A238693 -id:A238693 - cp's OEIS frontend

A238697 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-7} 2^(i-1)*binomial(i+2,3)/prime(n) (case 3).

0, 19, 197, 10481, 64027, 1980327, 259179061, 1257208799, 131286703021, 2756321451033, 12473384091267, 250290955437775, 21588599628845597, 1792050990708087027, 7763319803561678613, 620323392829436218475, 11365013042482773469559, 48487140450183407727097

Offset: 4

Vladimir Shevelev, Mar 03 2014

nonn

See comment in A238693.

V. Shevelev, Banach matchboxes problem and a congruence for primes, arXiv:1110.5686 [math.HO], 2011.

Cf. A238693.

Array[Sum[2^(i - 1)*Binomial[i + 2, 3]/#, {i, # - 7}] &@ Prime@ # &, 18, 4] (* Michael De Vlieger, Dec 06 2018 *)

a(n) = sum(i=1, prime(n)-7, 2^(i-1)*binomial(i+2,3))/prime(n); \\ Michel Marcus, Dec 06 2018

More terms from Peter J. C. Moses, Mar 03 2014

A238698 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-9} 2^(i-1)*binomial(i+3,4)/prime(n) (case 4).

Original entry on oeis.org

1, 27, 3599, 29157, 1362009, 271400395, 1469088801, 201573262419, 4910195172327, 23758960017789, 538608637491505, 54480012827209187, 5189654331623024397, 23446625614115858667, 2104894813684998321045, 41392675008326544152201, 182632116049323564469767

Offset: 5

Vladimir Shevelev, Mar 03 2014

nonn

See comment in A238693.

V. Shevelev, Banach matchboxes problem and a congruence for primes, arXiv:1110.5686

Cf. A238693, A238697.

k=4;(*case 4*)
Table[Sum[2^(i-1)Binomial[i+k-1,k],{i,p-(2k+1)}]/p,{p,Prime[Range[k+1,20]]}] (* Peter J. C. Moses, Mar 04 2014 *)

More terms from Peter J. C. Moses, Mar 03 2014

A238700 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-11} 2^(i-1)*binomial(i+4,5)/prime(n) (case 5).

Original entry on oeis.org

1, 625, 7451, 587687, 192856629, 1183808479, 220742818733, 6334029208601, 32973262995075, 853235644319439, 102411500363403805, 11294927679436544243, 53352132931526366997, 5415828333647578287211, 114722120087477391174007, 524320903831521291661817

Offset: 6

Vladimir Shevelev, Mar 03 2014

nonn

See comment in A238693.

V. Shevelev, Banach matchboxes problem and a congruence for primes, arXiv:1110.5686

Cf. A238693, A238697, A238698.

A238700:=n->sum(2^(i-1)*binomial(i+4,5)/ithprime(n), i=1..ithprime(n) - 11); seq(A238700(n), n=6..25); # Wesley Ivan Hurt, Mar 03 2014

Table[Sum[2^(i - 1)*Binomial[i + 4, 5]/Prime[n], {i, Prime[n] - 11}], {n, 6, 25}] (* Wesley Ivan Hurt, Mar 03 2014 *)

More terms from Peter J. C. Moses, Mar 03 2014

A239502 (Round(q^prime(n)) - 1)/prime(n), where q is the tribonacci constant (A058265).

Original entry on oeis.org

4, 10, 74, 212, 1856, 5618, 53114, 1630932, 5161442, 167427844, 1729192432, 5577731626, 58401766802, 2005139696964, 69737304018266, 228184540445268, 8043367476888770, 86866463049858250, 285815985033409648, 10225367934387562098, 111384745483589787826

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Mar 20 2014

nonn

For n>=3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

			For n=3, q^5 = 21.049..., so a(3) = (21 - 1)/5 = 4.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Tribonacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A058265.

A239544 (Round(c^prime(n)) - 1)/prime(n), where c is the tetranacci constant (A086088).

Original entry on oeis.org

14, 124, 390, 4118, 13690, 156122, 6351030, 22074820, 948652694, 11818395344, 41868809842, 528803858638, 24052859078262, 1108257471317098, 3982717894786008, 185987895674303758, 2422894681885464596, 8755616404517667662, 414985190213435939298

Offset: 4

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=4, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Tetranacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A086088, A239502.

A239564 a(n) = (round(c^prime(n)) - 1)/prime(n), where c is the pentanacci constant (A103814).

Original entry on oeis.org

154, 504, 5758, 19912, 245714, 11251030, 40679232, 1967728552, 26525975822, 97753187576, 1335948880418, 68398141417510, 3547322151373882, 13260715720748120, 697034813138756392, 9825603574709578482, 36935066391752894480, 1970457739485406707872

Offset: 5

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Pentanacci Constant

Cf. A000040, A007619, A007663, A238693, A238697, A238698, A238700, A103814, A239502, A239544.

A239565 (Round(c^prime(n)) - 1)/prime(n), where c is the hexanacci constant (A118427).

Original entry on oeis.org

6702, 23594, 301738, 14576792, 53653610, 2738173594, 38254296398, 143514673148, 2032676550562, 109797468019174, 6007838407290514, 22863415355711030, 1267938526864061370, 18523200405015238420, 70884650213591098558, 3989789924439684599434

Offset: 7

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Hexanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A118427, A239502, A239544, A239564.

A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

Original entry on oeis.org

7200, 25562, 332466, 16472758, 61145666, 3200477798, 45473543628, 172043098818, 2478186385762, 137291966046470, 7704742900338106, 29569459376703894, 1681851263230158754, 24987922624169214866, 96433670513455876108, 5566902760779797458210

Offset: 7

Peter J. C. Moses and Vladimir Shevelev, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Heptanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565.

All roots of the equation x^7-x^6-x^5-x^4-x^3-x^2-x-1 = 0
are the following: c=1.9919641966050350211,
-0.78418701799584451319 +/- 0.36004972226381653409*i,
-0.24065633852269642508 + /- 0.84919699909267892575*i,
  0.52886125821602342773 +/-  0.76534196109589443115*i.
Absolute values of all roots, except for septanacci constant c, are less than 1.
Conjecture. Absolute values of all roots of the equation x^n - x^(n-1) - ... -x - 1 = 0, except for n-bonacci constant c_n, are less than 1. If the conjecture is valid, then for sufficiently large k=k(n), for all m>=k, we have round(c_n^prime(m)) == 1 (mod 2*prime(m)) (cf. Shevelev link).

A238901 a(n) is the smallest k, 1<=k<=(p_n-3)/2, such that sum{i=1,...,p_n-2k-1} 2^(i-1) Binomial(k-1+i, k)/p_n is prime; a(n)=0, if such k does not exist.

Original entry on oeis.org

1, 1, 1, 6, 2, 5, 10, 1, 3, 11, 8, 13, 21, 5, 8, 29, 19, 19, 37, 0, 11, 11, 45, 42, 25, 11, 41, 7, 62, 39, 55, 70, 29, 60, 49, 24, 1, 0, 47, 73, 49, 78, 52, 11, 80, 80, 28, 32, 0, 92, 117, 112, 43, 102, 19, 97, 47, 38, 140, 51, 152, 44, 43, 141

Offset: 4

Vladimir Shevelev, Mar 06 2014

nonn

Cf. comment in A238693.

Peter J. C. Moses, Table of n, a(n) for n = 4..1003
V. Shevelev, Banach matchboxes problem and a congruence for primes, arXiv:1110.5686

Cf. A238693, A238697, A238698, A238700.

More terms from Peter J. C. Moses, Mar 06 2014

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

Original entry on oeis.org

3, 3, 4, 5, 7, 7, 10, 13, 14, 14, 19, 23, 23, 31, 34, 34, 46, 50, 60, 65, 73, 79, 88, 92, 107, 113, 126, 139, 149, 168, 182, 198, 210, 227, 244, 265, 276, 292, 317, 340, 369, 384, 408, 436, 444, 480, 516, 540, 565, 606, 628, 669, 704, 735, 759, 810, 829, 895, 925

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Mar 23 2014

nonn

The n-bonacci constant is a unique root x_1>1 of the equation x^n-x^(n-1)-...-x-1=0. So, for n=2 we have Fibonacci constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265); for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.

			Let n=2, then c_2 = phi (Fibonacci constant). We have round(c_2^2)=3 is not == 1 (mod 4), round(c_2^3)=4 is not == 1 (mod 6), while round(c_2^5)=11 == 1 (mod 10) and one can prove that for p>=5, we have round(c_2^p) == 1 (mod 2*p). Since 5=prime(3), then a(2)=3.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Cf. A001622, A007619, A007663, A058265, A086088, A103814, A118427, A118428, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565, A239566.

cp's OEIS Frontend

A238697 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-7} 2^(i-1)*binomial(i+2,3)/prime(n) (case 3).

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A238698 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-9} 2^(i-1)*binomial(i+3,4)/prime(n) (case 4).

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A238700 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-11} 2^(i-1)*binomial(i+4,5)/prime(n) (case 5).

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A239502 (Round(q^prime(n)) - 1)/prime(n), where q is the tribonacci constant (A058265).

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A239544 (Round(c^prime(n)) - 1)/prime(n), where c is the tetranacci constant (A086088).

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A239564 a(n) = (round(c^prime(n)) - 1)/prime(n), where c is the pentanacci constant (A103814).

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A239565 (Round(c^prime(n)) - 1)/prime(n), where c is the hexanacci constant (A118427).

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A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

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Formula

A238901 a(n) is the smallest k, 1<=k<=(p_n-3)/2, such that sum{i=1,...,p_n-2k-1} 2^(i-1) Binomial(k-1+i, k)/p_n is prime; a(n)=0, if such k does not exist.

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A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

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