A258409 -id:A258409 - cp's OEIS frontend

A280862 Numbers n such that A258409(n) * A002322(n) = A000010(n), where A258409(1) = 1.

1, 2, 4, 6, 10, 14, 15, 18, 21, 22, 26, 33, 34, 35, 38, 39, 45, 46, 50, 51, 54, 55, 57, 58, 62, 65, 69, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 98, 99, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 135, 141, 142, 143, 145, 146, 147, 153, 155, 158, 159, 161, 162, 166, 175, 177

Offset: 1

Thomas Ordowski and Altug Alkan, Jan 09 2017

nonn

Squarefree terms > 2 give A006881.
Even terms are even terms of A033948.
No terms of the form p^k > 4, where p is a prime.
The sequence has natural density 0. - Information from Carl Pomerance, Jan 09 2017

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

Cf. A000010, A002322, A006881, A033948, A258409.

{1}~Join~Select[Range@ 180, (GCD @@ (Divisors[#] - 1)) CarmichaelLambda@ # == EulerPhi@ # &] (* Michael De Vlieger, Jan 10 2017 *)

a258409(n) = if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[, 1]));
a002322(n) = lcm(znstar(n)[2]);
is(n) = a258409(n) * a002322(n) == eulerphi(n);

A284671 Numbers n > 2 such that A258409(n)*A002322(n) divides n-1.

Original entry on oeis.org

1729, 41041, 75361, 172081, 449065, 656601, 670033, 1050985, 2433601, 2704801, 3664585, 4903921, 6840001, 7995169, 8355841, 8719921, 9582145, 9613297, 9890881, 10402561, 11205601, 12945745, 13992265, 15888313, 16778881, 17586361, 17812081, 19683001, 20964961

Offset: 1

Thomas Ordowski and Altug Alkan, Apr 01 2017

nonn

A proper subset of the Carmichael numbers A002997.
If n is in the sequence, then n-1 is not squarefree.
Problem: are there infinitely many such numbers?

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

Cf. A002322, A002997, A258409.

Select[Range[3, 10^6], Divisible[#-1, CarmichaelLambda[#] * GCD @@ (Divisors[#] - 1)]  &] (* Amiram Eldar, Jun 26 2019 *)

A260300 Bisection of A258409: a(n) = A258409(2n+1).

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 2, 2, 58, 60, 2, 4, 66, 2, 70, 72, 2, 2, 78, 2, 82, 4, 2, 88, 6, 2, 2, 96, 2, 100, 102, 2, 106, 108, 2, 112, 2, 2, 2, 10, 2, 4, 126, 2, 130, 6, 2, 136, 138, 2, 2, 4, 2, 148, 150

Offset: 1

Michel Marcus, Nov 10 2015

nonn

Cf. A258409.

a(n) = gcd(apply(x->x-1, divisors(2*n+1)));

A049559 a(n) = gcd(n - 1, phi(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4

Offset: 1

Labos Elemer, Dec 28 2000

nonn

For prime n, a(n) = n - 1. Question: do nonprimes exist with this property?
Answer: No. If n is composite then a(n) < n - 1. - Charles R Greathouse IV, Dec 09 2013
Lehmer's totient problem (1932): are there composite numbers n such that a(n) = phi(n)? - Thomas Ordowski, Nov 08 2015
a(n) = 1 for n in A209211. - Robert Israel, Nov 09 2015

			a(9) = 2 because phi(9) = 6 and gcd(8, 6) = 2.
a(10) = 1 because phi(10) = 4 and gcd(9, 4) = 1.

Richard K. Guy, Unsolved Problems in Number Theory, B37.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A002322, A039766, A063994, A160595, A209211, A219428, A264012, A264024, A280262, A283656, A283872, A284089, A284440, A318827, A318829, A318830, A330747 (ordinal transform), A340195.

Cf. also A009195, A058515, A058663, A187730, A258409, A339964, A340071, A340081, A340087 for more or less analogous sequences.

[Gcd(n-1, EulerPhi(n)): n in [1..80]]; // Vincenzo Librandi, Oct 13 2018

seq(igcd(n-1, numtheory:-phi(n)), n=1..100); # Robert Israel, Nov 09 2015

Table[GCD[n - 1, EulerPhi[n]], {n, 93}] (* Michael De Vlieger, Nov 09 2015 *)

a(n)=gcd(eulerphi(n),n-1) \\ Charles R Greathouse IV, Dec 09 2013

from sympy import totient, gcd
print([gcd(totient(n), n - 1) for n in range(1, 101)]) # Indranil Ghosh, Mar 27 2017

a(p^m) = a(p) = p - 1 for prime p and m > 0. - Thomas Ordowski, Dec 10 2013
From Antti Karttunen, Sep 09 2018: (Start)
a(n) = A000010(n) / A160595(n) = A063994(n) / A318829(n).
a(n) = n - A318827(n) = A000010(n) - A318830(n).
(End)
a(n) = gcd(A000010(n), A219428(n)) = gcd(A000010(n), A318830(n)). - Antti Karttunen, Jan 05 2021

A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

Original entry on oeis.org

341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751

Offset: 1

Eric W. Weisstein

nonn

Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - T. D. Noe, Jan 11 2011
Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016
Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016
Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - Thomas Ordowski, Sep 12 2016
Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - Robert Israel, Sep 13 2016
Composite numbers n such that 2^A258409(n) == 1 (mod n). - Thomas Ordowski, Sep 15 2016

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

A214305 is a subsequence.
A065341 is a subsequence. - Thomas Ordowski, Nov 20 2016
Cf. A001220, A001567, A178997.

filter:= = proc(n)
    not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Sep 13 2016

Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]

is(n)=if(isprime(n), return(0)); fordiv(n,d, if(Mod(2,d)^d!=2, return(0))); n>1 \\ Charles R Greathouse IV, Aug 27 2016

A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 8, 12, 18, 27, 38, 53, 74, 102, 137, 184, 241, 317, 413, 536, 687, 880, 1112, 1405, 1765, 2215, 2755, 3424, 4229, 5216, 6402, 7847, 9572, 11662, 14148, 17139, 20688, 24940, 29971, 35969, 43044, 51438, 61311, 72985, 86678, 102807, 121675

Offset: 0

Gus Wiseman, Oct 09 2019

nonn

A partition is relatively prime if the GCD of its parts is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(2) = 1 through a(9) = 18 partitions:
  (2)  (21)  (22)   (32)    (42)     (43)      (62)       (54)
             (211)  (221)   (222)    (52)      (332)      (63)
                    (2111)  (321)    (322)     (422)      (72)
                            (2211)   (421)     (431)      (432)
                            (21111)  (2221)    (521)      (522)
                                     (3211)    (2222)     (621)
                                     (22111)   (3221)     (3222)
                                     (211111)  (4211)     (3321)
                                               (22211)    (4221)
                                               (32111)    (4311)
                                               (221111)   (5211)
                                               (2111111)  (22221)
                                                          (32211)
                                                          (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are given by A328168.
Partitions whose parts are relatively prime are A000837.
Partitions whose parts plus 1 are relatively prime are A318980.
The GCD of the prime indices of n, all minus 1, is A328167(n).
Cf. A007359, A018783, A258409, A289509, A328163, A328164.

Table[Length[Select[IntegerPartitions[n],GCD@@(#-1)==1&]],{n,0,30}]

seq(n)=Vec(sum(d=1, n-1, moebius(d)*(-1/(1-x) + 1/prod(k=0, n\d, 1 - x*x^(k*d) + O(x*x^n)))), -(n+1)) \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1/(1-x) + 1/(Prod_{k>=0} 1 - x^(k*d + 1))). - Andrew Howroyd, Oct 17 2019

A328168 Numbers whose prime indices minus 1 are relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 95, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 147

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

A multiset is relatively prime if the GCD of its elements is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of partitions whose parts minus one are relatively prime. The enumeration of these partitions by sum is given by A328170.

			The sequence of terms together with their prime indices begins:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}
   51: {2,7}
   54: {1,2,2,2}
   57: {2,8}

Positions of 1's in A328167.
Numbers whose prime indices are relatively prime are A289509.
The version for prime indices plus 1 is A318981.
The GCD of prime indices is A289508.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A056239, A112798, A258409, A281116, A290103, A328163, A328169.

q:= n-> igcd(map(i-> numtheory[pi](i[1])-1, ifactors(n)[2])[])=1:
select(q, [$1..150])[];  # Alois P. Heinz, Oct 13 2019

Select[Range[100],GCD@@(PrimePi/@First/@If[#==1,{},FactorInteger[#]]-1)==1&]

A328167 GCD of the prime indices of n, all minus 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 1, 0, 6, 1, 7, 2, 1, 4, 8, 1, 2, 5, 1, 3, 9, 1, 10, 0, 1, 6, 1, 1, 11, 7, 1, 2, 12, 1, 13, 4, 1, 8, 14, 1, 3, 2, 1, 5, 15, 1, 2, 3, 1, 9, 16, 1, 17, 10, 1, 0, 1, 1, 18, 6, 1, 1, 19, 1, 20, 11, 1, 7, 1, 1, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			85 has prime indices {3,7}, so a(85) = GCD(2,6) = 2.

Positions of 0's are A000079.
Positions of 1's are A328168.
Positions of records (first appearances) are A006005.
The GCD of the prime indices of n is A289508(n).
The GCD of the prime indices of n, all plus 1, is A328169(n).
Looking at divisors instead of prime indices gives A258409.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A001222, A007359, A056239, A112798, A289509, A318978, A318981, A328163, A328164.

Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]-1),{n,100}]

A328169 GCD of the prime indices of n, all plus 1.

Original entry on oeis.org

0, 2, 3, 2, 4, 1, 5, 2, 3, 2, 6, 1, 7, 1, 1, 2, 8, 1, 9, 2, 1, 2, 10, 1, 4, 1, 3, 1, 11, 1, 12, 2, 3, 2, 1, 1, 13, 1, 1, 2, 14, 1, 15, 2, 1, 2, 16, 1, 5, 2, 1, 1, 17, 1, 2, 1, 3, 1, 18, 1, 19, 2, 1, 2, 1, 1, 20, 2, 1, 1, 21, 1, 22, 1, 1, 1, 1, 1, 23, 2, 3, 2

Offset: 1

Gus Wiseman, Oct 09 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			85 has prime indices {3,7}, so a(85) = GCD(4,8) = 4.

Positions of 0's and 1's are A318981.
Positions of records (first appearances) appear to be A116974.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Cf. A000837, A056239, A112798, A258409, A289508, A289509, A290103, A328168.

Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]+1),{n,100}]

a(n) = A289508(A003961(n)).

A328219 LCM of the prime indices of n, all plus 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Sorted positions of first appearances are A328451.
LCM of prime indices is A290103.
LCM of prime indices minus 1 is A328456.
GCD of prime indices plus 1 is A328169.
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981,
Cf. A003961, A056239, A112798, A258409, A289508, A289509, A328167, A328168.

Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}]

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n)); \\ Antti Karttunen, Oct 18 2019

a(n) = A290103(A003961(n)).

If n = A000040(i_1) * ... * A000040(i_k), then a(n) = lcm(1+i_1,...,1+i_k).

cp's OEIS Frontend

A280862 Numbers n such that A258409(n) * A002322(n) = A000010(n), where A258409(1) = 1.

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A284671 Numbers n > 2 such that A258409(n)*A002322(n) divides n-1.

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A260300 Bisection of A258409: a(n) = A258409(2n+1).

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A049559 a(n) = gcd(n - 1, phi(n)).

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A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

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A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

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A328168 Numbers whose prime indices minus 1 are relatively prime.

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A328167 GCD of the prime indices of n, all minus 1.

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