A333951
Numbers k such that both k and k+1 are recursive abundant numbers (A333928).
Original entry on oeis.org
56924, 82004, 84524, 109395, 158235, 241604, 261260, 266475, 285075, 361844, 442035, 445004, 469755, 611324, 666315, 694484, 712844, 922635, 968715, 971684, 1102724, 1172115, 1190475, 1199835, 1239524, 1304324, 1338435, 1430715, 1442924, 1486275, 1523115, 1550835
Offset: 1
56924 is a term since A333926(56924) = 120960 > 2 * 56924, and A333926(56925) = 116064 > 2 * 56925.
-
recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); recAbQ[n_] := recDivSum[n] > 2*n; Select[Range[2*10^5], recAbQ[#] && recAbQ[# + 1] &]
A357608
Numbers k such that k and k+1 are both in A357605.
Original entry on oeis.org
76544, 104895, 126224, 165375, 170624, 174824, 201824, 245024, 257984, 271215, 273104, 316575, 338624, 387855, 447615, 469664, 477224, 540224, 618975, 633555, 641024, 659295, 705375, 752895, 770175, 842624, 843975, 862784, 870975, 893024, 913275, 957824, 1047375
Offset: 1
76544 is a term since 76544 and 76545 are both in A357605: A162296(76544) = 170688 > 2*76544 and A162296(76545) = 157248 > 2*76545.
-
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[2, 10^6], q[#] && q[#+1] &]
A335251
Numbers k such that k and k+1 have the same unitary abundance (A129468).
Original entry on oeis.org
1, 20, 35, 143, 208, 2623, 5183, 27796, 11177983, 69677008, 920158207, 1099508482048
Offset: 1
1 is a term since 1 and 2 have the same unitary abundance: A129468(1) = usigma(1) - 2*1 = 1 - 2 = -1, and A129468(2) = usigma(2) - 2*2 = 3 - 4 = -1.
A339937
Numbers k such that k and k+1 are both coreful abundant numbers (A308053).
Original entry on oeis.org
2282175, 33350624, 46734975, 86424975, 87152624, 105674624, 126114975, 169707824, 179762624, 214491375, 221370975, 235857824, 266022224, 270586575, 278524575, 297774224, 360021375, 372683024, 380858624, 395715375, 425840624, 470624175, 489873824, 503963775
Offset: 1
2282175 is a term since 2282175 and 2282176 are both coreful abundant numbers.
-
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); abQ[n_] := s[n] > 2*n; q1 = False; seq = {}; Do[q2 = abQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 10^8}]; seq
A348276
Numbers k such that k and k+1 are both noninfinitary abundant numbers (A348274).
Original entry on oeis.org
64198575, 84909824, 86424975, 110238975, 113223824, 191206575, 211266224, 224722575, 231058575, 231800624, 240069375, 240584175, 245383424, 262648575, 262911824, 279597824, 293893424, 297774224, 333773055, 338676975, 340250624, 340829775, 347244975, 372683024
Offset: 1
64198575 is a term since A348271(64198575) = 69470136 > 64198575 and A348271(64198576) = 65363424 > 64198576.
-
f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := DivisorSigma[1,n] - isigma[n] > n; seq = {}; q1 = q[1]; Do[q2 = q[n]; If[q1 && q2, AppendTo[seq, n-1]]; q1=q2 ,{n,2,10^8}]; seq
A348606
Numbers k such that k and k+1 are both nonexponential abundant numbers.
Original entry on oeis.org
21735, 76544, 170624, 301664, 345344, 348704, 382304, 739935, 862784, 1218944, 1262624, 1272704, 1314495, 1370655, 1376864, 1539615, 1558304, 1707615, 1718144, 1761375, 1845375, 1890944, 1926015, 2100735, 2132864, 2223584, 2415104, 2463615, 2581215, 2675295, 2747744
Offset: 1
21735 is a term since A160135(21735) = 21930 > 21735 and A160135(21736) = 23230 > 21736.
-
esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 3*10^6], q[#] && q[#+1] &]
A361935
Numbers k such that k and k+1 are both primitive unitary abundant numbers (definition 2, A302574).
Original entry on oeis.org
2457405145194, 2601523139214, 3320774552094, 3490250769005, 3733421997305, 3934651766045, 3954730124345, 4514767592334, 4553585751714, 4563327473705, 5226433847634
Offset: 1
-
f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
puabQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
Select[Import["https://oeis.org/A331412/b331412.txt", "Table"][[;; , 2]], puabQ[#] && puabQ[# + 1] &]
A380933
Numbers k such that k and k+1 are both in A380929.
Original entry on oeis.org
121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064
Offset: 1
121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.
-
q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]
-
isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));
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