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Revision History for A286652

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A286652

Unitary practical numbers: numbers n such that every 1 <= k <= usigma(n) is a sum of distinct unitary divisors of n.
(history; published version)

#17 by N. J. A. Sloane at Fri Jun 02 00:42:51 EDT 2017

STATUS

proposed

approved

#16 by Amiram Eldar at Tue May 30 11:35:17 EDT 2017

STATUS

editing

proposed

#15 by Amiram Eldar at Tue May 30 11:35:07 EDT 2017

LINKS

Amiram Eldar, <a href="/A286652/b286652.txt">Table of n, a(n) for n = 1..1000</a>

STATUS

approved

editing

#14 by N. J. A. Sloane at Sat May 27 07:59:01 EDT 2017

STATUS

proposed

approved

#13 by Amiram Eldar at Sat May 27 03:46:22 EDT 2017

STATUS

editing

proposed

#12 by Amiram Eldar at Sat May 27 03:35:17 EDT 2017

NAME

allocated for Amiram EldarUnitary practical numbers: numbers n such that every 1 <= k <= usigma(n) is a sum of distinct unitary divisors of n.

DATA

1, 2, 6, 30, 42, 66, 78, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1050, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1326, 1410, 1470, 1482, 1518, 1554, 1590, 1650, 1722, 1770, 1794, 1806, 1830, 1914, 1950, 1974, 2010, 2046, 2130

OFFSET

1,2

COMMENTS

The unitary version of A005153. The squarefree terms of both sequences are the same, A265501. The nonsquarefree terms of this sequence are in A287173.

MATHEMATICA

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[#, n/#] == 1 &]]; uPracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; r = Sort[p^e]; Do[If[r[[i]] > 1 + usigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[p]}]; ok]]]; Select[ Range[100000], uPracticalQ]

CROSSREFS

Cf. A005153, A034448, A265501, A287173.

KEYWORD

allocated

nonn

AUTHOR

Amiram Eldar, May 27 2017

STATUS

approved

editing

#11 by Amiram Eldar at Sat May 27 03:35:17 EDT 2017

NAME

allocated for Amiram Eldar

KEYWORD

recycled

allocated

#10 by N. J. A. Sloane at Fri May 26 21:35:05 EDT 2017

STATUS

proposed

approved

#9 by Joerg Arndt at Fri May 26 10:52:47 EDT 2017

STATUS

editing

proposed

#8 by Joerg Arndt at Fri May 26 10:52:43 EDT 2017

NAME

Gaps in each minimal positive integer arithmetic sequence where any two members have GCD=1.

DATA

1, 1, 2, 2, 6, 6, 6, 6, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 510510, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870

OFFSET

1,3

COMMENTS

a(n) is the gap between consecutive members of each arithmetic sequence of length N with smallest natural numbers not including 1. - Ophir Spector, 11 May 2017

For a coprime positive integer sequence of length n, no member can have the prime factors n/2 or less. As a result, the gap must be the product of the first primes up to n/2. - Ophir Spector, 11 May 2017

LINKS

<a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm">Extended Euclidean algorithm</a>

<a href="http://mathworld.wolfram.com/DiophantineEquation.html">Diophantine equations</a>

<a href="https://en.wikipedia.org/wiki/Modular_multiplicative_inverse">Modular arithmetic</a>

EXAMPLE

First sequence, of length 2, has a(1)=2 and gap=1, with the sequence {2,3}.

The next sequences are {3,4,5}, {5,7,9,11}, {5,7,9,11,13}, {7,13,19,25,31,37}.

CROSSREFS

A286650, A002110, A034386.

KEYWORD

nonn,easy,nice,changed

recycled

AUTHOR

Ophir Spector, May 11 2017

STATUS

proposed

editing