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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A333611 Sum of the iterated infinitary totient function iphi (A091732).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 6, 14, 10, 20, 9, 21, 9, 14, 29, 45, 14, 32, 21, 21, 20, 42, 9, 33, 21, 45, 32, 60, 14, 44, 29, 41, 45, 33, 33, 69, 32, 33, 21, 61, 21, 63, 44, 61, 42, 88, 44, 92, 33, 61, 69, 121, 45, 61, 32, 69, 60, 118, 33, 93, 44, 92, 106, 92, 41, 107
Offset: 1

Views

Author

Amiram Eldar, Mar 28 2020

Keywords

Examples

			a(3) = iphi(3) + iphi(iphi(3)) = 2 + 1 = 3.

Links

Crossrefs

Cf. A091732, A092693, A329153, A330273, A331273, A333609.

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); a[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

A362487 Infinitary highly totient numbers: numbers k that have more solutions x to the equation iphi(x) = k than any smaller k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 6, 12, 24, 48, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760, 967680, 1209600, 1451520, 1935360, 2419200, 2903040, 3628800
Offset: 1

Views

Author

Amiram Eldar, Apr 22 2023

Keywords

Comments

Indices of records of A362485.
The corresponding numbers of solutions are 2, 4, 6, 10, 14, 18, 22, ... (A362488).

Links

Crossrefs

Cf. A091732, A362484, A362485, A362486, A362488.
Similar sequences: A097942, A100827, A361968, A362183.

Programs

  • Mathematica
    solnum[n_] := Length[invIPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^4] (* using the function invIPhi from A362484 *)

A362666 a(n) is the largest m such that iphi(m) = n, where iphi is the infinitary totient function A091732, or a(n) = 0 if no such m exists.

Original entry on oeis.org

2, 6, 8, 10, 0, 24, 0, 30, 0, 22, 0, 42, 0, 0, 32, 54, 0, 56, 0, 66, 0, 46, 0, 120, 0, 0, 0, 58, 0, 96, 0, 102, 0, 0, 0, 168, 0, 0, 0, 110, 0, 86, 0, 138, 128, 94, 0, 216, 0, 0, 0, 106, 0, 152, 0, 174, 0, 118, 0, 264, 0, 0, 0, 270, 0, 184, 0, 0, 0, 142, 0, 312
Offset: 1

Views

Author

Amiram Eldar, Apr 29 2023

Keywords

Examples

			a(1) = 2 since there are two solutions to iphi(x) = 1: 1 and 2, and 2 is the larger of them.
a(6) = 24 since there are four solutions to iphi(x) = 6: 7, 12, 14 and 24, and 24 is the largest of them.

Links

Crossrefs

The infinitary version of A057635.
Cf. A091732, A362484, A362486 (positions of 0's), A362667 (record values), A362668 (indices of records).

Programs

  • Mathematica
    a[n_] := If[(inv = invIPhi[n]) == {}, 0, Max[inv]]; Array[a, 100] (* using the function invIPhi from A362484 *)

Formula

a(A362486(n)) = 0.

A362667 Infinitary sparsely totient numbers: numbers k such that m > k implies iphi(m) > iphi(k), where iphi is the infinitary totient function A091732.

Original entry on oeis.org

2, 6, 8, 10, 24, 30, 42, 54, 56, 66, 120, 168, 216, 264, 270, 312, 330, 384, 408, 456, 480, 510, 552, 840, 1080, 1320, 1560, 1920, 2040, 2280, 2376, 2760, 3000, 3192, 3480, 3720, 3864, 4440, 4920, 5160, 5208, 5640, 7560, 9240, 10920, 11880, 13440, 14280, 15960
Offset: 1

Views

Author

Amiram Eldar, Apr 29 2023

Keywords

Links

Crossrefs

The infinitary version of A036913.
Record values of A362666.
Cf. A091732, A362484, A362668.

Programs

  • Mathematica
    s[n_] := If[(inv = invIPhi[n]) == {}, 0, Max[inv]]; seq[kmax_] := Module[{v = {}, s1, sm = 0}, Do[s1 = s[k]; If[s1 > sm, sm = s1; AppendTo[v, s1]], {k, 1, kmax}]; v]; seq[3000] (* using the function invIPhi from A362484 *)

A362668 a(n) = A091732(A362667(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 36, 48, 60, 64, 72, 80, 90, 96, 108, 120, 128, 132, 144, 192, 240, 288, 360, 384, 432, 480, 528, 576, 648, 672, 720, 792, 864, 960, 1008, 1080, 1104, 1152, 1440, 1728, 1920, 2160, 2304, 2592, 2880, 3072, 3168, 3456, 3600, 3840
Offset: 1

Views

Author

Amiram Eldar, Apr 29 2023

Keywords

Links

Crossrefs

The infinitary version of A036912.
Indices of records of A362666.
Cf. A091732, A362484, A362667.

Programs

  • Mathematica
    s[n_] := If[(inv = invIPhi[n]) == {}, 0, Max[inv]]; seq[kmax_] := Module[{v = {}, s1, sm = 0}, Do[s1 = s[k]; If[s1 > sm, sm = s1; AppendTo[v, k]], {k, 1, kmax}]; v]; seq[4000] (* using the function invIPhi from A362484 *)

A327572 Partial sums of an infinitary analog of Euler's phi function: a(n) = Sum_{k=1..n} iphi(k), where iphi is A091732.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 19, 22, 30, 34, 44, 50, 62, 68, 76, 91, 107, 115, 133, 145, 157, 167, 189, 195, 219, 231, 247, 265, 293, 301, 331, 346, 366, 382, 406, 430, 466, 484, 508, 520, 560, 572, 614, 644, 676, 698, 744, 774, 822, 846, 878, 914, 966, 982, 1022, 1040
Offset: 1

Views

Author

Amiram Eldar, Sep 17 2019

Keywords

References

  • Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Links

Crossrefs

Cf. A091732 (iphi), A327575.
Cf. A002088 (sums of phi), A177754 (unitary), A306070 (bi-unitary).

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); Accumulate[Array[iphi, 52]]

Formula

a(n) ~ c * n^2, where c = 0.328935... (A327575).

A327575 Decimal expansion of the constant that appears in the asymptotic formula for average order of an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

3, 2, 8, 9, 3, 5, 8, 3, 8, 8, 4, 0, 3, 3, 5, 5, 1, 6, 3, 5, 5, 7, 4, 8, 4, 8, 7, 3, 6, 5, 2, 2, 0, 2, 2, 9, 5, 7, 7, 0, 6, 6, 5, 2, 3, 7, 9, 4, 6, 9, 4, 0, 4, 4, 8, 0, 8, 4, 0, 3, 7, 9, 8, 7, 5, 2, 8, 1, 2, 4, 0, 0, 7, 7, 3, 7, 9, 6, 8, 7, 4, 8, 8, 3, 9, 9, 7
Offset: 0

Views

Author

Amiram Eldar, Sep 17 2019

Keywords

Examples

			0.328935838840335516355748487365220229577066523794694...

References

  • Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Links

Crossrefs

Cf. A091732, A050376, A327572.
Cf. A104141 (corresponding constant for phi), A065463 (unitary), A306071 (bi-unitary).

Programs

  • Mathematica
    $MaxExtraPrecision = 1500; m = 1500; em = 10; f[x_] := Sum[Log[1 - x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals Limit_{k->oo} A327572(k)/k^2.
Equals (1/2) * Product_{P} (1 - 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

A333610 Least number that reaches 1 after n iterations of the infinitary totient function A091732.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 17, 47, 85, 227, 257, 919, 1229, 2459, 4369, 9839, 30865, 101503, 148157, 438499, 828297, 2201671, 3316617, 11055391, 35354993, 140810491, 188991053, 377982107, 848170377, 1704741139, 6933926513
Offset: 0

Views

Author

Amiram Eldar, Mar 28 2020

Keywords

Crossrefs

Cf. A003271, A007755, A091732, A333609.

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); numiter[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1; n = 0; seq = {}; Do[If[numiter[k] == n, AppendTo[seq, k]; n++], {k, 1, 1000}]; seq

Formula

A333609(a(n)) = n.

A333612 Numbers at which the sum of the iterated infinitary totient function (A091732) attains a record.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 11, 13, 16, 17, 29, 37, 47, 49, 53, 81, 101, 107, 113, 149, 173, 197, 257, 389, 401, 509, 529, 531, 557, 593, 677, 701, 747, 773, 829, 963, 977, 1109, 1297, 1493, 1675, 1733, 1901, 2417, 2761, 2837, 3089, 3313, 3329, 3413, 3467, 3677, 3803, 3989
Offset: 1

Views

Author

Amiram Eldar, Mar 28 2020

Keywords

Comments

Analogous to A181659 with the infinitary totient function A091732 instead of the Euler totient function phi (A000010).
The corresponding record values are 0, 1, 3, 6, 10, 14, 20, 21, 29, 45, ... (see the link for more values).

Links

Crossrefs

Cf. A091732, A181659, A330400, A331407, A333611.

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); s[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; seq = {}; smax = -1; Do[s1 = s[n];  If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A335327 Numbers k such that iphi(k) divides k, where iphi is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

1, 2, 6, 12, 24, 72, 120, 240, 480, 1440, 2880, 5760, 8640, 17280, 65280, 86400, 120960, 130560, 259200, 391680, 783360, 1566720, 2350080, 4700160, 23500800, 32901120, 47001600, 70502400, 94003200, 470016000, 1410048000, 2820096000, 4294901760, 5640192000, 8460288000
Offset: 1

Views

Author

Amiram Eldar, Jun 01 2020

Keywords

Examples

			6 is a term since iphi(6) = 2 is a divisor of 6.

Crossrefs

Cf. A007694, A091732, A319481.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1);  Select[Range[10^5], Divisible[#,a[#]] &]
Previous Showing 11-20 of 33 results. Next

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