Module[{s = ss, k = kk, r = rr},
If[s + w[[k]] >= mm && s + w[[k]] <= m, t = False;
Goto[done] (* Found *),
If[s + w[[k]] + w[[k + 1]] <= m,
subOfSum[s + w[[k]], k + 1, r - w[[k]]]];
If[s + r - w[[k]] >= m && s + w[[k + 1]] <= m,
subOfSum[s, k + 1, r - w[[k]] ]]]; t]; (* end subOfSum *)
greedyQ[ab_] := Module[{abn = ab, v, sum, s, j, jj, k}, tt = False;
jj = Length[w]; (* start search *)
Do[s = r; sum = 0; Do[v = w[[j]]; sum = sum + v;
If[sum > abn, sum = sum - v; Goto[nxt]];
If[sum == abn, tt = True; Goto[doneG]]; s = s - v;
Label[nxt], {j, jj, 1, -1}];
jj = jj - 1, {k, 1, jj - 1}]; Label[doneG];
(* True means found, False not found *) tt]; (* end greedyQ *)
cnt = 0;
Do[ If[ Mod[n, 3] == 0, Goto[agn]]; r = DivisorSigma[1, n];
m = r - 2*n;
If[m > 0, fi = FactorInteger[n]; largestP = fi[[Length[fi]]][[1]];
nn = n/largestP; If[m > 2*nn || Length[fi] < 3, Goto[agn]];
If[DivisorSigma[1, nn] > 2*nn, Goto[agn]]; t = True; r = r - n;
ww = Divisors[n]; lenW = Length[ww];
Do[ If[ ww[[i]] <= m, w = Drop[ww, i - lenW]; Break[],
r = r - ww[[i]]], {i, lenW - 1, 1, -1}];
If[r >= m,
If[ greedyQ[m], t = False, (* Powers of 2 dropped *)
exp2 = fi[[1]][[2]]; sig2 = 2^(exp2 + 1) - 1; mm = m - sig2;
lenW = Length[w]; ww = {};
If[exp2 > 1,
Do[ Do[ If[ w[[i]] == 2^ii, ww = AppendTo[ww, w[[i]]]],
{i, 1, lenW}], {ii, 0, exp2}];
w = Complement[w, ww]
(* end T if *), w = Drop[w, 2]];
(* end Pwr2 *) t = subOfSum[0, 1, r]]]; Label[done];
If[t, Print[++cnt, " ", n, " ", t]]];
Label[agn], {n, 2, 10000000, 2}]]
(* from Brent Baughn via http://mathematica.stackexchange.com/questions/73301/calculating-weird-numbers, Robert G. Wilson v, Nov 21 2015 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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