A000073 -id:A000073 - cp's OEIS frontend

A359848 a(n) is the smallest tribonacci number (A000073) with exactly n distinct prime factors.

1, 2, 24, 504, 35890, 8646064, 1697490356184, 120879712950776, 98079530178586034536500564, 748829299860308729347600, 119816209721856219780831547518850, 15418262617564622254988364568360573618470100684551892712710640455037970

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

nonn

			a(4) = 35890, because 35890 is a tribonacci number with 4 distinct prime factors {2, 5, 37, 97} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001221, A060319, A359849, A359850.

a(11) from Daniel Suteu, Jan 17 2023

A359850 a(n) is the index of the smallest tribonacci number (A000073) with exactly n distinct prime factors.

Original entry on oeis.org

2, 4, 8, 13, 20, 29, 49, 56, 101, 93, 124, 268, 221

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001221, A060320, A359848, A359851.

a(11)-a(12) from Daniel Suteu, Jan 17 2023

A366780 Number of distinct prime divisors of A000073(n) (tribonacci numbers).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 1, 3, 4, 3, 3, 2, 2, 2, 3, 3, 3, 5, 3, 4, 5, 3, 4, 3, 5, 4, 5, 3, 5, 3, 2, 4, 2, 4, 5, 4, 4, 6, 2, 5, 5, 6, 3, 5, 7, 5, 2, 3, 5, 4, 6, 5, 4, 7, 3, 2, 4, 4, 3, 3, 4, 5, 2, 6, 6, 6, 5, 3, 6, 5, 4, 2, 6, 3, 6, 1, 7

Offset: 2

Sean A. Irvine, Oct 22 2023

nonn

			a(8)=2 because the 8th tribonacci number 24 = 2^3*3 has 2 distinct prime factors.

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000073, A001221, A022307, A366781, A366782.

PrimeNu[LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 87]] (* Amiram Eldar, Oct 23 2023 *)

a(n) = A001221(A000073(n)).

A366781 Number of prime divisors of A000073(n) (tribonacci numbers) (counted with multiplicity).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 4, 3, 4, 1, 2, 6, 3, 3, 8, 5, 2, 3, 4, 4, 3, 6, 4, 3, 4, 4, 3, 8, 4, 5, 11, 6, 4, 4, 10, 5, 5, 5, 9, 4, 2, 4, 2, 6, 5, 4, 11, 11, 2, 6, 7, 9, 3, 5, 9, 6, 2, 3, 5, 8, 12, 5, 11, 12, 4, 2, 4, 6, 3, 3, 6, 6, 2, 10, 7, 8, 7, 5, 12, 7, 4, 2, 6, 4

Offset: 2

Sean A. Irvine, Oct 22 2023

nonn

			a(8)=4 because the 8th tribonacci number 24 = 2^3*3 has 4 prime factors.

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000073, A001222, A022307, A366780, A366782.

PrimeOmega[LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 84]] (* Amiram Eldar, Oct 23 2023 *)

a(n) = A001222(A000073(n)).

A366782 Number of divisors of A000073(n) (tribonacci numbers).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 8, 6, 5, 2, 4, 24, 6, 8, 21, 16, 3, 8, 16, 12, 8, 12, 8, 6, 12, 12, 8, 80, 12, 24, 168, 20, 16, 12, 144, 24, 32, 16, 128, 12, 4, 16, 4, 32, 32, 16, 96, 320, 4, 48, 72, 216, 8, 32, 256, 48, 4, 8, 32, 48, 384, 32, 96, 576, 12, 4, 16, 36, 8, 8

Offset: 2

Sean A. Irvine, Oct 22 2023

nonn

			a(8)=8 because the 8th tribonacci number 24 has divisors {1, 2, 3, 4, 6, 8, 12, 24}.

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000005, A000073, A366780, A366781, A366783, A063375.

DivisorSigma[0, LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 70]] (* Amiram Eldar, Oct 23 2023 *)

a(n) = A000005(A000073(n)).

A112305 Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that n divides T(k).

Original entry on oeis.org

1, 3, 7, 4, 14, 7, 5, 7, 9, 19, 8, 7, 6, 12, 52, 15, 28, 12, 18, 31, 12, 8, 29, 7, 30, 39, 9, 12, 77, 52, 14, 15, 35, 28, 21, 12, 19, 28, 39, 31, 35, 12, 82, 8, 52, 55, 29, 64, 15, 52, 124, 39, 33, 35, 14, 12, 103, 123, 64, 52, 68, 60, 12, 15, 52, 35, 100, 28, 117

Offset: 1

N. J. A. Sloane, Nov 30 2005

nonn

Brenner proves that every prime divides some tribonacci number T(n). The Mathematica program computes similar sequences for any n-step Fibonacci sequence.

			T(1), T(2), T(3), T(4), ... are 1,1,2,4,7,13,24,...; a(3) = 7 because 3 first divides T(7) = A000073(8) = 24.

Ed Pegg, Jr., Posting to Sequence Fan mailing list, Nov 30, 2005

T. D. Noe, Table of n, a(n) for n = 1..1000
J. L. Brenner, Linear Recurrence Relations, Amer. Math. Monthly, Vol. 61 (1954), 171-173.
Eric Weisstein's World of Mathematics, Tribonacci Number.

Cf. A000073.

Cf. A112312 (least k such that prime(n) divides T(k)).

n=3; Table[a=Join[{1}, Table[0, {n-1}]]; k=0; While[k++; s=Mod[Plus@@a, i]; a=RotateLeft[a]; a[[n]]=s; s!=0]; k, {i, 100}] (* T. D. Noe *)

A222407 Digital roots of tribonacci numbers A000073.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 4, 6, 8, 9, 5, 4, 9, 9, 4, 4, 8, 7, 1, 7, 6, 5, 9, 2, 7, 9, 9, 7, 7, 5, 1, 4, 1, 6, 2, 9, 8, 1, 9, 9, 1, 1, 2, 4, 7, 4, 6, 8, 9, 5, 4, 9, 9, 4, 4, 8, 7, 1, 7, 6, 5, 9, 2, 7, 9, 9, 7, 7, 5, 1, 4, 1, 6, 2, 9, 8, 1, 9, 9, 1, 1, 2, 4, 7, 4, 6, 8, 9, 5, 4, 9, 9, 4, 4, 8, 7, 1, 7, 6, 5, 9, 2, 7, 9, 9, 7, 7, 5, 1, 4, 1, 6, 2, 9, 8, 1, 9, 9, 1, 1

Offset: 0

N. J. A. Sloane, Feb 20 2013

nonn

From a(2) onwards, periodic with period length 39.

The period sums to 216 and contains no 3s. When divided into three sets of 13, further patterns are revealed in connection with A100402 (see link below). - Peter M. Chema, Dec 21 2016

Peter M. Chema, Period 39, divided into three sets of 13, with further digital root patterns

Cf. A000073, A010888, A210456.

f:=proc(n) option remember; if n <= 1 then 0; elif n=2 then 1; else f(n-3)+f(n-2)+f(n-1); fi; end; # A000073
P:=n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # A010888
[seq(P(f(n)),n=0..200)];

FixedPoint[Total@ IntegerDigits@ # &, #] & /@ CoefficientList[ Series[x^2/(1 - x - x^2 - x^3), {x, 0, 81}], x] (* Michael De Vlieger, Dec 22 2016 *)
droot[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; droot/@LinearRecurrence[{1,1,1},{0,0,1},150] (* or *) PadRight[{0,0},150,{9,9,1,1,2,4,7,4,6,8,9,5,4,9,9,4,4,8,7,1,7,6,5,9,2,7,9,9,7,7,5,1,4,1,6,2,9,8,1}] (* Harvey P. Dale, Aug 21 2024 *)

a(n) = A010888(A000073(n)). - Michel Marcus, Dec 19 2016
From Chai Wah Wu, Jan 30 2018: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-7) - a(n-9) + a(n-10) - a(n-12) + a(n-13) - a(n-15) + a(n-16) - a(n-18) + a(n-19) - a(n-21) + a(n-22) - a(n-24) + a(n-25) - a(n-27) + a(n-28) - a(n-30) + a(n-31) - a(n-33) + a(n-34) - a(n-36) + a(n-37) for n > 38.
G.f.: (-9*x^38 + 8*x^36 - 16*x^35 - x^34 + 15*x^33 - 20*x^32 + 4*x^31 + 12*x^30 - 17*x^29 + 10*x^27 - 17*x^26 - 2*x^25 + 10*x^24 - 15*x^23 + 3*x^22 + 3*x^21 - 11*x^20 + 2*x^19 + 2*x^18 - 5*x^17 - 4*x^16 + x^15 - x^14 - 4*x^13 - 4*x^12 - x^11 + x^10 - 5*x^9 - 5*x^8 + 2*x^7 - 3*x^6 - 3*x^5 - x^4 - x^2)/(x^37 - x^36 + x^34 - x^33 + x^31 - x^30 + x^28 - x^27 + x^25 - x^24 + x^22 - x^21 + x^19 - x^18 + x^16 - x^15 + x^13 - x^12 + x^10 - x^9 + x^7 - x^6 + x^4 - x^3 + x - 1). (End)

A254231 Product of tribonacci numbers A000073(2) * ... * A000073(n).

Original entry on oeis.org

1, 1, 2, 8, 56, 728, 17472, 768768, 62270208, 9278260992, 2542243511808, 1281290729951232, 1187756506664792064, 2025124843863470469120, 6350791510355843391160320, 36631365431732504680212725760, 388622155865250142152376807587840

Offset: 2

Vaclav Kotesovec, Jan 27 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..89

Cf. A000073, A003266, A058265, A126772, A135407, A254232.

Table[Product[SeriesCoefficient[x^2/(1-x-x^2-x^3),{x,0,k}],{k,2,n}], {n,2,20}]

A254231_list, a, b, c, d = [1], 0, 0, 1, 1
for _ in range(15):
    a, b, c = b, c, a+b+c
    d *= c
    A254231_list.append(d) # Chai Wah Wu, Jan 27 2015

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = A058265 = 1.839286755214161132551852564653286600424178746097592246778758639404203222... is the root of the equation r^3 - r^2 - r - 1 = 0, d = 0.061463687669952618841340986526101395138659648898940720192319213600612851... is the root of the equation -1 + 36*d - 440*d^2 + 1936*d^3 = 0, c = 4.156714772910304733054135311449211887936035199917470476143821433373978333... .

A304943 Number of ways to write n as the sum of a positive tribonacci number (A000073) and a positive odd squarefree number.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 3, 2, 2, 2, 2, 2, 3, 4, 2, 4, 2, 4, 3, 4, 2, 4, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 1, 3, 3, 3, 3, 4, 2, 3, 4, 3, 4, 3, 2, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3

Offset: 1

Zhi-Wei Sun, May 22 2018

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 6, 10, 11, 13, 29, 76, 1332, 25249.

			a(2) = 1 with 2 = 1 + 1, where 1 = A000073(2) = A000073(3) is a positive tribonacci number, and 1 is also odd and squarefree.
a(29) = 1 since 29 = A000073(8) + 5 with 5 odd and squarefree.
a(76) = 1 since 76 = A000073(6) + 3*23 with 3*23 odd and squarefree.
a(1332) = 1 since 1332 = A000073(7) + 1319 with 1319 odd and squarefree.
a(25249) = 1 since 25249 = A000073(4) + 25247 with 25247 odd and squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000073, A005117, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720, A304721.

f[0]=0;f[1]=0;f[2]=1;
f[n_]:=f[n]=f[n-1]+f[n-2]+f[n-3];
QQ[n_]:=QQ[n]=Mod[n,2]==1&&SquareFreeQ[n];
tab={};Do[r=0;k=3;Label[bb];If[f[k]>=n,Goto[aa]];If[QQ[n-f[k]],r=r+1];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,100}];Print[tab]

A337285 a(n) = Sum_{i=1..n} (i-1)^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

Original entry on oeis.org

0, 1, 5, 41, 297, 1522, 7606, 35830, 159734, 691175, 2911275, 11995471, 48573775, 193800376, 763577276, 2976338876, 11493413820, 44020618429, 167385941185, 632387189285, 2375420846885, 8876467428110, 33013780952786, 122261706093330, 451010242361106, 1657768413841731, 6073328651742855

Offset: 1

N. J. A. Sloane, Sep 12 2020

nonn

R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073.)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-9,-7,-56,96,108,252,-162,-114,-318,126,-16,136,-36,12,-21,3,-1,1).

Cf. A000073, A085697, A107239, A337282, A337283, A337284, A337286.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1 -2*x+15*x^2+62*x^3-97*x^4+96*x^5+73*x^6-64*x^7-57*x^8-194*x^9-127*x^10-138*x^11 -55*x^12-12*x^13-9*x^14-4*x^15)/((1-x)*(1+x+x^2-x^3)^3*(1-3*x-x^2 -x^3)^3) )); // G. C. Greubel, Nov 22 2021

T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
A337285[n_]:= Sum[j^2*T[j+1]^2, {j,0,n-1}];
Table[A337285[n], {n, 40}] (* G. C. Greubel, Nov 22 2021 *)

@CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337285(n): return sum( j^2*T(j+1)^2 for j in (0..n-1) )
[A337285(n) for n in (1..40)] # G. C. Greubel, Nov 22 2021

From G. C. Greubel, Nov 22 2021: (Start)
a(n) = A337286(n) - 2*A337283(n) + A107239(n).
a(n) = Sum_{j=0..n-1} j^2*A000073(j+1)^2.
G.f.: x^2*(1 -2*x +15*x^2 +62*x^3 -97*x^4 +96*x^5 +73*x^6 -64*x^7 -57*x^8 -194*x^9 -127*x^10 -138*x^11 -55*x^12 -12*x^13 -9*x^14 -4*x^15)/((1-x)*(1 +x +x^2 -x^3)^3*(1 -3*x -x^2 -x^3)^3). (End)

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A359848 a(n) is the smallest tribonacci number (A000073) with exactly n distinct prime factors.

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A359850 a(n) is the index of the smallest tribonacci number (A000073) with exactly n distinct prime factors.

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A366780 Number of distinct prime divisors of A000073(n) (tribonacci numbers).

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A366781 Number of prime divisors of A000073(n) (tribonacci numbers) (counted with multiplicity).

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A366782 Number of divisors of A000073(n) (tribonacci numbers).

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A112305 Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that n divides T(k).

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A222407 Digital roots of tribonacci numbers A000073.

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Maple

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A254231 Product of tribonacci numbers A000073(2) * ... * A000073(n).

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A304943 Number of ways to write n as the sum of a positive tribonacci number (A000073) and a positive odd squarefree number.

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