A002997 -id:A002997 - cp's OEIS frontend

A326690 Denominator of the fraction (Sum_{prime p | n} 1/p - 1/n).

1, 1, 1, 4, 1, 3, 1, 8, 9, 5, 1, 4, 1, 7, 15, 16, 1, 9, 1, 20, 7, 11, 1, 24, 25, 13, 27, 28, 1, 1, 1, 32, 33, 17, 35, 36, 1, 19, 13, 40, 1, 21, 1, 44, 45, 23, 1, 16, 49, 25, 51, 52, 1, 27, 11, 8, 19, 29, 1, 60, 1, 31, 63, 64, 65, 11, 1, 68, 69, 35, 1, 72

Offset: 1

Jonathan Sondow, Jul 18 2019

Theorem. If n is a prime or a Carmichael number, then a(n) = A309132(n) = denominator of (N(n-1)/n + D(n-1)/n^2), where B(k) = N(k)/D(k) is the k-th Bernoulli number. This is a generalization of Theorem 1 in A309132 that A309132(p) = 1 if p is a prime. The proof generalizes that in A309132. As an application of Theorem, for n a prime or a Carmichael number one can compute A309132(n) without calculating Bernoulli numbers; see A309268.
A composite number n is a Giuga number A007850 if and only if a(n) = 1. (In fact, Sum_{prime p | n} 1/p - 1/n = 1 for all known Giuga numbers n.)
Semiprimes m = pq such that 1/p + 1/q - 1/m = p/q are exactly A190275. - Amiram Eldar and Thomas Ordowski, Jul 22 2019
The preceding comment may be rephrased as "Semiprimes m = pq such that A326689(m) = p and a(m) = q are exactly A190275." - Jonathan Sondow, Jul 22 2019
More generally, semiprimes m = pq such that 1/p + 1/q - 1/m = P/Q are exactly A190273, where P <> Q are primes. In other words, semiprimes m such that A326689(m) is prime and a(m) is prime are exactly A190273. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

			-1/1, 0/1, 0/1, 1/4, 0/1, 2/3, 0/1, 3/8, 2/9, 3/5, 0/1, 3/4, 0/1, 4/7, 7/15, 7/16, 0/1, 7/9, 0/1, 13/20, 3/7, 6/11, 0/1, 19/24, 4/25, 7/13, 8/27, 17/28, 0/1, 1/1
a(12) = denominator of (Sum_{prime p | 12} 1/p - 1/12) = denominator of (1/2 + 1/3 - 1/12) = denominator of 3/4 = 4.
Computing A309132(561) involves numerator(B(560)) which has 865 digits. But 561 is a Carmichael number, so Theorem implies A309132(561) = a(561) = denominator(1/3 + 1/11 + 1/17 - 1/561) = denominator(90/187) = 187.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Bernoulli number
Wikipedia, Carmichael number
Wikipedia, Giuga number

Numerators are A326689. Quotients n/a(n) are A326691.
Cf. A069359, A007947 (denominator of Sum_{prime p | n} 1/p).
Cf. A002997, A007850, A190275, A309132, A309235, A309268, A309378, A326692.

[1] cat [Denominator(&+[1/p:p in PrimeDivisors(k)]-1/k):k in [2..72]]; // Marius A. Burtea, Jul 27 2019

A326690 := n -> denom((A069359(n)-1)/n):
seq(A326690(n), n=1..72); # Peter Luschny, Jul 22 2019

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[ f[n], {n, 100}]

a(n) = denominator(sumdiv(n, d, isprime(d)/d) - 1/n); \\ Michel Marcus, Jul 19 2019

p = lambda n: [n//f[0] for f in factor(n)]
A326690 = lambda n: ((sum(p(n)) - 1)/n).denominator()
[A326690(n) for n in (1..72)] # Peter Luschny, Jul 22 2019

a(n) = 1 if n is a prime or a Giuga number A007850.
a(n) = denominator of (N(n-1)/n + D(n-1)/n^2) if n is a Carmichael number A002997.
a(n) = denominator((A069359(n) - 1)/n). - Peter Luschny, Jul 22 2019

A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 5, 13, 3, 7, 29, 15, 31, 2, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 1, 23, 47, 1, 7, 5, 17, 13, 53, 3, 55, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 5, 19, 77, 13, 79, 5, 3, 41, 83, 21

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 25 2019

Does not contain any elements of A324315, and thus none of the Carmichael numbers A002997.

See the section on Bernoulli polynomials in Kellner and Sondow 2019.

			For p = 2 and 3, the sum of the base p digits of 6 is 1+1+0 = 2 >= 2 and 2+0 = 2 < 3, respectively, so a(6) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A002997, A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324404, A324405.

f:= n -> convert(select(p -> convert(convert(n,base,p),`+`)Robert Israel, Apr 26 2020

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD3[n_] := Times @@ Select[LP[n], SD[n, #] < # &];
Table[DD3[n], {n, 1, 100}]

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) < p)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324369(n) = A007947(n) = radical(n).

a(n) * A195441(n) = a(n) * A324369(n) * A324370(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

A050992 4-Knödel numbers.

Original entry on oeis.org

6, 8, 12, 16, 20, 24, 28, 40, 44, 48, 52, 60, 68, 76, 80, 92, 112, 116, 120, 124, 148, 154, 164, 172, 188, 208, 212, 236, 240, 244, 264, 268, 280, 284, 292, 316, 332, 340, 356, 364, 388, 404, 412, 428, 436, 452, 508, 520, 524, 548, 556, 596, 604, 628, 652

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Knödel Numbers.

Cf. A002997, A050990, A033553, A050993, A208154-A208158.

Select[Range[6, 1000, 2], Divisible[# - 4, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

A050993 5-Knödel numbers.

Original entry on oeis.org

25, 65, 85, 145, 165, 185, 205, 265, 305, 365, 445, 485, 505, 545, 565, 685, 745, 785, 825, 865, 905, 965, 985, 1085, 1145, 1165, 1205, 1285, 1345, 1385, 1405, 1465, 1565, 1585, 1685, 1745, 1765, 1865, 1925, 1945, 1985, 2005, 2045, 2105, 2165

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
John H. Castillo and Jhony Fernando Caranguay Mainguez, The set of k-units modulo n, arXiv:1708.06812 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Knödel Numbers

Cf. A002997, A050990, A033553, A050992, A208154-A208158.

Select[Range[10, 2500, 5], Divisible[# - 5, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

A112428 Carmichael numbers equal to the product of 5 primes.

Original entry on oeis.org

825265, 1050985, 9890881, 10877581, 12945745, 13992265, 16778881, 18162001, 27336673, 28787185, 31146661, 36121345, 37167361, 40280065, 41298985, 41341321, 41471521, 47006785, 67371265, 67994641, 69331969, 74165065

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

A subsequence is given by (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) with n in A206349. - M. F. Hasler, Apr 14 2015

			a(1)=825265=5*7*17*19*73

R. J. Mathar, Table of n, a(n) for n = 1..14938

Cf. A002997, A014614, A206349.

Cf. A087788, A141711, A074379, A112429 - A112432, A006931.

is_A112428(n)=bigomega(n)==5&&is_A002997(n) \\ M. F. Hasler, Apr 14 2015

A112428 = A002997 intersect A014614. - M. F. Hasler, Apr 14 2015

Crossrefs added by M. F. Hasler, Apr 14 2015

A067256 Numbers n such that n, 2n+1, 3n+2 are primes.

Original entry on oeis.org

3, 5, 23, 29, 83, 89, 173, 233, 239, 293, 419, 659, 953, 1013, 1223, 1409, 1559, 1583, 1889, 2003, 2129, 2339, 2549, 2693, 2939, 3359, 3389, 3593, 3803, 4349, 4373, 4409, 4919, 4943, 5333, 6113, 6173, 8093, 8273, 8513, 9059, 9479, 9539, 10163, 10313

Offset: 1

Benoit Cloitre, Feb 20 2002

nonn

a(n)*(2a(n)+1)*(3a(n)+2) are Lucas-Carmichael numbers for n > 1. Analogous to A174734 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers). - Amiram Eldar, Aug 11 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A005384, A006972, A067257, A067258, A101767, A101768, A101769, A101770, A174734, A216925.

[n: n in [1..15000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(3*n+2)]; // Vincenzo Librandi, Oct 31 2014

Select[Prime[Range[10^4]],PrimeQ[2*#+1]&&PrimeQ[3*#+2] &] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

is(n)=isprime(3*n+2) && isprime(2*n+1) && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

A182816 Number of values b in Z/nZ such that b^n = b.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 4, 13, 4, 9, 2, 17, 4, 19, 4, 9, 4, 23, 4, 5, 4, 3, 8, 29, 8, 31, 2, 9, 4, 9, 4, 37, 4, 9, 4, 41, 8, 43, 4, 15, 4, 47, 4, 7, 4, 9, 8, 53, 4, 9, 4, 9, 4, 59, 8, 61, 4, 9, 2, 25, 24, 67, 4, 9, 16, 71, 4, 73, 4, 9, 8, 9, 8, 79, 4, 3, 4, 83, 8, 25, 4, 9, 4, 89, 8, 49, 4, 9, 4, 9, 4, 97, 4, 9, 4, 101, 8, 103, 4, 45, 4, 107, 4, 109, 8, 9, 8, 113

Offset: 1

M. F. Hasler, Dec 05 2010

nonn

a(n) is the number of nonnegative bases b < n such that b^n == b (mod n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063994.

f:= n -> mul(1+igcd(n-1,p[1]-1), p = ifactors(n)[2]):
map(f, [$1..200]); # Robert Israel, Sep 05 2018

Table[Times @@ Map[(1 + GCD[n - 1, # - 1]) &, FactorInteger[n][[All, 1]] ], {n, 113}] (* Michael De Vlieger, Sep 01 2020 *)

PARI
```
A182816(n)=sum(a=1,n,Mod(a,n)^n==a);
```

{ A182816(n) = my(p=factor(n)[,1]); prod(j=1,#p,1+gcd(n-1,p[j]-1)); } \\ Max Alekseyev, Dec 06 2010

a(n) = n for primes A000040 and Carmichael numbers A002997.
a(n) = Product_{i=1..m} (1 + gcd(n-1, p_i-1)), where p_1, p_2, ..., p_m are all distinct primes dividing n. - Max Alekseyev, Dec 06 2010
a(p^k) = p for prime p with k > 0. - Thomas Ordowski, Sep 05 2018

A208154 6-Knödel numbers.

Original entry on oeis.org

8, 10, 12, 18, 24, 30, 36, 42, 66, 72, 78, 84, 90, 102, 114, 126, 138, 168, 174, 186, 210, 222, 234, 246, 252, 258, 282, 318, 354, 366, 390, 396, 402, 426, 438, 456, 474, 498, 504, 534, 546, 582, 606, 618, 630, 642, 654, 678, 762, 786, 798, 822, 834, 894, 906

Offset: 1

Paolo P. Lava, Feb 24 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Knödel Numbers.

Cf. A002997, A050990, A033553, A050992, A050993, A208155-A208158.

with(numtheory);
knodel:= proc(i,k)
local a,n,ok;
for n from k+1 to i do
  ok:=1;
  for a from 1 to n do
     if gcd(a,n)=1 then  if (a^(n-k) mod n)<>1 then ok:=0; break; fi; fi;
  od;
  if ok=1 then print(n); fi;
od;
end:
knodel(10000,6);

knodelQ[m_Integer?PrimeQ, n_Integer] := False; knodelQ[m_Integer, n_Integer] := Module[{i = n + 1}, While[i < m && (GCD[i, m] > 1 || Mod[i^(m - n), m] == 1), i++]; (i == m)]; Select[Range[1000], knodelQ[#, 6] &] (* Alonso del Arte, Feb 24 2012 *)

A208158 10-Knödel numbers.

Original entry on oeis.org

12, 24, 28, 30, 50, 70, 110, 130, 150, 170, 190, 230, 290, 310, 330, 370, 410, 430, 442, 470, 530, 532, 550, 590, 610, 670, 710, 730, 790, 830, 890, 910, 970, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1650, 1670, 1730, 1790

Offset: 1

Paolo P. Lava, Feb 24 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Knödel Numbers

Cf. A002997, A050990, A033553, A050992, A050993, A208154-A208157.

with(numtheory);
knodel:=proc(i,k)
local a,n,ok;
for n from k+1 to i do
  ok:=1;
  for a from 1 to n do
     if gcd(a,n)=1 then  if (a^(n-k) mod n)<>1 then ok:=0; break; fi; fi;
  od;
  if ok=1 then print(n); fi;
od;
end:
knodel(10000,10)

Select[Range[12, 1790, 2], Divisible[# - 10, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

A324319 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also hexagonal numbers (A000384) with index equal to their largest prime factor.

Original entry on oeis.org

231, 561, 3655, 5565, 8911, 10585, 13695, 23653, 32131, 45451, 59685, 74305, 108345, 115921, 157641, 243253, 248865, 302253, 314821, 334153, 371091, 392055, 417241, 458403, 505515, 546535, 688551, 702705, 795691, 821121, 915981, 932295, 1004653, 1145341, 1181953

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

561, 8911, and 10585 are also Carmichael numbers (A002997).
The smallest primary Carmichael number (A324316) in the sequence is 8801128801 = 181 * 733 * 66337 = A000384(66337).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(1) = 231 = 3 * 7 * 11 = 11 * (2 * 11 - 1) = A000384(11), so 231 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000384, A002997, A195441, A324315, A324316, A324317, A324318, A324320, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
HN[n_] := n(2n - 1);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[HN@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

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A326690 Denominator of the fraction (Sum_{prime p | n} 1/p - 1/n).

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A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

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A050992 4-Knödel numbers.

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A050993 5-Knödel numbers.

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A112428 Carmichael numbers equal to the product of 5 primes.

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A067256 Numbers n such that n, 2n+1, 3n+2 are primes.

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A182816 Number of values b in Z/nZ such that b^n = b.

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