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.

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A348523 Numbers that are both infinitary and noninfinitary abundant numbers.

Original entry on oeis.org

960, 1440, 1800, 2016, 2400, 2940, 3240, 3528, 3780, 4536, 4860, 6720, 7260, 8640, 10080, 10140, 10560, 12096, 12480, 12600, 13860, 14784, 15120, 15360, 15840, 16320, 16380, 16800, 17472, 17640, 18240, 18480, 18720, 18900, 19008, 19800, 20160, 21420, 21600, 21840
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Comments

Apparently, the smallest odd term is 9170790153525.

Examples

			960 is a term since A049417(960) = 2040 > 2*960 = 1920 and A348271(960) = 1008 > 960.

Links

Crossrefs

Intersection of A129656 and A348274.
Subsequence of A068403.
Cf. A049417, A348271.

Programs

  • Mathematica
    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_] := (i = isigma[n]) > 2*n && DivisorSigma[1, n] - i > n; Select[Range[10^4], q]

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

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Examples

			64198575 is a term since A348271(64198575) = 69470136 > 64198575 and A348271(64198576) = 65363424 > 64198576.

Crossrefs

Cf. A348271.
Subsequence of A096399 and A348274.
Similar sequences: A318167, A327635, A327942, A331412.

Programs

  • Mathematica
    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

A348527 Noninfinitary Zumkeller numbers: numbers whose set of noninfinitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

48, 80, 96, 112, 150, 180, 240, 252, 294, 336, 360, 396, 432, 468, 480, 486, 504, 528, 560, 600, 612, 624, 630, 672, 684, 720, 726, 768, 792, 810, 816, 828, 864, 880, 912, 936, 960, 1008, 1014, 1040, 1044, 1050, 1056, 1104, 1116, 1120, 1134, 1176, 1200, 1232, 1248
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Comments

The smallest odd term is a(104) = 2475.

Examples

			48 is a term since its set of noninfinitary divisors, {2, 4, 6, 8, 12, 24}, can be partitioned into the two disjoint sets, {2, 6, 8, 12} and {4, 24}, whose sums are equal: 2 + 6 + 8 + 12 = 4 + 24 = 28.

Links

Crossrefs

Cf. A348271, A348274, A348341.
Similar sequences: A083207, A290466, A335215, A335142, A335197, A335218.

Programs

  • Mathematica
    nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; nizQ[n_] := Module[{d = nidiv[n], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1250], !IntegerQ@ Log2@ DivisorSigma[0, #] && nizQ[#] &]

A348922 Numbers that are both infinitary and noninfinitary harmonic numbers.

Original entry on oeis.org

45, 60, 54600, 257040, 1801800, 2789640, 4299750, 47297250, 1707259680, 4093362000
Offset: 1

Views

Author

Amiram Eldar, Nov 04 2021

Keywords

Comments

a(11) > 10^10.
For each term the two sets of infinitary and noninfinitary divisors both contain more than one element. The only number with a single infinitary divisor is 1 which does not have noninfinitary divisors. Numbers with a single noninfinitary divisor are the squares of primes which are not infinitary harmonic numbers. Therefore, this sequence is a subsequence of A348715.
Nonsquarefree numbers k such that A049417(k) divides k*A037445(k) and A348271(k) divides k*A348341(k). The sequence also includes: 18779856480, 44425017000, 13594055202000, 27188110404000, 299069214444000, 6824215711404000. - Daniel Suteu, Nov 06 2021

Examples

			45 is a term since the infinitary divisors of 45 are 1, 5, 9 and 45, and their harmonic mean is 3, and the noninfinitary divisors of 45 are 3 and 15, and their harmonic mean is 5.

Crossrefs

Intersection of A063947 and A348918.
Subsequence of A348715.
Cf. A348923.
Cf. A037445, A049417, A348271, A348341.

Programs

  • Mathematica
    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]; id[1] = 1; id[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1]& /@ FactorInteger[n][[;; , 2]] ]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[3*10^5], !pow2Q[DivisorSigma[0, #]] && IntegerQ[# * (d = id[#])/(s = isigma[#])] && IntegerQ[# * (DivisorSigma[0, #] - d)/(DivisorSigma[1, #] - s)] &]
Previous Showing 11-14 of 14 results.

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