Jonathan Sondow - cp's OEIS frontend

A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.

0, 1, 1, 1, 1, -37, -211, 2311, 37153, -818946931, 277930363757, -711223555487930419, -6367871182840222481, 35351107998094669831, 12690449182849194963361, -15116334304443206742413679091, 1431925649981017658678758915153153, -19921854762028779869513196624259348280501

Offset: 1

Jonathan Sondow, Aug 15 2019

sign

a(n) is an integer, as conjectured by Thomas Ordowski and proved by the author in A309132 and A326690.

Ordowski also conjectured that the sequence is a subsequence of A174341.

			Prime(6) = 13 and B(12) = -691/2730, so a(6) = -691/13 + 2730/13^2 = -37.

Wikipedia, Bernoulli number

Cf. A174341, A309132, A326690.

a[n_] := With[{p = Prime[n]}, With[{b = BernoulliB[p - 1]}, (p  Numerator[b] + Denominator[b])/p^2]];
Table[a[n], {n, 1, 18}]

a(n) = my(p = prime(n), b = bernfrac(p-1)); numerator(b)/p + denominator(b)/p^2; \\ Michel Marcus, Aug 16 2019

A326716 3-term arithmetic progressions of primes whose indices are also primes in arithmetic progression.

Original entry on oeis.org

5, 11, 17, 461, 617, 773, 401, 599, 797, 877, 1087, 1297, 1471, 1597, 1723, 1217, 1847, 2477, 3001, 3259, 3517, 3001, 3637, 4273, 2417, 3407, 4397, 2081, 3299, 4517, 4339, 4549, 4759, 3733, 4801, 5869, 7193, 8117, 9041, 11927, 12203, 12479, 13103, 13217, 13331

Offset: 1

Jonathan Sondow, Aug 11 2019

3-term arithmetic progressions are ordered first by highest term, then by middle term, and last by lowest term.

Is there a proof that the sequence is infinite?

			The indices of 5,11,17 form the arithmetic progression of primes 3,5,7.
The indices of 461,617,773 form the arithmetic progression of primes 89,113,137.

Cf. A000040, A000720, A006450, A231406.

l:= NULL: nn:= 2000:  # nn = upper limit for index of largest prime found
for n from 3 to nn do
  if isprime(n) then
    for i from iquo(n-1, 2) to 1 by -1 do
      if isprime(n-i) and isprime(n-2*i) then
        p, q, r:= map(ithprime, [seq(n-i*j, j=0..2)])[];
        if p-q = q-r then l:= l, r, q, p
fi fi od fi od: l;  # Alois P. Heinz, Aug 12 2019

a(3*k+2) - a(3*k+1) = a(3*k+3) - a(3*k+2) for k >= 0.
pi(a(3*k+2)) - pi(a(3*k+1)) = pi(a(3*k+3)) - pi(a(3*k+2)) for k >= 0.
a(n) = prime(pi(a(n))) = A000040(A000720(a(n))).
pi(a(n)) = prime(pi(pi(a(n)))).

More terms from Alois P. Heinz, Aug 12 2019

A326715 Values of n for which the denominator of (Sum_{prime p | n} 1/p - 1/n) is 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

n is in the sequence iff either n = 1 or n is a prime or n is a Giuga number, by one definition of Giuga numbers A007850.

			a(30) = denominator(Sum_{prime p | 30} 1/p - 1/30) = denominator(1/2 + 1/3 + 1/5 - 1/30) = denominator(1/1) = 1, and 30 is a Giuga number.

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

Cf. A007850, A326689, A326690, A326691, A326692.

filter:= proc(n) local p;
   denom(add(1/p, p = numtheory:-factorset(n))-1/n)=1
end proc:
select(filter, [$1..300]); # Robert Israel, Dec 15 2020

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == 1 &]

n such that A326690(n) = 1.

A326691 a(n) = n/denominator(Sum_{prime p | n} 1/p - 1/n).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 3, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 1, 29, 30, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 2, 43, 1, 1, 2, 47, 3, 1, 2, 1, 1, 53, 2, 5, 7, 3, 2, 59, 1, 61, 2, 1, 1, 1, 6, 67, 1, 1, 2, 71, 1, 73, 2, 3, 1, 1, 2, 79

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

Denominator(Sum_{prime p | n} 1/p - 1/n) is a factor of n, since all primes in the sum divide n. So a(n) is an integer.

			a(18) = 18/denominator(Sum_{prime p | 18} 1/p - 1/18) = 18/denominator(1/2 + 1/3 - 1/18) = 18/denominator(7/9) = 18/9 = 2.
a(30) = 30/denominator(Sum_{prime p | 30} 1/p - 1/30) = 30/denominator(1/2 + 1/3 + 1/5 - 1/30) = 30/denominator(1/1) = 30/1 = 30, and 30 is a Giuga number.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Christian Krause, et al, LODA, an assembly language, a computational model and a tool for mining integer sequences
Wikipedia, Giuga number

Cf. A003415, A007850, A326689, A326690, A326692, A326715.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[n/f[n], {n, 79}]

A326691(n) = (n/A326690(n)); \\ Antti Karttunen, Mar 15 2021

a(n) = n/A326690(n).
a(n) = n > 1 iff n is either a prime or a Giuga number A007850.
a(n) = gcd(n, 1+((n-1)*A003415(n))). [Conjectured, after an empirical formula found by LODA miner. This holds at least up to n=2^27] - Antti Karttunen, Mar 15 2021

A326692 Values of k for which the denominator of (Sum_{prime p | k} 1/p - 1/k) is k.

Original entry on oeis.org

1, 4, 8, 9, 15, 16, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 49, 51, 52, 60, 63, 64, 65, 68, 69, 72, 76, 77, 81, 85, 87, 88, 91, 92, 95, 96, 99, 100, 104, 108, 112, 115, 116, 117, 119, 121, 123, 124, 125, 128, 133, 135, 136, 140, 141, 143, 144, 145, 148

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

Any prime power p^k with k > 1 is a term, as 1/p - 1/p^k = (p^(k-1) - 1)/p^k which is in reduced form and has denominator p^k.

Are there infinitely many Carmichael numbers A002997 in the sequence?

			1/3 + 1/5 - 1/15 = 7/15 has denominator 15, so 15 is a term.

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

Cf. A002997, A326689, A326690, A326691, A326715.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == # &]

is(k) = {my(p = factor(k)[,1]); denominator(sum(i = 1, #p, 1/p[i]) - 1/k) == k;} \\ Amiram Eldar, Apr 26 2024

Solutions of A326690(x) = x. That is, fixed points of A326690.

A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

Original entry on oeis.org

-1, 0, 0, 1, 0, 2, 0, 3, 2, 3, 0, 3, 0, 4, 7, 7, 0, 7, 0, 13, 3, 6, 0, 19, 4, 7, 8, 17, 0, 1, 0, 15, 13, 9, 11, 29, 0, 10, 5, 27, 0, 20, 0, 25, 23, 12, 0, 13, 6, 17, 19, 29, 0, 22, 3, 5, 7, 15, 0, 61, 0, 16, 29, 31, 17, 10, 0, 37, 25, 29, 0, 59, 0, 19, 13, 41

Offset: 1

Jonathan Sondow, Jul 18 2019

See Comments on denominators in A326690.

			-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

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

Denominators are A326690. Cf. also A007850, A309132, A309235, A309378.

Cf. A028235.

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

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

a(p) = 0 if p is a prime.

a(g) = 1 if g is a known Giuga number (see my 2nd comment in A007850).

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

Original entry on oeis.org

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

A324977 Denominator(Bernoulli_{m-1}) / m, where m is the n-th Carmichael number.

Original entry on oeis.org

26805565070, 76004922, 702286000198710990, 302278602666, 5360679390, 423023231634556544606744470770, 582934735516230690164248578, 106515855804560422705933720818, 8763422623117673428800595536306232967379299351012370, 9231375124608836430, 94422948020637332890056101961518875879389605546105043450762033482730

Offset: 1

Jonathan Sondow, Mar 28 2019

nonn

a(n) is an integer, because an odd composite number m is a Carmichael number iff m divides the denominator of Bernoulli_{m-1} (by Korselt's criterion and the von Staudt-Clausen theorem). See Pomerance, Selfridge, & Wagstaff, page 1006, and Kellner & Sondow, section on Bernoulli numbers.

			The 1st Carmichael number is 561, and the denominator of Bernoulli_560 is 15037922004270, so a(1) = 15037922004270 / 561 = 26805565070.

Amiram Eldar, Table of n, a(n) for n = 1..358
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 preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Carl Pomerance, John L. Selfridge, and Samuel S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., 35 (1980), 1003-1026.
Index entries for sequences related to Bernoulli numbers.
Index entries for sequences related to Carmichael numbers.

Cf. A027642, A002997.

with(numtheory): A324977 := proc(n) local C, Fc;
if n = 1 or irem(n,2) = 0 or isprime(n) then return NULL fi;
Fc := select(isprime, map(i->i+1, divisors(n-1)));
C := mul(i, i=Fc); if irem(C, n) <> 0 then NULL else C/n fi end:
seq(A324977(n), n=1..40000); # Peter Luschny, May 21 2019

carnum = Cases[Range[1, 100000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
Table[Denominator[BernoulliB[m - 1]]/m, {m, carnum}]

a(n) = A027642(A002997(n)-1)/A002997(n).

A324976 Rank of the n-th primary Carmichael number.

Original entry on oeis.org

12, 8, 18, 12, 52, 52, 20, 32, 16, 54, 8, 36, 124, 34, 12, 72, 96, 26, 28, 76, 98, 1804, 108, 124, 18, 72, 172, 120, 10, 104, 32, 244, 130, 376, 18, 92, 780, 36, 172, 92, 284, 24, 198, 12, 244, 64, 234, 340, 100, 284, 24, 124, 44, 518, 364, 16, 82, 148, 8, 206

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a primary Carmichael number A324316 by Kellner and Sondow 2019.

			If m = A324316(1) = 1729 = 7*13*19, then p = 19, so a(1) = 2+2*((1729/19)-1)/(19-1) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Wikipedia, Polygonal number

Subsequence of A324975 (rank of the n-th Carmichael number A002997) and of A324974 (rank of the n-th special polygonal number A324973).

Cf. also A324316, A324972.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
T = Select[Range[1, 10^7, 2], TestCP[#] &];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

a(n) = 2+2*((m/p)-1)/(p-1), where m = A324316(n) and p is its greatest prime factor. Hence a(n) is even; see Formula in A324975.

More terms from Amiram Eldar, Mar 27 2019

A324973 Special polygonal numbers.

Original entry on oeis.org

6, 15, 66, 70, 91, 190, 231, 435, 561, 703, 715, 782, 861, 946, 1045, 1105, 1426, 1653, 1729, 1770, 1785, 1794, 1891, 2035, 2278, 2465, 2701, 2821, 2926, 3059, 3290, 3367, 3486, 3655, 4371, 4641, 4830, 5005, 5083, 5151, 5365, 5551, 5565, 5995, 6441, 6545, 6601

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

nonn

Squarefree polygonal numbers P(r,p) = (p^2*(r-2)-p*(r-4))/2 whose greatest prime factor is p >= 3, and whose rank (or order) is r >= 3 (see A324974).

The Carmichael numbers A002997 and primary Carmichael numbers A324316 are subsequences. See Kellner and Sondow 2019.

			P(3,5) = 15 is squarefree, and its greatest prime factor is 5, so 15 is a member.
More generally, if p is an odd prime and P(3,p) is squarefree, then P(3,p) is a member, since P(3,p) = (p^2+p)/2 = p*(p+1)/2, so p is its greatest prime factor.
CAUTION: P(6,7) = 91 = 7*13 is a member even though 7 is NOT its greatest prime factor, as P(6,7) = P(3,13) and 13 is its greatest prime factor.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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 preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
Wikipedia, Polygonal number.

Subsequence of A324972 = intersection of A005117 and A090466.
A002997, A324316, A324319 and A324320 are subsequences.
Cf. A324974, A324975, A324976.

GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
T = Select[Flatten[Table[{p, (p^2*(r - 2) - p*(r - 4))/2}, {p, 3, 150}, {r, 3, 100}], 1], SquareFreeQ[Last[#]] && First[#] == GPF[Last[#]] &];
Take[Union[Table[Last[t], {t, T}]], 47]

is(k) = if(issquarefree(k) && k>1, my(p=vecmax(factor(k)[, 1]), r); p>2 && (r=2*(k/p-1)/(p-1)) && denominator(r)==1, 0); \\ Jinyuan Wang, Feb 18 2021

Several missing terms inserted by Jinyuan Wang, Feb 18 2021

cp's OEIS Frontend

User: Jonathan Sondow

A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.

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A326716 3-term arithmetic progressions of primes whose indices are also primes in arithmetic progression.

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A326715 Values of n for which the denominator of (Sum_{prime p | n} 1/p - 1/n) is 1.

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A326691 a(n) = n/denominator(Sum_{prime p | n} 1/p - 1/n).

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A326692 Values of k for which the denominator of (Sum_{prime p | k} 1/p - 1/k) is k.

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A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

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

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