Search: a335543 -id:a335543
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A335544
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Numbers with more abundant divisors than deficient divisors.
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+10
3
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216, 240, 288, 360, 432, 480, 504, 540, 576, 600, 648, 672, 720, 792, 840, 864, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1728, 1800, 1872, 1920, 1944, 2016, 2112, 2160, 2240, 2268, 2304
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OFFSET
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1,1
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COMMENTS
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This sequence is infinite. For example, 216*p is a term for all primes p.
The least odd term of this sequence is a(16317321) = 638512875.
Apparently, this sequence has an asymptotic density of about 0.025.
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LINKS
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FORMULA
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EXAMPLE
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216 is a term since it has 8 abundant divisors, {12, 18, 24, 36, 54, 72, 108, 216}, and only 7 deficient divisors, {1, 2, 3, 4, 8, 9, 27}.
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MATHEMATICA
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ab[n_] := DivisorSigma[1, n] - 2n; moreAbQ[n_] := Count[(abs = ab/@Divisors[n]), _?(# > 0 &)] > Count[abs, _?(# < 0 &)]; Select[Range[50000], moreAbQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A357460
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Numbers whose number of deficient divisors is equal to their number of nondeficient divisors.
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+10
3
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72, 108, 120, 168, 180, 252, 420, 528, 560, 624, 1188, 1224, 1368, 1400, 1404, 1632, 1656, 1824, 1836, 1960, 1980, 2040, 2052, 2088, 2208, 2232, 2280, 2340, 2484, 2664, 2760, 2772, 2784, 2856, 2952, 2976, 3060, 3096, 3132, 3192, 3200, 3276, 3348, 3384, 3420, 3432
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OFFSET
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1,1
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COMMENTS
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This sequence is infinite: if p >= 17 is a prime then 72*p is a term.
The least odd term of this sequence is a(36126824) = A357461(1) = 3010132125.
Since the number of divisors of any term is even, none of the terms are squares.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 10, 131, 1172, 12003, 120647, 1199147, 11992293, 120089446, ... . Apparently, the asymptotic density of this sequence exists and is equal to about 0.012.
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LINKS
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EXAMPLE
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72 is a term since it has 12 divisors, 6 of them (1, 2, 3, 4, 8 and 9) are deficient and 6 (6, 12, 18, 24, 36 and 72) are not.
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MATHEMATICA
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q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, 1, -1] &] == 0; Select[Range[3500], q]
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PROG
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(PARI) is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A357462
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Numbers whose sum of deficient divisors is equal to their sum of nondeficient divisors.
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+10
3
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6, 28, 30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 308, 318, 330, 354, 364, 366, 390, 402, 426, 438, 462, 474, 476, 496, 498, 510, 532, 534, 546, 570, 582, 606, 618, 642, 644, 654, 678, 690, 714, 726, 750, 762, 786, 798, 812, 822, 834
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OFFSET
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1,1
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COMMENTS
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All the terms are nondeficient numbers (A023196).
All the perfect numbers (A000396) are terms.
This sequence is infinite: if k = 2^(p-1)*(2^p-1) is an even perfect number and q > 2^p-1 is a prime, then k*q is a term.
Since the total sum of divisors of any term is even, none of the terms are squares or twice squares.
Are there odd terms in this sequence? There are none below 10^10.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 6, 63, 605, 6164, 61291, 614045, 6139193, 61382607, 613861703, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06138... .
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LINKS
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EXAMPLE
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6 is a term since the sum of its deficient divisors, 1 + 2 + 3 is equal to 6, its only nondeficient divisor.
30 is a term since the sum of its deficient divisors, 1 + 2 + 3 + 5 + 10 + 15 = 36 is equal to the sum of its nondeficient divisors, 6 + 30 = 36.
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MATHEMATICA
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q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, #, -#] &] == 0; Select[Range[1000], q]
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PROG
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(PARI) is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, d, -d)) == 0;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A357461
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Odd numbers whose number of deficient divisors is equal to their number of nondeficient divisors.
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+10
2
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3010132125, 4502334375, 5065535475, 6456074625, 8813660625, 9881746875, 15395254875, 15452011575, 16874983125, 18699305625, 19814169375, 19909992375, 21380506875, 25366375125, 26643400875, 26746594875, 28943578125, 31562182575, 33074966925, 34315506225, 35300640375
(list;
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refs;
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OFFSET
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1,1
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COMMENTS
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If there are no odd perfect numbers, then this sequence is also the subsequence of the odd terms of A335543.
The first 100 terms are all divisible by 4725 = 3^3 * 5^2 * 7.
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LINKS
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PROG
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(PARI) is(n) = n%2 && sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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