A001917 -id:A001917 - cp's OEIS frontend

A319350 Filter sequence which records the number of cyclotomic cosets of 2 mod p for odd primes p, and for any other number assigns a unique number.

1, 2, 3, 4, 3, 5, 6, 7, 8, 9, 3, 10, 3, 11, 12, 13, 6, 14, 3, 15, 16, 17, 6, 18, 19, 20, 21, 22, 3, 23, 24, 25, 26, 27, 28, 29, 3, 30, 31, 32, 6, 33, 34, 35, 36, 37, 6, 38, 39, 40, 41, 42, 3, 43, 44, 45, 46, 47, 3, 48, 3, 49, 50, 51, 52, 53, 3, 54, 55, 56, 6, 57, 58, 59, 60, 61, 62, 63, 6, 64, 65, 66, 3, 67, 68, 69, 70, 71, 58, 72, 73, 74, 75, 76, 77, 78, 6, 79, 80, 81, 3, 82, 6

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A006694((n-1)/2) when n is an odd prime, otherwise -n.
For all i, j:
  a(i) = a(j) => A305801(i) = A305801(j),
  a(i) = a(j) => A319351(i) = A319351(j).

			a(3) = a(5) = a(11) = a(13) = a(19) = a(29) = a(37) because 3, 5, 11, 13, 19, 29, 37 are primes p for which A006694((p-1)/2) = 1 (are in A001122).
a(7) = a(17) = a(23) = a(41) = a(47) because 7, 17, 23, 41, 47 are primes p for which A006694((p-1)/2) = 2 (are in A115591).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001917, A006694, A286573, A305801, A319351.
Cf. A001122 (positions of 3's), A115591 (positions of 6's).
Cf. also A319704, A319705, A319706.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A319350aux(n) = if((n<=2)||!isprime(n),n,-((n-1)/znorder(Mod(2, n))));
v319350 = rgs_transform(vector(up_to,n,A319350aux(n)));
A319350(n) = v319350[n];

A002323 ((2^m - 1) / p) mod p, where p = prime(n) and m = ord(2,p).

Original entry on oeis.org

1, 3, 1, 5, 3, 15, 3, 20, 1, 1, 1, 32, 37, 22, 36, 8, 36, 10, 1, 7, 49, 48, 23, 77, 92, 81, 13, 95, 49, 1, 17, 95, 30, 96, 66, 132, 67, 107, 3, 50, 148, 25, 52, 175, 167, 109, 143, 201, 99, 30, 13, 207, 200, 255, 64, 260, 190, 208, 159, 208, 78, 98, 243, 60

Offset: 2

N. J. A. Sloane

a(n) = 0 if and only if prime(n) is a Wieferich prime (A001220). - Eric M. Schmidt, Feb 23 2015

			For p = prime(3) = 5, we find that m = 4 is the smallest positive integer for which 2^m - 1 is divisible by p. So a(3) = ((2^4 - 1) / 5) mod 5 = 3. - _Eric M. Schmidt_, Jun 21 2013

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
W. Meißner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p=1093, Sitzungsberichte der Königlich Preußischen Akadamie der Wissenschaften, Berlin, 35 (1913), 663-667.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric M. Schmidt, Table of n, a(n) for n = 2..10000
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Cf. A001220, A001917.

Table[p = Prime[n]; Mod[(2^MultiplicativeOrder[2, p] - 1)/p, p], {n, 2, 100}] (* T. D. Noe, Jun 21 2013 *)

a(n) = my(p=prime(n));(lift(Mod(2,p^2)^znorder(Mod(2,p)))-1)/p \\ Jeppe Stig Nielsen, May 30 2023

def A002323(n) : p = nth_prime(n); return (2^(Mod(2,p).multiplicative_order()) - 1) // p % p # Eric M. Schmidt, Jun 21 2013

Proper definition added by and more terms from Eric M. Schmidt, Jun 21 2013

A170820 Let p = n-th prime; a(n) = (p-1)/(order of (p+3)/2 mod p).

Original entry on oeis.org

2, 1, 1, 3, 1, 6, 2, 4, 1, 1, 1, 2, 2, 4, 1, 5, 2, 10, 2, 3, 1, 1, 12, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 5, 2, 2, 4, 3, 42, 1, 1, 1, 1, 2, 8, 1, 1, 2, 4, 1, 1, 7, 2, 4, 6, 2, 2, 4, 30, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 25, 4, 11, 1, 10, 2, 3, 1, 1, 8, 10, 33, 1, 2, 3, 1, 6, 2, 4, 1, 2, 1, 2, 2, 1

Offset: 3

N. J. A. Sloane, Dec 24 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..65536
I. Anderson and D. A. Preece, Combinatorially fruitful properties of 3*2^(-1) and 3*2^(-2) modulo p, Discr. Math., 310 (2010), 312-324.

Cf. A014664, A001917, A170821, A170822.

with(numtheory); [seq((ithprime(n)-1)/order((ithprime(n)+3)/2,ithprime(n)),n=3..130)];

a[n_] := Module[{p=Prime[n]}, (p-1)/MultiplicativeOrder[(p+3)/2, p]]; Array[a, 100, 3] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(p=prime(n)); (p-1)/znorder(Mod((p+3)/2, p)); \\ Michel Marcus, Dec 03 2018

A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

Original entry on oeis.org

0, 1, 1, 3, 4, 8, 10, 15, 19, 26, 31, 39, 46, 56, 64, 75, 85, 98, 109, 123, 136, 152, 166, 183, 199, 218

Offset: 3

Kival Ngaokrajang, May 25 2013

From the drawings as shown in links, it can be separated the distinct quadrilaterals into 3 types:
Q1: Quadrilaterals which have at least one side equal to n-gon sides length.
Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than n-gon sides length.
Q3: Quadrilaterals which have no parallel sides and all sides are longer than n-gon side length.
Q1(n) = A004652(n-3); Q2(n) = A001917(n-6), Q2(3) = 0, Q2(4) = 0; Q3(n) = A005232(n-10), Q3(3) = 0, Q3(4) = 0, Q3(5) = 0, Q3(6) = 0, Q3(7) = 0, Q3(8) = 0, Q3(9) = 0.
a(n) = Q1(n) + Q2(n). The total distinct quadrilaterals is Q1 + Q2 + Q3. Also the total distinct quadrilaterals = A005232(n-4), for n>=4. Also a(n) = A005232(n-4) - A005232(n-10), for n>=10.

			For a pentagon, there are 5 quadrilaterals which are the same size and shape. Therefore a(5) = 1.

Kival Ngaokrajang, The distinct quadrilaterals for n = 4..9
Kival Ngaokrajang, The distinct quadrilaterals for n = 10
Kival Ngaokrajang, Q1 & Q2 for n = 23
Kival Ngaokrajang, Q3 for n = 23

Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n-3) is number of distinct triangles in an n-gon.

Empirical g.f.: -x^4*(x^2-x+1)^2*(x^2+x+1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Oct 31 2013

A319351 Filter sequence which records the number of cyclotomic cosets of 2 mod p^k for powers of odd primes p, and for any other number assigns a unique number.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 6, 8, 3, 9, 3, 10, 11, 12, 6, 13, 3, 14, 15, 16, 6, 17, 6, 18, 19, 20, 3, 21, 22, 23, 24, 25, 26, 27, 3, 28, 29, 30, 6, 31, 19, 32, 33, 34, 6, 35, 36, 37, 38, 39, 3, 40, 41, 42, 43, 44, 3, 45, 3, 46, 47, 48, 49, 50, 3, 51, 52, 53, 6, 54, 55, 56, 57, 58, 59, 60, 6, 61, 36, 62, 3, 63, 64, 65, 66, 67, 55, 68, 69, 70, 71, 72, 73, 74, 6, 75, 76, 77, 3, 78, 6

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to the number of cyclotomic cosets of 2 mod p^k, while all other numbers occur in singular equivalence classes of their own.
Restricted growth sequence transform of function f defined as f(n) = A006694((n-1)/2) when n is an odd prime power > 1, otherwise -n.
For all i, j: a(i) = a(j) => A305976(i) = A305976(j). (See also A305975).

			a(7) = a(9) = a(17) = a(23) = a(25) = a(41) = ... because n = 7, 9, 17, 23, 25, 41, ... are such powers of odd primes for which A006694((n-1)/2) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001917, A006694, A286573, A305801, A319350.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006694(n) = (sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1); \\ From A006694
A319351aux(n) = if((n<=2)||!(n%2)||!isprimepower(n),n,-(A006694((n-1)/2)));
v319351 = rgs_transform(vector(up_to,n,A319351aux(n)));
A319351(n) = v319351[n];

A367318 Lesser of twin primes p such that p and p+2 are both in A115591.

Original entry on oeis.org

191, 311, 1487, 1871, 2711, 2999, 3167, 3767, 4967, 5519, 7559, 8087, 10271, 11351, 11831, 13679, 15647, 18311, 18911, 21647, 22271, 22367, 23687, 25799, 26711, 27239, 27527, 27791, 29399, 29879, 31727, 31847, 33287, 34367, 35591, 38447, 38567, 40127, 40847, 42071

Offset: 1

Amiram Eldar, Nov 14 2023

nonn

Primes p such that p+2 is also a prime and (p-1)/ord(2, p) = (p+1)/ord(2, p+2) = 2, where ord(2,k) is the multiplicative order of 2 modulo k.
Equivalently, lesser of twin primes p such that ord(2, p+2) = ord(2, p) + 1,
Equal consecutive values in A001917 that correspond to twin primes (p, p+2) are either 1 if p is in A319248, or 2 if p is in this sequence.
Terms are congruent to 23 modulo 24. - Jianing Song, Nov 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001359 and A115591.

Cf. A001122, A001917, A002326, A014664, A319248, A333743.

Select[Prime[Range[2, 4400]], PrimeQ[# + 2] && MultiplicativeOrder[2, # + 2] == MultiplicativeOrder[2, #] + 1 &]

is(n) = isprime(n) && isprime(n+2) && znorder(Mod(2, n + 2)) == znorder(Mod(2, n)) + 1;

A170822 Let p = n-th prime; a(n) = (p-1)/(order of A170821(n) mod p).

Original entry on oeis.org

1, 3, 2, 2, 1, 1, 2, 1, 1, 12, 1, 1, 2, 1, 2, 4, 1, 14, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 10, 1, 3, 1, 1, 4, 9, 2, 1, 2, 18, 2, 16, 1, 1, 1, 1, 2, 2, 1, 2, 6, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 10, 12, 1, 1, 42, 2, 12, 1, 2, 1, 4, 27, 2, 1, 4, 1, 6, 2, 6, 10, 4, 3, 2, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 1, 5

Offset: 3

N. J. A. Sloane, Dec 24 2009

nonn

			n=3: p=5, A170821(n)=2, order of 2 mod 5 = 4, (5-1)/4 = 1 = a(3).

I. Anderson and D. A. Preece, Combinatorially fruitful properties of 3*2^(-1) and 3*2^(-2) modulo p, Discr. Math., 310 (2010), 312-324.

Cf. A001917, A170820, A170821.

f(n) = my(p=prime(n), k=0); while(Mod(4*k, p) != 3, k++); k; \\ A170821
a(n) = my(p=prime(n)); (p-1)/znorder(Mod(f(n), p)); \\ Michel Marcus, Dec 04 2018

A182109 Records in A094593.

Original entry on oeis.org

1, 2, 4, 5, 6, 12, 16, 39, 84, 156, 350, 358, 589, 984, 2030, 2682, 3312, 4364, 19152, 61320, 61632, 142066, 353998, 702794, 1063044, 2056526, 2866334, 5479152, 8751462, 43544486

Offset: 1

Vassilis Papadimitriou, Apr 12 2012

It is for A094593 what A152598 is for A001917.

			First few terms of A094593 are 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 5, 1, 2, 1, 2, 6, 3, 2, 6, 1, 2, 1, 2, 1, 3, 2, 4, 1, 1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 2, 4, 2, 12, so a(1) to a(6) are 1, 2, 4, 5, 6, 12.

Cf. A094593.

a(26)-a(30) from Chai Wah Wu, Jan 15 2020

A174437 Successive maximal values of A174435.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 16, 31, 36, 48, 105, 129, 315, 316, 387, 538, 1285, 1542, 2048, 3855, 4599, 4864, 7760, 13797, 18166, 24417, 60787, 104694

Offset: 1

Vassilis Papadimitriou, Mar 19 2010

			First terms of A174435 are: 1,1,1,1,1,2,1,1,1,1,1,2,1,1,3,1,1,1,2,2,3,1,1,1,1,3,1,2,1,1,1,2,1,1,2,1,1,3,2,4,2,1,1,1,1,1,1,1,4,2,2,1,1,1,1,1,1,1,1,2,3,2,1,4,3,1,2,1,1,9, so a(1)=1, a(2)=2, a(3)=3, a(4)=4, a(5)=9.

It is for A174435 what A152598 is for A001917.

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

Original entry on oeis.org

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, 16664, 24263, 36168, 130439, 406583

Offset: 1

Eric Chen, Mar 13 2015

Here Phi_n is the n-th cyclotomic polynomial.
Is this sequence infinite?
Phi_n(2)/(2n+1) is only a probable prime for n > 16664.
a(33) > 2000000.
Subsequence of A005097 (2 * a(n) + 1 are all primes)
Subsequence of A081858.
2 * a(n) + 1 are in A115591.
Primes in this sequence are listed in A239638.
A085021(a(n)) = 2.
All a(n) are congruent to 0 or 3 (mod 4). (A014601)
All a(n) are congruent to 0 or 2 (mod 3). (A007494)
Except the term 20, all even numbers in this sequence are divisible by 8.

			Phi_11(2) = 23 * 89 and 23 = 2 * 11 + 1, so 11 is in this sequence.
Phi_35(2) = 71 * 122921 and 71 = 2 * 35 + 1, so 35 is in this sequence.
Phi_48(2) = 97 * 673 and 97 = 2 * 48 + 1, so 48 is in this sequence.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Will Edgington, Factorization of completely factored Phi_n(2) [from Internet Archive Wayback Machine]
Henri Lifchitz and Renaud Lifchitz, PRP records. Search for (2^a-1)/b
Samuel Wagstaff, The Cunningham project

Cf. A239638, A085724, A072226, A005384, A005097, A081858, A014601, A014664, A001917, A115591.

Select[Range[10000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[Cyclotomic[#, 2]/(2*#+1)] &]

isok(n) = if (((x=polcyclo(n, 2)) % (2*n+1) == 0) && (omega(x) == 2), print1(n, ", ")); \\ Michel Marcus, Mar 13 2015

cp's OEIS Frontend

A319350 Filter sequence which records the number of cyclotomic cosets of 2 mod p for odd primes p, and for any other number assigns a unique number.

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A002323 ((2^m - 1) / p) mod p, where p = prime(n) and m = ord(2,p).

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A170820 Let p = n-th prime; a(n) = (p-1)/(order of (p+3)/2 mod p).

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A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

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A319351 Filter sequence which records the number of cyclotomic cosets of 2 mod p^k for powers of odd primes p, and for any other number assigns a unique number.

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A367318 Lesser of twin primes p such that p and p+2 are both in A115591.

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A170822 Let p = n-th prime; a(n) = (p-1)/(order of A170821(n) mod p).

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A182109 Records in A094593.

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