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A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 69, 1497, 259465, 4852747709, 3429487924785490781, 305153651313989042415043589313598477, 21932475414742921908206321699222250910796483151080020353252738457741771

Offset: 1

Aaron Meyerowitz, Jonathan Vos Post, and Robert G. Wilson v, Jun 09 2013

The semiprime analogous to A181503.
Because the semiprimes are sparser than the primes in the beginning, the sequence contains more of the lesser semiprimes than the analogous sequence of primes. In fact, one has to get to the seventeenth semiprime before it, 49,is not present, whereas in A181503, one only has to get to the sixth prime before it, 13, is not present.
If you change 1/(a(n)+1) to simply 1/a(n) the sequence becomes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 355, 16627, 76723511, 17218740226618333, 374886275842473712491638217368219, 9036922116709843444667289331349853231276337589593114741410804131,....

			1/(4+1) + 1/(6+1) + 1/(9+1) + … 1/(46+1) + 1/(69+1) is still less than 1. Instead of 1/69, if one were to use any semiprime between 46 and 69, {} the sum would then exceed 1.

Cf. A181503, A226527.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2 (* For those who have Mmca v or later, you could use PrimeOmega@ n == 2 *) NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextSemiPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]

A226527 Slowest-growing sequence of 3-almost primes (trientprimes) where 1/(tp+1) sums to 1 without actually reaching it.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244, 245, 246, 255, 258, 261, 266, 268, 273, 275, 279, 282, 284, 285, 286, 290, 292, 310, 316, 318, 322, 325, 332, 333, 338, 343, 345, 354, 356, 357, 363, 366, 369, 370, 374, 385, 387, 388, 399, 402, 404, 406, 410, 412, 418, 423, 425, 426, 428, 429, 430, 434, 435, 436, 8662, 44335708, 1251938572491943, 1505273212784203338150808798466, 680617602541158152398258079780439819108542271775727566330763

Offset: 1

Aaron Meyerowitz, Jonathan Vos Post, and Robert G. Wilson v, Jun 09 2013

See comments in A226526.

Deviates from A014612 after the 110th term.

Cf. A181503, A226526.

kAlmostPrimeQ[n_, k_: 2] := Plus @@ Last /@ FactorInteger@ n == k (* For those who have Mmca v or later, you could use PrimeOmega@ n == k *) NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextkAlmostPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]

A190591 The coefficient of t^n in the power series solution of u in the equation -t+(1-t+t^2+t^3)u-(t+t^4)u^2+(t^3+t^5)u^3-t^4u^4=0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 4, 7, 12, 23, 47, 96, 195, 402, 843, 1781, 3772, 8020, 17143, 36810, 79304, 171368, 371450, 807516, 1760145, 3845770, 8421528, 18480552, 40634154, 89507024, 197496651, 436469232, 966043263, 2141158866, 4751978668

Offset: 0

This sequence was derived by Dr. Aaron Meyerowitz and submitted by Shanzhen Gao, May 13 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..750
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.

s:= solve(-t+(1-t+t^2+t^3)*u-(t+t^4)*u^2+(t^3+t^5)*u^3-t^4*u^4, u):
a:= n-> coeff(series(s, t, n+1), t, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 03 2011

A181503 Slowest-growing sequence of primes p where 1/(p+1) sums to 1 without actually reaching it.

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 127, 1931, 309121, 47777896349, 76090912606600214447, 120621395443859821620817698234224534627, 63813688766771960235613705494151343867425896610637722399417500492543759703

Offset: 1

Aaron Meyerowitz, Oct 24 2010

The sum of 1/(p+1) over p = 2, 3, 5, 7, 11, 23 = A046689 is exactly 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..17 . [From _Robert G. Wilson v_, Oct 27 2010]

Similar to A075442. See also A046689.

a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 1; Array[a, 15] (* Robert G. Wilson v, Oct 27 2010 *)

a(12) onwards from Robert G. Wilson v, Oct 27 2010

A177794 G.f. A satisfies -x+(1+x^3-x)A+(x^4-x^2)A^2+(x^5-x^3)A^3-x^4A^4 = 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 69, 145, 306, 651, 1398, 3026, 6590, 14425, 31720, 70040, 155229, 345193, 770002, 1722487, 3863274, 8685608, 19570860, 44188976, 99965361, 226548082, 514275345, 1169255837, 2662319778, 6070294053, 13858727891, 31678845485

Offset: 1

This sequence was derived by Dr. Aaron Meyerowitz and submitted by Shanzhen Gao, May 13 2010

nonn

Used in the enumeration of prudent self-avoiding walks.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. Gao, H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).

Cf. A178035.

m = 36; A[_] = 0;
Do[A[x_] = (x + A[x]^2*x^2 + A[x]^3*x^3 + A[x]^2*(-1 + A[x]^2)*x^4 - A[x]^3*x^5)/(1 - x + x^3) + O[x]^m, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 03 2019 *)

/* verification */
V177794=[1, 1, 1, 1, 2, 4, 8, 16, 33, 69, 145];
A=x*Ser(V177794); /*  = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + ... */
-x+(1+x^3-x)*A+(x^4-x^2)*A^2+(x^5-x^3)*A^3-x^4*A^4 /* = O(x^12) = "zero" */
/* Joerg Arndt, May 14 2011 */

A156712 Star numbers (A003154) that are also triangular numbers (A000217).

Original entry on oeis.org

1, 7, 91, 1261, 17557, 244531, 3405871, 47437657, 660721321, 9202660831, 128176530307, 1785268763461, 24865586158141, 346332937450507, 4823795538148951, 67186804596634801, 935791468814738257, 13033893758809700791, 181538721154521072811, 2528508202404485318557

Offset: 1

Aaron Meyerowitz, Feb 14 2009

From Colin Barker, Jan 06 2015: (Start)
Also indices of centered square numbers (A001844) which are also centered triangular numbers (A005448).
Also indices of centered octagonal numbers (A016754) which are also centered hexagonal numbers (A003215).
Also positive integers y in the solutions to 3*x^2-4*y^2-3*x+4*y = 0, the corresponding values of x being A001922.
(End)

Colin Barker, Table of n, a(n) for n = 1..875
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 30, 56.
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Wikipedia, Star Numbers
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A000217, A001570, A001844, A001922, A003154.

Cf. A003215, A005448, A007655, A016754.

[(Evaluate(ChebyshevSecond(n+1),7) - 13*Evaluate(ChebyshevU(n), 7) + 1)/2: n in [1..30]]; // G. C. Greubel, Oct 07 2022

f:= gfun[rectoproc]({a(n+3)=15*a(n+2)-15*a(n+1)+a(n),a(1)=1,a(2)=7,a(3)=91},a(n),'remember'):
seq(f(n), n=1..30); # Robert Israel, Jan 01 2015

f[n_] := (Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)] + 4)/8; Array[f, 17] (* Robert G. Wilson v, Oct 28 2010 *)

Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-14*x+1)) + O(x^100)) \\ Colin Barker, Jan 01 2015

def A156712(n): return (1 + chebyshev_U(n, 7) - 13*chebyshev_U(n-1, 7))/2
[A156712(n) for n in range(1,31)] # G. C. Greubel, Oct 07 2022

a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
If x^2 - 3*y^2 = 1 with x even then a(y) = (y+2)/4 evidently related to A001570 by: add 1 and halve.
G.f.: x*(1 - 8*x + x^2)/((1-x)*(1 - 14*x + x^2)). - Alexander R. Povolotsky, Feb 15 2009
a(n) = (4 + (2 + sqrt(3))*(7 - 4*sqrt(3))^n + (2 - sqrt(3))*(7 + 4*sqrt(3))^n)/8. - Colin Barker, Mar 05 2016
a(n) = (1/2)*( 1 + ChebyshevU(n, 7) - 13*ChebyshevU(n-1, 7) ). - G. C. Greubel, Oct 07 2022

a(11) onwards from Robert G. Wilson v, Oct 28 2010

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User: Aaron Meyerowitz

A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.

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A226527 Slowest-growing sequence of 3-almost primes (trientprimes) where 1/(tp+1) sums to 1 without actually reaching it.

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A190591 The coefficient of t^n in the power series solution of u in the equation -t+(1-t+t^2+t^3)*u-(t+t^4)*u^2+(t^3+t^5)*u^3-t^4*u^4=0.

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A181503 Slowest-growing sequence of primes p where 1/(p+1) sums to 1 without actually reaching it.

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A177794 G.f. A satisfies -x+(1+x^3-x)*A+(x^4-x^2)*A^2+(x^5-x^3)*A^3-x^4*A^4 = 0.

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A156712 Star numbers (A003154) that are also triangular numbers (A000217).

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A190591 The coefficient of t^n in the power series solution of u in the equation -t+(1-t+t^2+t^3)u-(t+t^4)u^2+(t^3+t^5)u^3-t^4u^4=0.

A177794 G.f. A satisfies -x+(1+x^3-x)A+(x^4-x^2)A^2+(x^5-x^3)A^3-x^4A^4 = 0.