A002174 -id:A002174 - cp's OEIS frontend

A074738 Decimal expansion of d = 1-(1+log(log(2)))/log(2) = 0.08607133....

0, 8, 6, 0, 7, 1, 3, 3, 2, 0, 5, 5, 9, 3, 4, 2, 0, 6, 8, 8, 7, 5, 7, 3, 0, 9, 8, 7, 7, 6, 9, 2, 2, 6, 7, 7, 7, 6, 0, 5, 9, 1, 1, 0, 9, 5, 3, 0, 3, 3, 3, 1, 7, 3, 4, 9, 2, 0, 2, 0, 2, 3, 6, 6, 6, 5, 4, 2, 2, 6, 3, 5, 8, 1, 4, 6, 2, 2, 8, 7, 9, 7, 9, 9, 3, 8, 0, 5, 3, 4, 6, 0, 2, 5, 2, 8, 7, 6, 8, 0, 7, 1, 6, 3

Offset: 0

Benoit Cloitre, Sep 05 2002

An Erdős constant: let s(x) denotes the number of numbers < x expressible as a product of 2 numbers less than or equal to sqrt(x). Erdős showed that S(x) is x/(log x)^(d+o(1)) where d is this constant.
Ford finds that, if H(x,y,z) is the number of integers n <= x which have a divisor in the interval (y,z] and for 3 <= y <= sqrt(x), H(x,y,2y) = x/(((log y)^delta)(log log y)^(3/2)) where delta is the Erdős constant whose decimal digits are A074738. - Jonathan Vos Post, Jul 19 2007
Occurs, citing Ford, in p.2 of Koukoulopoulos. - Jonathan Vos Post, May 18 2010
Luca & Pomerance call this the Erdős-Tenenbaum-Ford constant and show its relationship to the reduced totient function A002174. - Charles R Greathouse IV, Dec 28 2013

G. C. Greubel, Table of n, a(n) for n = 0..10000
Kevin Ford, Integers with a divisor in (y,2y], arXiv:math/0607473 [math.NT], 2006-2013.
Andrew Granville, Cihan Sabuncu, and Alisa Sedunova, The multiplication table constant and sums of two squares, arXiv:2308.14911 [math.NT], 2023.
Dimitris Koukoulopoulos, Divisors of shifted primes, arXiv:0905.0163 [math.NT], 2009-2010; International Mathematics Research Notices, 2010:24, pp. 4585-4627.
Florian Luca and Carl Pomerance, On the range of Carmichael's universal-exponent function, Acta Arithmetica 162 (2014), pp. 289-308.
G. Tenenbaum, Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Mathematica, 51 no. 2 (1984), p. 243-263 (see Theorem 1).

1-(1+Log(Log(2)))/Log(2); // G. C. Greubel, Apr 16 2018

evalf(1-(1+log(log(2)))/log(2), 119);  # Alois P. Heinz, Aug 30 2023

Join[{0}, RealDigits[1 - (1 + Log[Log[2]])/Log[2], 10, 100][[1]]] (* G. C. Greubel, Apr 16 2018 *)

1-(1+log(log(2)))/log(2) \\ Michel Marcus, Mar 14 2013

A304480 a(n) is the least m such that lambda(k) >= n for all k >= m where lambda is A002322, the Carmichael lambda function.

Original entry on oeis.org

1, 3, 25, 25, 241, 241, 505, 505, 505, 505, 505, 505, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 131041, 131041, 131041, 131041, 131041, 131041, 171865, 171865, 171865, 171865, 171865, 171865, 138181681, 138181681, 138181681, 138181681, 138181681, 138181681

Offset: 1

Michel Marcus, May 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Project Euler, Problem 533 - Minimum values of the Carmichael function
Wikipedia, Carmichael function

Cf. A002174, A002322, A002396, A141162, A143407, A335116, A335117.

minvl(n, v) = {vgt = select(x->(x>=n), v, 1); first = vgt[1]; vgtd = vector(#vgt-1, k, vgt[k+1] - vgt[k]); vgtdr = Vecrev(vgtd); vokdiff = select(x->x!=1, vgtdr, 1); if (#vokdiff, #v - vokdiff[1]+1, first);}
lista(nn) = {v = read("v002322.txt"); for (n=1, nn, print1(minvl(n, v), ", "););}

a(32) and beyond from Seiichi Manyama, May 24 2020

A141162 Smallest k such that lambda(k) = n, or 0 if there is no such k.

Original entry on oeis.org

1, 3, 0, 5, 0, 7, 0, 32, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 224, 0, 0, 0, 29, 0, 31, 0, 128, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 115, 0, 47, 0, 119, 0, 0, 0, 53, 0, 81, 0, 928, 0, 59, 0, 61, 0, 0, 0, 256, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 187, 0, 83, 0, 203, 0, 0, 0, 89, 0, 209, 0, 235, 0, 0, 0, 97, 0

Offset: 1

Michel Lagneau, Mar 17 2011

nonn

Sequence A002174 gives the n such that a(n) > 0. Removing the zeros from this sequence produces A002396. Note that some n appear only for large k. For example, 728 does not appear until k=49184. See A143407 for the largest k that produces a particular value of the lambda function. See A143408 for the number of times each value occurs. - T. D. Noe, Mar 17 2011

			a(8) = 32 because lambda(32) = 8.

Cf. A002174, A002322 (Carmichael lambda function), A002396, A143407, A143408.

with(numtheory):for k from 1 to 100 do:id:=0:for n from 1 to 1000 while(id=0)
  do: if lambda(n) = k then id:=1:printf(`%d, `,n):else fi:od:if id=0 then printf(`%d, `,0):else fi:od:

nn = 100; t = Table[0, {nn}]; Do[c = CarmichaelLambda[k]; If[c <= nn && t[[c]] == 0, t[[c]] = k], {k, 1000}]; t

a(A002174(n)) = A002396(n).

A270564 Terms of A143407, sorted.

Original entry on oeis.org

2, 24, 240, 264, 480, 504, 552, 1128, 1416, 1992, 2568, 4008, 4296, 5448, 5520, 5736, 6312, 6960, 8328, 8616, 9192, 10632, 11208, 11280, 11496, 12072, 12408, 12720, 13200, 13512, 13920, 14088, 14160, 15528, 15576, 15816, 16320, 17256, 18744, 19848, 19920, 20136, 20712, 21288, 21912, 22560, 23592

Offset: 1

Joerg Arndt, Mar 19 2016

nonn

Numbers m such that r is the maximal order in the multiplicative group modulo m and there is no M > m with the same maximal order modulo M.

Cf. A143407, A002322, A002174, A270562.

A302099 Decompose the multiplicative group of integers modulo N as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) is the smallest N such that the product contains a copy of C_{2n}.

Original entry on oeis.org

3, 5, 7, 32, 11, 13, 1247, 17, 19, 25, 23, 224, 4187, 29, 31, 128, 14111, 37, 43739, 41, 43, 115, 47, 119, 15251, 53, 81, 928, 59, 61, 116003, 256, 67, 70555, 71, 73, 33227, 174269, 79, 187, 83, 203, 74563, 89, 209, 235, 186497, 97, 67571, 101, 103

Offset: 1

Jianing Song, Apr 01 2018

nonn

a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there must exist two primes with the form 2a*n + 1 and 2b*n + 1 where at least one of a,b is coprime to 2n, then the multiplicative group of integers modulo (2a*n + 1)(2b*n + 1) is isomorphic to C_{2*n} x C_{2ab*n}.
Factorizations of a(n) where 2n is not a term in A002174: a(7) = 29*43, a(13) = 53*79, a(17) = 103*137, a(19) = 191*229, a(25) = 101*151, a(31) = 311*373, a(34) = 5*103*137, a(37) = 149*223, a(38) = 229*761, a(43) = 173*431, a(47) = 283*659, a(49) = 7^3*197. - Jianing Song, Apr 29 2018 [Corrected on Sep 15 2018]
It may appear that for odd n, A046072(a(n)) = 1 or 2, but this is not generally true. The smallest counterexample is a(85) = 1542013, as the multiplicative group of integers modulo 1542013 is isomorphic to C_2 x C_170 x C_4080. - Jianing Song, Sep 15 2018

			For n = 7 the multiplicative group of integers modulo 1247 is isomorphic to C_14 x C_84, and 1247 is the smallest number that contains a copy of C_14 in the product of cyclic groups, so a(7) = 1247.
For n = 34 the multiplicative group of integers modulo 70555 is isomorphic to C_2 x C_68 x C_408, and 70555 is the smallest number that contains a copy of C_68 in the product of cyclic groups, so a(34) = 70555. - _Jianing Song_, Sep 15 2018

Jianing Song, Table of n, a(n) for n = 1..200
Jianing Song, Group structure of the multiplicative group of integers modulo a(1) to a(200)
Jianing Song, Factorizations of a(1) to a(200)
Wikipedia, Multiplicative group of integers modulo n

Cf. A002174, A002322, A002396.

a(n)=my(i=3, Z=[2]); while(prod(j=1, #Z, 1-(Z[j]==2*n)), i++&&Z=znstar(i)[2]); i \\ Jianing Song, Sep 15 2018

Some terms corrected by Jianing Song, Apr 29 2018

Some terms corrected by Jianing Song, Sep 15 2018

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A074738 Decimal expansion of d = 1-(1+log(log(2)))/log(2) = 0.08607133....

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A304480 a(n) is the least m such that lambda(k) >= n for all k >= m where lambda is A002322, the Carmichael lambda function.

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A141162 Smallest k such that lambda(k) = n, or 0 if there is no such k.

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A270564 Terms of A143407, sorted.

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A302099 Decompose the multiplicative group of integers modulo N as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) is the smallest N such that the product contains a copy of C_{2n}.

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