A327942
Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).
Original entry on oeis.org
165375, 893024, 1047375, 1576575, 2282175, 2304224, 2858624, 3614624, 4068224, 4096575, 4597424, 4975424, 6591375, 7574175, 8555624, 9511424, 10446975, 10749375, 10872224, 11477024, 12535424, 13773375, 13946624, 14277375, 15926624, 16041375, 16505775, 16769024
Offset: 1
165375 is in the sequence since both 165375 and 165376 are nonunitary abundant: nusigma(165375) = 179280 > 165375, and nusigma(165376) = 183600 > 165376 (nusigma is the sum of nonunitary divisors, A048146).
-
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; s = {}; q1 = False; Do[q2 = nuabQ[n]; If[q1 && q2, AppendTo[s, n - 1]]; q1 = q2, {n, 2, 10^7}]; s
A307821
The number of exponential abundant numbers below 10^n.
Original entry on oeis.org
0, 0, 1, 12, 102, 1045, 10449, 104365, 1043641, 10436775, 104367354
Offset: 1
Below 10^3 there is only one exponential abundant number, A129575(1) = 900, thus a(3) = 1.
-
fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ esigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq
A307820
The number of infinitary abundant numbers below 10^n.
Original entry on oeis.org
0, 12, 114, 1270, 12518, 125634, 1257749, 12570993, 125716733, 1256921422, 12570417639
Offset: 1
Below 10^2 there are 12 infinitary abundant numbers, 24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, and 96, thus a(2) = 12.
-
fun[p_, e_] := Module[{ b = IntegerDigits[e, 2]}, m=Length[b]; Product[If[b[[j]] > 0, 1+p^(2^(m-j)), 1], {j, 1, m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ isigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq
A322287
The number of odd abundant numbers below 10^n.
Original entry on oeis.org
0, 0, 1, 23, 210, 1996, 20661, 205366, 2048662, 20502004, 204951472
Offset: 1
945 is the only odd abundant number below 10^3, thus a(3) = 1.
- C. W. Anderson, Density of Deficient Odd Numbers, The American Mathematical Monthly, Vol. 82, No. 10 (1975), pp. 1018-1020.
- Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
-
abQ[n_] := DivisorSigma[1, n] > 2 n; c = 0; k = 1; s = {}; Do[While[k < 10^n, If[abQ[k], c++]; k += 2]; AppendTo[s, c], {n, 1, 5}]; s
A308054
The number of coreful abundant numbers (A308053) below 10^n.
Original entry on oeis.org
0, 1, 24, 259, 2614, 26222, 262220, 2622178, 26221610, 262215860, 2622158194
Offset: 1
Below 10^2 there is only one coreful abundant number, 72, hence a(2) = 1.
-
f[p_, e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); cpQ[n_] := csigma[n] > 2*n; s={0}; c=0; p=100; Do[If[k==p, AppendTo[s, c]; p*=10]; If[cpQ[k], c++], {k, 1, 1000001}]; s
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