A007663 -id:A007663 - cp's OEIS frontend

A164740 (2^p-(p+2))/p as p runs through the primes.

0, 1, 5, 17, 185, 629, 7709, 27593, 364721, 18512789, 69273665, 3714566309, 53634713549, 204560302841, 2994414645857, 169947155749829, 9770521225481753, 37800705069076949, 2202596307308603177

Offset: 1

Tanin (Mirza Sabbir Hossain Beg) (mirzasabbirhossainbeg(AT)yahoo.com), Aug 24 2009

nonn

f[n_] := Block[{p = Prime@n}, (2^p - p - 2)/p]; Array[f, 15]
Table[(2^p-p-2)/p,{p,Prime[Range[20]]}] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = 2*A007663(n)-1. [T. D. Noe, Aug 24 2009]

Edited by N. J. A. Sloane, Aug 24 2009

Extended and offset changed by T. D. Noe and Robert G. Wilson v, Aug 24 2009

A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)...a(n-1)*p).

Original entry on oeis.org

3, 5, 13, 37, 73, 109, 181, 541, 1621, 4861, 9721, 19441, 58321, 87481, 379081, 408241, 2041201, 2449441, 7348321, 14696641, 22044961, 95528161, 382112641, 2292675841, 8024365441, 40121827201, 481461926401, 722192889601, 2888771558401, 7944121785601, 55608852499201, 111217704998401, 889741639987201, 1779483279974401

Offset: 1

Manuel Valdivia, Mar 15 2010

nonn

Conjecture: a(n) is the smallest integer k > 1 such that 2^(k-1) == 1 (mod a(0)*...*a(n-1)*k), with a(0) = 1. - Thomas Ordowski, Mar 13 2019

Either a(n) > a(n-1), or a(n) = a(n-1) is a Wieferich prime (A001220). - Max Alekseyev, Sep 29 2024

Max Alekseyev, Table of n, a(n) for n = 1..1000

Cf. A007663, A096060, A002200, A005109, A306826.

i=1;Do[p=Prime[n];If[Mod[2^(p-1)-1,p*i]==0,Print[p];i=p*i],{n,2,78498}]

findprime(prd) = {forprime(p=2, , if (Mod(2, p*prd)^(p-1) == 1, return (p)););}
lista(nn) = {my(prd = 1, na); for (n=1, nn, na = findprime(prd); print1(na, ", "); prd *= na;);} \\ Michel Marcus, Mar 14 2019

{ a175257_first_terms(N=1000) = my(P,L,t); P=[3]; L=2; for(n=#P,N, print(n," ",P[n]); forstep(p=P[n],oo,Mod(1,L), if(p==P[n], if(Mod(2,p^2)^(p-1)==1, error("Wieferich prime!"), next)); if(ispseudoprime(p), P=concat(P,[p]); t=Mod(2,p)^L; fordiv((p-1)\L,d, if(t^d==1, L*=d; break)); break))); P; } \\ Max Alekseyev, Sep 29 2024

a(17)-a(26) from Amiram Eldar, Feb 03 2019
Name corrected by Thomas Ordowski, Mar 13 2019
a(27) from Hans Havermann, Mar 29 2019
Eliminated a(0)=1 in the definition (empty products equal 1). - R. J. Mathar, Jun 19 2021
Terms a(28) onward from Max Alekseyev, Sep 29 2024

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).

A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

Original entry on oeis.org

1, 5, 7, 71, 379, 2659

Offset: 1

Alexander Adamchuk, Mar 10 2007

For k > 1 all 5 listed terms are primes.
The only known numbers k such that k divides the numerator of alternating Harmonic number H'((k-1)/2) = A058313((k-1)/2) are the Wieferich primes (A001220): 1093 and 3511.
An odd prime p = prime(n) belongs to this sequence iff Fermat quotient A007663(n) == A130912(n) == 2*(-1)^((p+1)/2) (mod p). - Max Alekseyev, Nov 30 2022

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A007663, A001220, A058313, A125854, A121999, A130912.

f=0; Do[ f = f + (-1)^(n+1)*1/n; g = Numerator[f]; If[ IntegerQ[ g/(2n-1) ], Print[2n-1]], {n,1,3000} ]

Edited by Max Alekseyev, Nov 30 2022

A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

Original entry on oeis.org

2, 4, 16, 68, 1476, 7280, 189120, 986244, 27676612, 4346071600, 23696518916, 3930960079760, 120508933265760, 669708812842692, 20814182249890948, 3654563002853231440, 650000099752136709444, 3664265812073801505200, 660535426260570501876228

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663, for a similar sequence for golden ratio, see A064723.

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213328.

A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

Original entry on oeis.org

4, 11, 78, 612, 46374, 428040, 38948910, 380144556, 37367223558, 38467601033550, 392545092308724, 426897839167539480, 45841425452161683630, 476794964068892779068, 51906117696097060014342, 59746844088106673671809870, 69664778857791165966384195366

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663. For similar sequences for golden ratio and silver ratio, see A064723 and A213327. Note that phi_3 is called "bronze ratio".

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213327.

A219539 T(n,k) is the number of k-points on the left side of a crosscut of simple symmetric n-Venn diagram.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 1, 1, 4, 11, 19, 23, 19, 11, 4, 1, 1, 5, 17, 38, 61, 71, 61, 38, 17, 5, 1, 1, 7, 33, 107, 257, 471, 673, 757, 673, 471, 257, 107, 33, 7, 1, 1, 8, 43, 161, 451, 977, 1675, 2303, 2559, 2303, 1675, 977, 451, 161, 43, 8, 1

Offset: 5

Michel Marcus, Nov 22 2012

A crosscut of a Venn diagram is defined as a segment of a curve which sequentially "cuts" (i.e., intersects) every other curve without repetition.
For n=2 and 3, there are 4 and 6 crosscuts respectively.
For n>3, there are either n crosscuts or none.
A k-point in a simple monotone Venn diagram is defined as being an intersection point that is incident to two k-regions.
The corresponding row sums are 3, 9, 93, .... (that is A007663).

			T(n, k) is defined for n>=5 being prime:
  5:  1, 1, 1,
  7:  1, 2, 3, 2, 1,
  11: 1, 4, 11, 19, 23, 19, 11, 4, 1,
  ...

K. Mamakani and F. Ruskey, A New Rose: The First Simple Symmetric 11-Venn Diagram, arXiv:1207.6452 [cs.CG], 2012.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 11.

Cf. A000108, A007663.

a(m) = {for (n=5, m, if (isprime(n), for (k=1, n-2,if (k==1, rk =1, rk = (binomial(n-1, k)+ (-1)^(k+1))/n);print1(rk, ", "););););}

For 1<=k=5 being prime.

T(n, k) - T(n, k-1) = (A000108(k-1) + 2*(-1)^(k+1))/n.

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2n-1)-1)/(2n-1), where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 1, 3, 9, 7, 93, 315, 1, 3855, 13797, 3, 182361, 41943, 9709, 9256395, 34636833, 31, 117, 1857283155, 105, 26817356775, 102280151421, 91, 1497207322929, 89756051247, 1285, 84973577874915, 19065, 4599, 4885260612740877, 18900352534538475, 1, 63, 1101298153654301589

Offset: 1

Amiram Eldar, Nov 08 2019

nonn

			a(3) = (2^lambda(2*3 - 1) - 1)/(2*3 - 1) = (2^lambda(5) - 1)/5 = (2^4 - 1)/5 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..1671
Florian Luca, Min Sha, and Igor E. Shparlinski On two functions arising in the study of the Euler and Carmichael quotients, Colloquium Mathematicum, Vol. 149, No. 2 (2017), pp. 179-192, arXiv preprint, arXiv:1705.00388 [math.NT] (2017).
Min Sha, The arithmetic of Carmichael quotients, Periodica Mathematica Hungarica, Vol. 71, No. 1 (2015), pp. 11-23, Correction to: The arithmetic of Carmichael quotients, ibid., Vol. 76, No. 2 (2018), pp. 271-273, preprint, arXiv:1108.2579v7 [math.NT] (2011-2017).
Chenhuang Wu, Zhixiong Chen, and Xiaoni Du, Binary threshold sequences derived from Carmichael quotients with even numbers modulus, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. 95, No. 7 (2012), pp. 1197-1199, alternative link.

Cf. A001226, A002322, A007663.

a[n_] := (2^CarmichaelLambda[n] - 1)/n; Table[a[n], {n, 1, 67, 2}]

A104137 Number of distinct necklaces with p beads of two possible colors, allowing turning over, p being a prime greater than 2.

Original entry on oeis.org

4, 8, 18, 126, 380, 4112, 14310, 184410, 9272780, 34669602, 1857545300, 26818405352, 102282248574, 1497215711538, 84973644983780, 4885261149611790, 18900353608280300, 1101298162244236182, 16628051030379615882

Offset: 1

Lekraj Beedassy, Mar 07 2005

nonn

For the general necklace problem, see A000029.

Martin Gardner, The Colossal Book of Mathematics, pp. 19, W. W. Norton & Co., NY 2001 (or, New Mathematical Diversions, pp. 243-4 MAA Washington DC 1995).

Cf. A000029.

for p from 2 to 30 do printf(`%d,`,(2^(ithprime(p)-1)-1)/ithprime(p) + 2^((ithprime(p)-1)/2) + 1) od: # James Sellers, Apr 10 2005

a(n) = (2^(p-1) - 1)/p + 2^{(p-1)/2} + 1 = A007663(n) + A061285(n) + 1.

More terms from James Sellers, Apr 10 2005

cp's OEIS Frontend

A164740 (2^p-(p+2))/p as p runs through the primes.

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A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)*...*a(n-1)*p).

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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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A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

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A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

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A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

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A219539 T(n,k) is the number of k-points on the left side of a crosscut of simple symmetric n-Venn diagram.

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Author

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Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2*n-1)-1)/(2*n-1), where lambda is the Carmichael lambda function (A002322).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A104137 Number of distinct necklaces with p beads of two possible colors, allowing turning over, p being a prime greater than 2.

Views

A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)...a(n-1)*p).

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2n-1)-1)/(2n-1), where lambda is the Carmichael lambda function (A002322).