cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A333928 Recursive abundant numbers: numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 168, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 240, 246, 252, 258, 260, 270, 276, 282, 294, 300, 306, 308, 318, 324, 330, 336, 340, 342, 348
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2020

Keywords

Examples

			12 is a term since A333926(12) = 28 > 2 * 12.

Links

Crossrefs

Cf. A333926, A333927.
Analogous sequences: A005101, A034683 (unitary), A064597 (nonunitary), A129575 (exponential), A129656 (infinitary), A292982 (bi-unitary).

Programs

  • Mathematica
    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]); Select[Range[350], recDivSum[#] > 2*# &]

A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Examples

			1575 is a term since it is odd and A333926(1575) = 3224 > 2 * 1575.

Links

Crossrefs

Intersection of A005408 and A333928.
Cf. A333926.
Analogous sequences: A005231, A094889 (nonunitary), A129485 (unitary), A127666 (infinitary), A293186 (bi-unitary), A321147 (exponential).

Programs

  • Mathematica
    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]); Select[2*Range[15000] + 1, recDivSum[#] > 2*# &]

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2020

Keywords

Comments

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

Examples

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Crossrefs

Cf. A049603, A282446, A333926.
Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

Programs

  • Mathematica
    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]); Select[Range[10^5], recDivSum[#] == 2*# &]

A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597, 274533, 288765
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Examples

			14 is a term since A333926(14) = A333926(15) = 24.

Links

Crossrefs

Cf. A333926.
Analogous sequences: A002961, A064115 (nonunitary), A064125 (unitary), A293183 (bi-unitary), A306985 (infinitary).

Programs

  • Mathematica
    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]); Select[Range[10^5], recDivSum[#] == recDivSum[# + 1] &]

A333929 Lesser of recursive amicable numbers pair: numbers m < k such that m = s(k) and k = s(m), where s(k) = A333926(k) - k is the sum of proper recursive divisors of k.

Original entry on oeis.org

220, 366, 2620, 3864, 5020, 16104, 16536, 26448, 29760, 43524, 63020, 67344, 69615, 100485, 122265, 142290, 142310, 196248, 196724, 198990, 239856, 240312, 280540, 308620, 309264, 319550, 326424, 341904, 348840, 366792, 469028, 522405, 537744, 580320, 647190, 661776
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2020

Keywords

Comments

The larger counterparts are in A333930.

Examples

			220 is a terms since A333926(220) - 220 = 284 and A333926(284) - 284 = 220.

Links

Crossrefs

Cf. A333926, A333927, A333930.
Analogous sequences: A002025, A002952 (unitary), A126165 (exponential), A126169 (infinitary), A292980 (bi-unitary).

Programs

  • Mathematica
    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]); s[n_] := recDivSum[n] - n; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 10^5}]; seq

A333930 Larger of recursive amicable numbers pair: numbers m < k such that m = s(k) and k = s(m), where s(k) = A333926(k) - k is the sum of proper recursive divisors of k.

Original entry on oeis.org

284, 378, 2924, 4584, 5564, 16632, 16728, 28752, 30912, 53692, 76084, 69552, 87633, 124155, 139815, 179118, 168730, 225096, 202444, 256338, 245904, 266568, 365084, 389924, 320016, 430402, 391656, 353616, 387720, 393528, 486178, 525915, 555216, 642720, 814698, 682896
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2020

Keywords

Comments

The terms are ordered according to their lesser counterparts (A333929).

Examples

			284 is a terms since A333926(284) - 284 = 220 and A333926(220) - 220 = 284.

Links

Crossrefs

Cf. A333926, A333927, A333929.
Analogous sequences: A002046, A002953 (unitary), A126166 (exponential), A126170 (infinitary), A292981 (bi-unitary).

Programs

  • Mathematica
    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]); s[n_] := recDivSum[n] - n; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 10^5}]; seq

A349284 Numbers k such that A051378(k) > 2*k and A333926(k) <= 2*k.

Original entry on oeis.org

126720, 134400, 149760, 188160, 195840, 456960, 510720, 549120, 618240, 718080, 748800, 779520, 802560, 833280, 940800, 979200, 994560, 1094400, 1102080, 1155840, 1263360, 1324800, 1393920, 1424640, 1585920, 1639680, 1670400, 1785600, 1800960, 1908480, 1946880
Offset: 1

Views

Author

Amiram Eldar, Nov 13 2021

Keywords

Comments

(1+e)-abundant numbers are numbers k such that A051378(k) > 2*k, i.e., numbers k whose sum of (1+e)-divisors exceeds 2*k.
Since all the recursive divisors (see A282446) of a number are also its (1+e)-divisors, the sequence of (1+e)-abundant numbers includes all the recursive abundant numbers (A333928). The first 21387 (1+e)-abundant numbers are also recursive abundant numbers. Therefore, this sequence includes only the (1+e)-abundant numbers that are not recursive abundant numbers.

Examples

			126720 is a term since A051378(126720) = 261144 > 2*126720 = 253440 and A333926(126720) = 246168 < 253440.

Links

Crossrefs

Cf. A049599, A049603, A051378, A282446, A333926, A333928.

Programs

  • Mathematica
    oesigma[1] = 1; oesigma[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; 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]]; recsigma[1] = 1; recsigma[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], oesigma[#] > 2*# && recsigma[#] <= 2*# &]

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

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Examples

			56924 is a term since A333926(56924) = 120960 > 2 * 56924, and A333926(56925) = 116064 > 2 * 56925.

Links

Crossrefs

Subsequence of A333928.
Cf. A333926, A333950.
Analogous sequences: A096399, A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary), A331412 (unitary).

Programs

  • Mathematica
    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] &]
Showing 1-8 of 8 results.

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