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 1-6 of 6 results.

A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

3, 42, 188970, 998670, 51670374, 91397154, 236280786, 259172826, 792554574, 1106710914, 1468869930, 1957827498, 2467823442, 2496238590, 3324585210, 4055970282, 4183629690, 4384566870, 13479861630, 20681058270, 29343074178, 43449285210, 68705958690, 71418085926
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			42 is a term since s(42) = 96 and s(40) + s(41) = 54 + 42 = 96.

Links

Crossrefs

The unitary version of A065900.
Cf. A034448, A065557, A075565, A076136, A145469, A291126, A291176, A292033, A294995, A332974.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3, 10^8], usigma[#] == usigma[# - 1] + usigma[# - 2] &]
  • PARI
    usigma(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)); \\ A034448
isok(k) = usigma(k) == usigma(k-1) + usigma(k-2); \\ Jinyuan Wang, Mar 08 2020

Extensions

Terms a(22) and beyond from Giovanni Resta, Mar 10 2020

A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

3, 24, 360, 5016, 28440, 42066, 50568, 60456, 187176, 998670, 1454706, 12055512, 14365608, 25726728, 27896424, 51670374, 91702962, 141084774, 236280786, 249854952, 386668344, 439362504, 792554574, 1115866152, 1931976696, 2467823442, 2496238590, 2655297558, 2715505440
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			24 is a term since isigma(24) = 60 and isigma(22) + isigma(23) = 36 + 24 = 60.

Links

Crossrefs

The infinitary version of A065900.
Cf. A049417, A065557, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995, A332976.

Programs

  • Mathematica
    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]; Select[Range[3, 10^5], isigma[#] == isigma[# - 1] + isigma[# - 2] &]

A332971 Infinitary phibonacci numbers: solutions k of the equation iphi(k) = iphi(k-1) + iphi(k-2) where iphi(k) is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

3, 4, 7, 23, 121, 2857, 5699, 6377, 9179, 46537, 63209, 244967, 654497, 1067873, 1112009, 3435929, 3831257, 6441593, 7589737, 7784507, 8149751, 14307856, 22434089, 24007727, 24571871, 44503417, 44926463, 56732729, 128199059, 140830367, 190145936, 401767631, 403152737
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			7 is a term since iphi(7) = 6 and iphi(5) + iphi(6) = 4 + 2 = 6.

Links

Crossrefs

An infinitary version of A065557.
Cf. A065900, A075565, A076136, A076251, A091732, A145469, A291126, A291176, A292033, A294995.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) - 1); Select[Range[3, 10^5], iphi[#] == iphi[# - 1] + iphi[# - 2] &]

A332972 Solutions k of the equation cototient(k) = cototient(k-1) + cototient(k-2) where cototient(k) is A051953.

Original entry on oeis.org

3, 4, 105, 165, 195, 2205, 2835, 38805, 131145, 407925, 936495, 1025505, 1231425, 1276905, 1788255, 1925565, 2521695, 2792145, 2847585, 3289935, 5003745, 5295885, 5710089, 6315309, 6986889, 13496385, 17168085, 19210065, 20171385, 22348365, 26879685, 27798705
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			3 is a term since cototient(3) = 1 and cototient(1) + cototient(2) = 0 + 1 = 1.
105 is a term since cototient(105) = 57 and cototient(103) + cototient(104) = 1 + 56 = 57.

Links

Crossrefs

Cf. A051953, A065557, A065900, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995.

Programs

  • Mathematica
    cotot[n_] := n - EulerPhi[n]; Select[Range[3, 10^6], cotot[#] == cotot[# - 1] + cotot[# - 2] &]

A332974 Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = usigma(k) - k is the sum of proper unitary divisors of k (A063919).

Original entry on oeis.org

3, 21, 321, 1257, 3237, 146139, 268713, 584835, 26749089, 9988999095, 25997557299, 54449485353, 935628578283, 2105722150095, 3921293253003, 8234992646643
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Comments

a(17) > 10^13. - Giovanni Resta, May 09 2020

Examples

			21 is a term since s(21) = 11 and s(19) + s(20) = 1 + 10 = 11.

Crossrefs

The unitary version of A291176.
Cf. A063919, A065557, A065900, A075565, A076136, A145469, A291126, A292033, A294995, A332973.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s[n_] := usigma[n] - n; Select[Range[3, 6*10^5], s[#] == s[# - 1] + s[# - 2] &]

Extensions

a(12)-a(16) from Giovanni Resta, May 09 2020

A332976 Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = isigma(k) - k is the sum of proper infinitary divisors of k (A126168).

Original entry on oeis.org

3, 8, 10, 21, 3237, 7377, 146139, 584835, 9988999095, 25997557299
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			8 is a term since s(8) = 7 and s(6) + s(7) = 6 + 1 = 7.

Crossrefs

The infinitary version of A291176.
Cf. A065557, A065900, A075565, A076136, A076251, A126168, A145469, A291126, A292033, A294995, A332975.

Programs

  • Mathematica
    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]; s[n_] := isigma[n] - n; Select[Range[3, 6*10^5], s[#] == s[# - 1] + s[# - 2] &]
Showing 1-6 of 6 results.

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