David Terr - cp's OEIS frontend

A358404 Multipliers involving Fibonacci-like sequences and Pythagorean triples.

2, 3, 5, 8, 13, 21, 23, 34, 41, 61, 85, 89, 144, 233, 255, 264, 377, 383, 397, 443, 610, 762, 875, 987

Offset: 1

David Terr, Nov 14 2022

A positive integer m is an element of this sequence if and only if there exists a Pythagorean triple of the form (m*G_0, m*G_1, G_n), where (G_k) is a Fibonacci-like sequence, i.e., a sequence with arbitrary positive integer starting values G_0 and G_1 and satisfying the recurrence G_k = G_{k-1} + G_{k-2} for every index k > 1.

			m = m(5, 0) = 2, since the Fibonacci-like sequence (G_n) with G_0 = 4 and G_1 = 3 has G_3 = 10 and (m*G_0, m*G_1, G_3) = (8, 6, 10) is a Pythagorean triple. Since m = 2 is the smallest positive integer with this property, m(1) = 2.

Cf. A000045.

(* Fibonacci entry point *)
T[m_] :=
  Module[{fi = FactorInteger[m], lenN, i, fi2, p, e, q, n1, divs,
    nDivs, d, found, preres, result = 1},
   If[m == 1, Return[1]];
   len = Length[fi];
   {p, e} = fi[[1]];
   q = p^e;
   If[len == 1,
    If[p == 5, Return[q]];
    If[e == 1,
     result = p - JacobiSymbol[p, 5];
     While[EvenQ[result] && Mod[Fibonacci[result], m] == 0,
      result /= 2];
     If[Mod[Fibonacci[result], m] != 0, result *= 2];
     fi2 = FactorInteger[result];
     If[EvenQ[result], Drop[fi2, 1]];
     n1 = Product[fi2[[i, 1]]^fi2[[i, 2]], {i, Length[fi2]}];
     divs = Divisors[n1];
     nDivs = Length[divs];
     found = False;
     For[i = 2, i <= nDivs && ! found, i++,
      d = divs[[i]];
      If[Mod[Fibonacci[d], m] == 0,
       found = True;
       result = d;
       Return[result];
       ];
      ],
     result = p^(e - 1 - If[p == 2 && e > 2, 1, 0])*T[p];
     Return[result];
     ],
    result = LCM[T[q], T[m/q]];
    ];
   result
   ];
(* Good moduli *)
GoodQ[m_] :=
  Module[{fi, len, i, p, t},
   If[m < 5, Return[False]];
   fi = FactorInteger[m];
   len = Length[fi];
   For[i = 1, i <= len, i++,
    p = fi[[i, 1]];
    If[Mod[p, 4] != 1, Return[False]];
    ];
   True
   ];
(* Great moduli *)
GreatQ[m_] := GoodQ[m] && OddQ[T[m]];
(* Fibonacci modular ratio *)
R[c_, k_] :=
  Module[{f0 = Fibonacci[k], f1},
   If[GCD[f0, c] > 1, Return[$Failed]];
   f1 = Fibonacci[k + 1];
   Mod[f1*PowerMod[f0, -1, c], c]
   ];
(* Starting pair for Fibonacci-like sequence *)
StartingPair[c_] :=
  Module[{pr, len, i, r0, t, n, r, u, v, g0, g1, preres},
   If[! GreatQ[c], Return[$Failed]];
   t = T[c];
   n = (t + 1)/2;
   r0 = R[c, n - 1];
   pr = PowersRepresentations[c, 2, 2];
   len = Length[pr];
   For[i = 1, i <= len, i++,
    {u, v} = pr[[i]];
    If[GCD[u, v] == 1,
     r = Mod[v*PowerMod[u, -1, c], c];
     preres = {Abs[u^2 - v^2], 2 u*v};
     If[r == c - r0, Return[preres]];
     If[r == r0, Return[Reverse[preres]]];
     ];
    ];
   $Failed
   ];
(* Great modulus multiplier *)
M[c_, j_] :=
  Module[{t, n0, n, g0, g1, result},
   If[! GreatQ[c], Return[$Failed]];
   {g0, g1} = StartingPair[c];
   t = T[c];
   n0 = (t + 1)/2;
   n = n0 + j*t;
   (g0*Fibonacci[n - 1] + g1*Fibonacci[n])/c
   ];
(* Master table *)
MasterTable[mMax_] :=
  Module[{c, j, m, g0, g1, t, n0, n, done, result = {}},
   For[c = 5, c <= GoldenRatio*mMax^2, c += 4,
    While[! GreatQ[c], c += 4];
    If[c <= GoldenRatio*mMax^2,
     {g0, g1} = StartingPair[c];
     t = T[c];
     n0 = (t + 1)/2;
     For[j = 0, j <= JMax[mMax, n0], j++,
      n = n0 + j*t;
      m = M[c, j];
      If[m <= mMax, AppendTo[result, {g0, g1, c, m, n}]];
      ];
     ];
    ];
   result
   ];
(* Multiplier list *)
MList[mMax_] := Union[MasterTable[mMax][[All, 4]]];

Each element of this list has a unique representation of the form m = m(c, j) = G_n / c, where j is an arbitrary nonnegative integer and c is "good", meaning that all of its prime divisors are of the form 4k + 1 and the Fibonacci entry point t of c is odd, in which case n = ((2j + 1)t + 1)/2 and (G_0, G_1, c) is the unique primitive Pythagorean triple such that G_0/G_1 is congruent to F_n/F_{n-1} modulo c.

A340864 Numbers k such that both sigma_{-1}(k) > 2 and sigma_0(k)/sigma_{-1}(k) are integers.

Original entry on oeis.org

672, 30240, 32760, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720, 14182439040, 43861478400, 51001180160, 66433720320, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904

Offset: 1

David Terr, Jan 24 2021

nonn

			a(1) = 672 is the smallest number k that is both an Ore number and multiperfect such that sigma(k)/k > 2.

Wikipedia, Multiply perfect number
Wikipedia, Harmonic divisor number.

Intersection of A001599 and A166069.

Cf. A007691, A325025.

Module[{a166069 = {120, 672, 30240, 32760, 523776, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720}, i, n, result = {}}, For[i = 1, i <= Length[a166069], i++, n = a166069[[i]]; If[Mod[DivisorSigma[0, n], DivisorSigma[-1, n]] == 0, AppendTo[result, n]]]; result]

Name changed by and more terms from Jinyuan Wang, Feb 11 2021

A307320 a(n) is the base-2 logarithm of the denominator of sigma_{-1}(P(n)), where P(n) = 2^(n-1)*M(n), where M(n) = 2^n - 1 is the n-th Mersenne number.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 3, 4, 0, 6, 6, 0, 2, 3, 10, 0, 8, 0, 9, 12, 13, 17, 16, 17, 8, 21, 13, 22, 14, 0, 25, 22, 12, 18, 22, 30, 14, 17, 27, 36, 29, 32, 32, 25, 36, 40, 37, 40, 34, 18, 30, 47, 44, 40, 39, 29, 46, 53, 40, 0, 26, 51, 55, 41, 50, 62, 42, 57, 44, 61

Offset: 1

David Terr, Apr 02 2019

nonn

a(n) = 0 if and only if P(n) is multiperfect. In particular, a(n) = 0 if M(n) is prime.

			a(6) = 2 since P(6) = 2016 and sigma_{-1}(2016) = 13/2^2.

M[n_] := 2^n - 1;
P[n_] := 2^(n - 1) M[n];
A[n_] := Log[2, Denominator[DivisorSigma[-1, P[n]]]];

a(n) = logint(denominator(sigma(2^(n-1)*(2^n-1),-1)), 2); \\ Michel Marcus, Apr 02 2019

More terms from Felix Fröhlich, Sep 29 2019

A192322 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 4-group, C2 x C2.

Original entry on oeis.org

84, 120, 132, 168, 195, 228, 280, 312, 340, 372, 408, 435, 483, 520, 532, 555, 595, 627, 708, 715, 760, 795, 1012, 1435

Offset: 1

David Terr, Jun 28 2011

Added keyword "full" - This sequence is a subsequence of A013658, whose last term is 1555. I have verified that the terms above are complete and correct. - Rick L. Shepherd, May 06 2013

Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler, Principally Polarized Squares of Elliptic Curves with Field of Moduli Equal To Q, arXiv:1806.03826 [math.NT], 2018 (see table 1 page 4).
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Subsequence of A013658.

ok(n)={isfundamental(-n) && [2, 2] == quadclassunit(-n).cyc} \\ Andrew Howroyd, Jul 20 2018

A100748 Years of recorded appearances of Halley's Comet (negative years are B.C.).

Original entry on oeis.org

-240, -164, -86, -11, 66, 141, 218, 295, 374, 451, 530, 607, 684, 760, 837, 912, 989, 1066, 1145, 1222, 1301, 1378, 1456, 1531, 1607, 1682, 1759, 1835, 1910, 1985

Offset: 1

David Terr, Jan 03 2005

sign

This sequence is not well-defined for many reasons (negative years are ill-defined; calendars change often, so when does a year begin?; what does "observed" mean?; what if the comet was observed both at the end of one year and at the beginning of another, as in 1985/1986?; etc.). - N. J. A. Sloane, Jun 24 2007

			a(30) = 1985 because the 30th recorded appearance of Halley's comet was in 1985.

Joseph M. Laufer, Halley's Comet, Its 30th Recorded Return, 1985-1986.

See A072235 for another version.

A098895 Number of divisors of the n-th superior highly composite number.

Original entry on oeis.org

2, 4, 6, 12, 16, 24, 48, 60, 120, 240, 288, 384, 576, 1152, 2304, 2688, 5376, 8064, 16128, 20160, 40320, 46080, 92160, 184320, 368640, 737280, 983040, 1966080, 3932160, 4423680, 6635520, 13271040, 15925248, 31850496, 63700992, 127401984

Offset: 1

David Terr, Oct 14 2004

nonn

Sequence A002201 gives the values of the n-th superior highly composite number N(n) and A000705 gives the values of the (prime) ratio N(n)/N(n-1).

			a(8)=60 because the eighth superior highly composite number, 5040, has 60 divisors.

S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

Amiram Eldar, Table of n, a(n) for n = 1..3401 (calculated using the b-file at A000705)
S. Ramanujan, Highly Composite Numbers.

Cf. A000705, A002201.

a(n) = a(n-1) * (1 + 1/k(n)), where k(n) is the p(n)-adic valuation of the n-th superior highly composite number N(n), with p(n) = N(n)/N(n-1) and N(0)=1.

A098896 p(n)-adic valuation of the n-th superior highly composite number N(n), where p(n) = N(n)/N(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 1, 1, 5, 3, 2, 1, 1, 6, 1, 2, 1, 4, 1, 7, 1, 1, 1, 1, 3, 1, 1, 8, 2, 1, 5, 1, 1, 1, 1, 2, 1, 1, 9, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 6, 4, 1, 1, 2, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

David Terr, Oct 14 2004

nonn

(1+1/a(n)) appears in the denominators of the log arguments of the denominators of the numbers in the table of the reference, pp. 115-117.

			a(8) = 4 since N(8)=5040 has 2-adic valuation of 4 and N(8)/N(7)=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using the b-file at A000705)
S. Ramanujan, Highly Composite Numbers.
S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115-117.

Cf. A000705, A002201.

More terms from Amiram Eldar, Aug 12 2019

A082990 Rent due for landing on undeveloped Monopoly properties starting at Mediterranean Avenue and finishing at Boardwalk, not including railroads or utilities.

Original entry on oeis.org

2, 4, 6, 6, 8, 10, 10, 12, 14, 14, 16, 18, 18, 20, 22, 22, 24, 26, 26, 28, 35, 50

Offset: 0

David Terr, May 29 2003

Cf. A060225.

A083556 n-th Payam number E_{+}(n), defined as the smallest positive odd integer k such that for every positive integer n, the number k*2^n+1 is not divisible by any primes p such that the multiplicative order of 2 mod p is less than or equal to e.

Original entry on oeis.org

3, 9, 15, 105, 105, 105, 105, 105, 165, 165, 75075, 75075, 75075, 75075, 75075, 75075, 855855, 855855, 5583435, 5583435, 5583435, 18625035, 18625035, 18625035, 18625035, 18625035, 27183585, 27183585, 27183585, 27183585, 27183585

Offset: 2

David Terr, Jun 10 2003

Payam numbers are good candidates for looking for Proth primes, i.e. primes of the form k*2^n+1

			E_{+}(3) = 9 because 9 is the smallest odd integer k such that for every nonnegative integer n, k*2^n+1 is not divisible by 3 or 7, the only primes p for which the multiplicative order of 2 mod p is less than or equal to 3.

Author?, Title?
Eric Weisstein's World of Mathematics, Payam Number

Cf. A080076.

Cf. A083391.

A083391 n-th Payam number E_{-}(n), defined as the smallest positive odd integer k such that for every positive integer n, the number k*2^n-1 is not divisible by any primes p such that the multiplicative order of 2 mod p is less than or equal to e.

Original entry on oeis.org

3, 3, 45, 45, 45, 45, 45, 45, 2145, 2805, 92235, 92235, 92235, 92235, 92235, 92235, 529815, 529815, 529815, 529815, 529815, 529815, 529815, 529815, 1426425, 1426425, 247016055, 247016055, 247016055, 247016055

Offset: 2

David Terr, Jun 11 2003

nonn

Payam numbers yield many primes of the form k*2^n+1 (Proth primes) and k*2^n-1.

			E_{-}(4) = 45 because 45 is the smallest odd integer k such that for every nonnegative integer n, k*2^n-1 is not divisible by 3, 5, or 7, the only primes p for which the multiplicative order of 2 mod p is less than or equal to 4.

Robert Smith, A coordinated search for primes in the Payam number series.
Eric Weisstein's World of Mathematics, Payam Number

Cf. A083556.

Cf. A080076.

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User: David Terr

A358404 Multipliers involving Fibonacci-like sequences and Pythagorean triples.

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A340864 Numbers k such that both sigma_{-1}(k) > 2 and sigma_0(k)/sigma_{-1}(k) are integers.

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A307320 a(n) is the base-2 logarithm of the denominator of sigma_{-1}(P(n)), where P(n) = 2^(n-1)*M(n), where M(n) = 2^n - 1 is the n-th Mersenne number.

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A192322 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 4-group, C2 x C2.

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A100748 Years of recorded appearances of Halley's Comet (negative years are B.C.).

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A098895 Number of divisors of the n-th superior highly composite number.

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A098896 p(n)-adic valuation of the n-th superior highly composite number N(n), where p(n) = N(n)/N(n-1).

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A082990 Rent due for landing on undeveloped Monopoly properties starting at Mediterranean Avenue and finishing at Boardwalk, not including railroads or utilities.

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A083556 n-th Payam number E_{+}(n), defined as the smallest positive odd integer k such that for every positive integer n, the number k*2^n+1 is not divisible by any primes p such that the multiplicative order of 2 mod p is less than or equal to e.

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