Displaying 51-60 of 73 results found.
Odd numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.
+10
2
245025, 540225, 893025, 2205225, 3080025, 4862025, 6125625, 6890625, 7868025, 10989225, 13505625, 14402025, 19847025, 22896225, 23474025, 26471025, 27720225, 29648025, 43758225, 45765225, 55130625, 57836025, 60140025, 65367225, 70812225, 72335025, 76475025, 77000625, 94770225, 121550625, 153140625, 156125025
COMMENTS
Odd numbers k such that A033880(k) is positive but A342926(k) is negative. This is a subsequence of A156942, "odd abundant numbers whose abundance is odd". Proof: If sigma(k) > 2*k, and sigma(k) were even, then sigma(k)/2 would be an integer and a divisor of sigma(k), and we could compute A003415(sigma(k)) as A003415(2)*(sigma(k)/2) + 2* A003415(sigma(k)/2) by the definition of the arithmetic derivative. But that value is certainly larger than k, because sigma(k)/2 > k, therefore sigma(k) must be an odd number, with also its abundance sigma(k)-(2k) odd. This also entails that all terms are squares. See A347891 for the square roots. The first term that is not a multiple of 25 is a(146) = 6800806089 = 82467^2.
This is not a subsequence of A325311. The first term that is not present there is a(5) = 3080025.
PROG
(PARI)
\\ Using the program given in A347891 would be much faster than this: A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347890(n) = ((n%2)&&( A003415(sigma(n))<n)&&(sigma(n)>(2*n)));
155925, 225225, 259875, 294525, 297675, 363825, 405405, 429975, 496125, 552825, 562275, 571725, 606375, 628425, 694575, 760725, 765765, 921375, 945945, 987525, 1044225, 1167075, 1195425, 1334025, 1412775, 1447875, 1486485, 1507275, 1526175, 1611225, 1640925, 1645875
COMMENTS
Since any positive odd multiple of a term of A347936 is also a term of A347936, the sequence A347936 consists of the positive odd multiple of this sequence.
EXAMPLE
The first 8 terms of this sequence are the same as those of A347936. But A347936(9) = 467775 = 3 * 155925 = 3 * A347936(1) is not a term of this sequence.
MATHEMATICA
abQ[n_] := DivisorSigma[1, n] > 2*n; s[n_] := DivisorSum[n, # &, abQ[#] &]; q[n_] := s[n] > 2*n && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1, 300000, 2], q]
Odd numbers k such that A183097(k) > 2*k.
+10
2
3472875, 10418625, 17364375, 24310125, 31255875, 52093125, 72930375, 86821875, 93767625, 121550625, 156279375, 170170875, 202145625, 218791125, 260465625, 281302875, 364651875, 420217875, 434109375, 468838125, 510512625, 586915875, 606436875, 607753125, 656373375
EXAMPLE
3472875 is a term since A183097(3472875) = 7002474 > 2*3472875 = 6945750.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 2.5*10^7, 2], q]
Odd numbers whose number of deficient divisors is equal to their number of nondeficient divisors.
+10
2
3010132125, 4502334375, 5065535475, 6456074625, 8813660625, 9881746875, 15395254875, 15452011575, 16874983125, 18699305625, 19814169375, 19909992375, 21380506875, 25366375125, 26643400875, 26746594875, 28943578125, 31562182575, 33074966925, 34315506225, 35300640375
COMMENTS
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.
PROG
(PARI) is(n) = n%2 && sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;
Odd numbers k such that A162296(k) > 2*k.
+10
2
4725, 6615, 7875, 8505, 11025, 14175, 15435, 17325, 19845, 20475, 22275, 23625, 24255, 25515, 26775, 28665, 29925, 31185, 33075, 36225, 36855, 37125, 37485, 38115, 39375, 40425, 41895, 42525, 46305, 47775, 48195, 50715, 51975, 53235, 53865, 55125, 57915, 59535
COMMENTS
The least term that is not divisible by 3 is a(89047132) = 134785275625.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 4, 60, 640, 6650, 66044, 660230, 6604594, 66073470, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000660... .
EXAMPLE
4725 is a term since it is odd, and A162296(4725) = 9728 > 2*4725.
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[3, 60000, 2], q]
Odd cubefree abundant numbers.
+10
2
1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775
COMMENTS
First differs from A333950 at n = 1258. Terms that are not in A333950 include 8564325, 8565795, 8567325, ... and terms of A333950 that are not here include 1126125, 2096325, 2207205, ... . The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 16, 125, 1127, 11734, 116911, 1162781, 11638566, 116342286, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00116... .
EXAMPLE
1575 = 3^2 * 5^2 * 7 is a term since it is odd and cubefree and sigma(1575) = 3224 > 2*1575.
MATHEMATICA
f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = 0; q[n_] := AllTrue[(fct = FactorInteger[n])[[;; , 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[1, 30000, 2], q]
PROG
(PARI) is(n) = {my(f); if(n%2 == 0, return(0)); f = factor(n); (n==1 || vecmax(f[, 2]) < 3) && sigma(f, -1) > 2};
Odd numbers k such that A246601(k) > 2*k.
+10
2
4095, 16777215, 33550335, 67096575, 134189055, 268374015, 536743935, 1073483775, 2146963455, 4293922815, 8587841535, 17175678975, 34351353855, 68702703615, 68719476735, 137405403135, 137422176255, 137438949375, 274810802175, 274827575295, 274844348415, 274877894655
COMMENTS
These are the odd terms of A359084 and also its primitive terms, since if m is a term then m*2^k is a term of A359084 for all k >= 0. The least term that is not divisible by 4095 is a(29) = 1099511627775 = 2^40 - 1.
MATHEMATICA
s[n_] := DivisorSum[n, # &, BitAnd[n, #] == # &]; Select[Range[1, 2^24, 2], s[#] > 2*# &]
PROG
(PARI) is(n) = n%2 && sumdiv(n, d, d * (bitor(n, d) == n)) > 2*n;
Numbers k such that A360327(k) > 2*k.
+10
2
7425, 8415, 22275, 25245, 37125, 42075, 46035, 66825, 75735, 76725, 81675, 92565, 101475, 111375, 126225, 138105, 143055, 182655, 185625, 200475, 210375, 227205, 230175, 245025, 260865, 277695, 304425, 334125, 345015, 355725, 378675, 383625, 408375, 414315, 429165
COMMENTS
Analogous to abundant numbers ( A005101) with divisors that are restricted to numbers that have only prime-indexed prime factors. The abundancy index of numbers in A076610 (i.e., numbers whose prime factors are only prime-indexed primes) is bounded by P = Product_{p in A006450} p/(p-1) which seems to be less than 4 (see A076610). Therefore, there are no terms k of A076610 with sigma(k) >= 4*k, or equivalently, no even terms in this sequence, and all the terms of this sequence are in A076610. Also, assuming that P < 15/4 = 3.75, there are no terms in this sequence that are coprime to 15. Since P > 3 there are terms that are not divisible by 3. The least of them must be larger than Product_{k=1..21826870} A006450(k) = 3 * 5 * 11 * ... * 8958801613 > 10^206662375, because Product_{k=2..m} A006450(k)/( A006450(k)-1) > 2 only for m >= 21826870. The least term that is not divisible by 5 is 789909738655399955305165431.
The least term that is not divisible by 11 is a(30) = 355725.
The least squarefree term is 14093057715.
MATHEMATICA
f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], s[#] > 2*# &]
PROG
(PARI) is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n; }
Odd numbers k such that A360522(k) > 2*k.
+10
2
15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655
EXAMPLE
15015 is a term since A360522(15015) = 32256 > 2*15015.
MATHEMATICA
f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 10^5, 2], q]
PROG
(PARI) isab(n) = { my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + f[i, 2]) > 2*n; }
is(n) = n%2 && isab(n);
944, 1574, 2204, 2834, 3464, 4094, 4724, 5354, 5774, 5984, 6434, 6614, 6824, 7244, 7424, 7874, 8084, 8414, 8504, 8924, 9134, 9554, 9764, 10394, 11024, 11654, 12284, 12704, 12914, 13544, 14174, 14804, 15014, 15434, 16064, 16694, 17324, 17954, 18584, 19214, 19304
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