A007733 -id:A007733 - cp's OEIS frontend

A014662 Primes p such that order of 2 mod p (=A007733(p)) is even.

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419

Offset: 1

N. J. A. Sloane

nonn

Apart from the first term, identical to A091317. - Charles R Greathouse IV, Feb 13 2009

Dirichlet density is 5/24 (Fein, Gordon, & Smith); they show a result on expressing -1 as the sum of two squares relating to this sequence. - Charles R Greathouse IV, May 15 2024

P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000
Burton Fein, Basil Gordon, and John H. Smith, On the representation of -1 as a sum of two squares in an algebraic number field J. Num. Theor. (1971) Vol. 3, Issue 3, 310-315.
Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

The prime terms of A296243.

Cf. A091317.

[ p: p in PrimesInInterval(3, 419) | IsEven(Modorder(2, p)) ]; // Klaus Brockhaus, Dec 09 2008

select(t -> isprime(t) and numtheory:-order(2,t)::even, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

Select[Prime[Range[80]], EvenQ[MultiplicativeOrder[2, #/(2^IntegerExponent[ #, 2])]]&] (* Jean-François Alcover, Sep 02 2018 *)

isok(p) = isprime(p) && !(znorder(Mod(2, p/2^valuation(p, 2))) % 2); \\ Michel Marcus, Sep 02 2018

is(n)=n>2 && Mod(2,n)^(n>>valuation(n-1,2))!=1 && isprime(n) \\ Charles R Greathouse IV, May 07 2024

More terms from Klaus Brockhaus, Dec 09 2008

A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 14, 23, 9, 29, 42, 40, 65, 80, 90, 31, 40, 121, 44, 142, 189, 109, 61, 115, 77, 302, 273, 148, 318, 94, 434, 532, 20, 497, 115, 86, 148, 826, 702, 271, 148, 355, 230, 601, 119, 220, 265, 131, 299, 1178, 297, 485, 86, 265, 1430, 838, 320, 328, 271, 556, 1769, 1957, 1890, 50, 142, 2017, 148, 751, 2277, 179, 373, 832, 665, 2932, 54, 856, 485

Offset: 1

Antti Karttunen, May 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A007733, A046523, A286160, A286161.

A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from Michel Marcus, Apr 11 2015
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286573(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n));

from sympy import divisors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a002326(n):
    m=1
    while True:
        if (2**m - 1)%(2*n + 1)==0: return m
        else: m+=1
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a007733(n): return a002326((a000265(n) - 1)/2)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a007733(n), a046523(n)) # Indranil Ghosh, May 26 2017

a(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n)).

A256757 Number of iterations of A007733 required to reach 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 1, 3, 2, 3, 3, 4, 4, 3, 2, 3, 3, 4, 3, 3, 4, 5, 2, 4, 3, 2, 3, 4, 4, 3, 3, 4, 4, 5, 2, 3, 3, 3, 1, 3, 3, 4, 2, 4, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 2, 5, 3, 4, 3, 2, 4, 4, 3, 4, 3, 3, 4, 3, 5, 4, 2, 3, 4, 3, 3

Offset: 1

Ivan Neretin, Apr 09 2015

In other words, the minimal height (not counting k) of the power tower 2^(2^(...^(2^k)...)) required to make it eventually constant modulo n (=A245970(n)) for sufficiently large k.

a(n) <= A227944(n) + 1. - Max Alekseyev, Oct 11 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007733, A256607 (second iteration), A256758 (positions of records), A003434, A227944 (similarly built upon the totient function).

a256757 n = fst $ until ((== 1) . snd)
            (\(i, x) -> (i + 1, fromIntegral $ a007733 x)) (0, n)
-- Reinhard Zumkeller, Apr 13 2015

A007733 = Function[n, MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]];
a = Function[n, k = 0; m = n; While[m > 1, m = A007733[m]; k++]; k];
Table[a[n], {n, 100}] (* Ivan Neretin, Apr 13 2015 *)

a(n) = {if (n==1, return(0)); nb = 1; while((n = znorder(Mod(2, n/2^valuation(n, 2)))) != 1, nb++); nb;} \\ Michel Marcus, Apr 11 2015

For n>1, a(n) = a(A007733(n)) + 1.

A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 7, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 27, 4, 28, 15, 29, 8, 30, 16, 31, 1, 32, 17, 33, 9, 34, 7, 35, 5, 36, 18, 37, 10, 38, 19, 39, 3, 40, 20, 41, 11, 42, 21, 43, 6, 44, 22, 45, 12, 46, 23, 47, 2, 48, 24, 49, 13, 50

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A007733(n), A278222(n)].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A003602, A007733, A278222, A286622, A324400, A336933, A336934.

up_to = 65537;
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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux336935(n) = [A007733(n), A278222(n)];
v336935 = rgs_transform(vector(up_to, n, Aux336935(n)));
A336935(n) = v336935[n];

A336933 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 3, 1, 8, 5, 9, 3, 5, 6, 10, 2, 11, 7, 9, 4, 12, 3, 13, 1, 6, 8, 7, 5, 14, 9, 7, 3, 11, 5, 15, 6, 7, 10, 16, 2, 17, 11, 8, 7, 18, 9, 11, 4, 9, 12, 19, 3, 20, 13, 5, 1, 7, 6, 21, 8, 22, 7, 23, 5, 24, 14, 11, 9, 25, 7, 26, 3, 27, 11, 28, 5, 8, 15, 12, 6, 10, 7, 7, 10, 6, 16, 14, 2, 29, 17, 25, 11, 30, 8, 31, 7, 7

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of A007733.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007733, A336934, A336935, A336936.

up_to = 65537;
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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
v336933 = rgs_transform(vector(up_to, n, A007733(n)));
A336933(n) = v336933[n];

A336934 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 18, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 27, 4, 28, 15, 29, 8, 30, 16, 31, 1, 18, 17, 32, 9, 33, 18, 34, 5, 35, 19, 36, 10, 37, 18, 38, 3, 39, 20, 40, 11, 25, 21, 41, 6, 12, 22, 18, 12, 17, 23, 42, 2, 43, 24, 44, 13, 45, 25, 46, 7, 47

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A007733(n), A336158(n)].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007733, A336158, A336933, A336935, A336936.

up_to = 65537;
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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
Aux336934(n) = [A007733(n), A336158(n)];
v336934 = rgs_transform(vector(up_to, n, Aux336934(n)));
A336934(n) = v336934[n];

A336936 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A007733(n), A329697(n), A331410(n)], for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 10, 4, 14, 8, 15, 1, 16, 9, 17, 5, 18, 10, 17, 3, 19, 11, 20, 6, 21, 12, 22, 2, 23, 13, 24, 7, 25, 10, 26, 4, 27, 14, 28, 8, 29, 15, 30, 1, 21, 16, 31, 9, 32, 17, 33, 5, 34, 18, 35, 10, 36, 17, 37, 3, 38, 19, 39, 11, 40, 20, 41, 6, 42, 21, 43, 12, 44, 22, 45, 2, 46, 23, 47, 13

Offset: 1

Antti Karttunen, Aug 11 2020

nonn

Restricted growth sequence transform of the triplet [A007733(n), A329697(n), A331410(n)], or equally, of the ordered pair [A007733(n), A335880(n)].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003602, A007733, A329697, A331410, A335880.

Cf. also A324400, A336920, A336933, A336934, A336935.

up_to = 65537;
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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
Aux336936(n) = [A007733(n), A329697(n), A331410(n)];
v336936 = rgs_transform(vector(up_to, n, Aux336936(n)));
A336936(n) = v336936[n];

A351453 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j) and A007733(i) = A007733(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 4, 2, 9, 6, 10, 4, 11, 7, 12, 3, 13, 8, 14, 5, 15, 4, 16, 2, 7, 9, 17, 6, 18, 10, 8, 4, 19, 11, 20, 7, 21, 12, 22, 3, 23, 13, 9, 8, 24, 14, 25, 5, 10, 15, 26, 4, 27, 16, 11, 2, 8, 7, 28, 9, 29, 17, 30, 6, 31, 18, 13, 10, 32, 8, 33, 4, 34, 19, 35, 11, 9, 20, 15, 7, 36, 21, 8, 12, 37, 22, 38, 3, 39, 23, 32, 13, 40, 9, 41, 8, 17

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered pair [A006530(n), A007733(n)].

For all i, j >= 1: A324400(i) = A324400(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A007733.

Cf. also A324400, A351452, A351454, A351460.

up_to = 65537;
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; };
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from A007733
Aux351453(n) = [A006530(n), A007733(n)];
v351453 = rgs_transform(vector(up_to, n, Aux351453(n)));
A351453(n) = v351453[n];

A296244 Odd numbers k such that A007733(k) is even but 2^x == -1 (mod k) is insoluble.

Original entry on oeis.org

15, 21, 35, 39, 45, 51, 55, 63, 69, 75, 77, 85, 87, 91, 93, 95, 105, 111, 115, 117, 119, 123, 133, 135, 141, 143, 147, 153, 155, 159, 165, 175, 183, 187, 189, 195, 203, 207, 213, 215, 219, 221, 225, 231, 235, 237, 245, 247, 253, 255, 259, 261, 267, 273, 275, 279, 285, 287

Offset: 1

Max Alekseyev, Dec 09 2017

nonn

Set difference of A296243 and A014657.

Cf. A002326, A007733.

A333745 Numbers k such that the binary representations of 1/k and 1/(k+1) have the same period (A007733).

Original entry on oeis.org

1, 90, 104, 164, 286, 457, 665, 702, 740, 836, 975, 1458, 1469, 1628, 2071, 2146, 2625, 2849, 3800, 4441, 4575, 5233, 5284, 5418, 5715, 6039, 6073, 6387, 6458, 6601, 6649, 7384, 7417, 8029, 8521, 9817, 10136, 11306, 11439, 11497, 11642, 12402, 12651, 13050, 13322

Offset: 1

Amiram Eldar, Apr 03 2020

nonn

Numbers k such that A007733(k) = A007733(k+1).

			1 is a term since A007733(1) = A007733(2) = 1.
90 is a term since A007733(90) = A007733(91) = 12.

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

Cf. A007733, A333746.

f[n_] := MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]; Select[Range[10^4], f[#] == f[# + 1] &]

cp's OEIS Frontend

A014662 Primes p such that order of 2 mod p (=A007733(p)) is even.

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A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A256757 Number of iterations of A007733 required to reach 1.

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A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(j), for all i, j >= 1.

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A336933 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j), for all i, j >= 1.

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A336934 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A336936 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A007733(n), A329697(n), A331410(n)], for all i, j >= 1.

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A351453 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j) and A007733(i) = A007733(j) for all i, j >= 1.

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