A045979 -id:A045979 - cp's OEIS frontend

A156543 Multiplicative closure of primes that are not Sophie Germain primes (A053176).

1, 7, 13, 17, 19, 31, 37, 43, 47, 49, 59, 61, 67, 71, 73, 79, 91, 97, 101, 103, 107, 109, 119, 127, 133, 137, 139, 149, 151, 157, 163, 167, 169, 181, 193, 197, 199, 211, 217, 221, 223, 227, 229, 241, 247, 257, 259, 263, 269, 271, 277, 283, 289, 301, 307, 311

Offset: 1

Reinhard Zumkeller, Feb 09 2009

nonn

a(n) mod 6 = 1 or = 5; subsequence of A007310;
A156542(a(n)) = 0;
A045979 is a subsequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007310, A045979, A053176, A156541, A156542.

A202318 Let (n)_p denote the exponent of prime p in the prime power factorization of n. Then a(n) is defined by the formulas a(1)=1; for n >= 2, (a(n))_2 = (n)_2, (a(n))_3 = (n)_3 and, for p >= 5, (a(n))_p = 1 + ((2n)/(p-1))_p if p-1|2*n, and (a(n))_p = 0 otherwise.

Original entry on oeis.org

1, 10, 21, 20, 11, 2730, 1, 680, 1197, 550, 23, 5460, 1, 290, 7161, 1360, 1, 5757570, 1, 45100, 6321, 230, 47, 185640, 11, 530, 3591, 580, 59, 283933650, 1, 2720, 32361, 10, 781, 840605220, 1, 10, 1659, 1533400, 83, 23830170, 1, 40940, 408177, 470, 1, 36014160, 1, 277750, 2163, 1060, 107, 1882725390

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2011

nonn

a(n)=1 iff n has form 6n+-1 and, if d >= 5 is a divisor of n, then 2*d+1 is not prime. The places of 1's form sequence A045979.

If p is an odd prime and p^n is the side length of the odd leg of a primitive Pythagorean triangle (PPT) it constrains the other leg and hypotenuse to be (p^(2n)-1)/2 and (p^(2n)+1)/2 and the area to be (p^n-1)p^n(p^n+1)/4. Now consider the term (p^n-1)p^n(p^n+1): it must at least be divisible by 24 for all odd primes p because the area of a PPT is divisible by 6 (see A127922 for n=1). a(n) equals the common divisor of the term (p^n-1)p^n(p^n+1)/24 for all odd primes p. - Frank M Jackson, Dec 09 2017

			Let n=6. Since 2*6+1=13 is prime, the max p that should be considered is 13. We have
  (a(6))_2  = (a(6))_3 = 1,
  (a(6))_5  = (12/4)_5 + 1 = 1,
  (a(6))_7  = (12/6)_7 + 1 = 1,
  (a(6))_13 = (12/12)_13 + 1 = 1.
Thus a(6) = 2*3*5*7*13 = 2730.

Frank M Jackson, Table of n, a(n) for n = 1..10000

Cf. A053657, A045979, A002445, A127922.

Table[Numerator[Exp[Re[Limit[Zeta[s] (Zeta[-1]^(s - 1) - Zeta[-(2*n - 1)]^(s - 1)), s -> 1]]]], {n, 1, 54}] (* Mats Granvik, Feb 05 2016 *)
Table[(lst=Table[p=Prime[m+1]; (p^n-1)p^n(p^n+1), {m, 1, 10}]; GCD@@lst/24), {n, 1, 100}] (* Frank M Jackson, Dec 09 2017 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; Array[a[2#+1]/(24 a[2#-1]) &, 100] (* using Jean-François Alcover's program A053657 *)(* Frank M Jackson, Dec 16 2017 *)

a(n) = {my(r = 1); forprime(p=2, 2*n+1, if (p<=3, r *= p^valuation(n, p), if (! (2*n % (p-1)), r *= p^(1+valuation((2*n)/(p-1), p))););); r;} \\ Michel Marcus, Feb 06 2016

a(n) = (1/24)*b(2n+1)/b(2n-1), where b(n) = A053657(n).
a(p) = A002445(p)/6, for prime p >= 5.
a(n) = numerator of e^(real(lim_{s -> 1} (zeta(s)*(zeta(-1)^(s-1) - zeta(-(2*n-1))^(s-1))))). - Mats Granvik, Feb 05 2016
a(n) = A036283(n)/6. - Hugo Pfoertner, Dec 18 2022

A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

Original entry on oeis.org

7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, 55, 57, 59, 61, 62, 64, 67, 71, 73, 76, 77, 79, 80, 84, 85, 87, 91, 92, 93, 94, 97, 101, 103, 104, 107, 109, 110, 115, 117, 118, 121, 122, 123, 124, 127, 129, 132, 133, 137, 139, 142, 143, 144, 145, 147

Offset: 1

Benoit Cloitre, Feb 03 2002

nonn

A045979(n), Bernoulli numbers with denominators 6, are included in the sequence.

Also numbers n such that both n+1 and 2n+1 are not prime. - Alexander Adamchuk, Oct 05 2006

Alexander Adamchuk, Oct 05 2006, Table of n, a(n) for n = 1..565

Cf. A000330, A001044.

Cf. A166602. - R. J. Mathar, Feb 14 2010

Select[Range[2,1000],Numerator[ #(#+1)(2#+1)/6/#!^2]==1&] (* Alexander Adamchuk, Oct 05 2006 *)
Select[Range[1000],!PrimeQ[ #+1]&&!PrimeQ[2#+1]&] (* Alexander Adamchuk, Oct 05 2006 *)

Also numbers n>1 such that A000330[n] = Sum[k^2,{k,1,n}] = n(n+1)(2n+1)/6 divides A001044[n] = Product[k^2,{k,1,n}] = (n!)^2. Also numbers n>1 such that Numerator[n(n+1)(2n+1)/6 /(n!)^2] = 1. - Alexander Adamchuk, Oct 05 2006

A081863 Largest integer m such that m divides (sigma_(2n+1)(2k-1)-sigma(2k-1)) for all k>=1.

Original entry on oeis.org

24, 240, 168, 480, 264, 21840, 24, 16320, 3192, 2640, 552, 43680, 24, 6960, 57288, 32640, 24, 15353520, 24, 216480, 7224, 5520, 1128, 1485120, 264, 12720, 3192, 13920, 1416, 454293840, 24, 65280, 258888, 240, 18744, 2241613920, 24, 240, 13272, 7360320, 1992

Offset: 1

Benoit Cloitre, Apr 12 2003

nonn

a(n)==0 mod 24. It seems that a(n)==0 (mod 2n+1) if and only if 2n+1 is an odd prime.

It appears that a(n)=24 for n in A045979, a(n)=168 for n in A051227, a(n)=264 for n in A051229, and a(n)=240 or 480 if n is in A051225. - Michel Marcus, Dec 07 2013

Cf. A000203.

ds(n, k) = sigma(2*k-1, 2*n+1) - sigma(2*k-1);
a(n) = {my(m = ds(n, 1)); for (k=2, 100, m = gcd(m, ds(n, k));); m;} \\ Script computes gcd of 100 terms; for current data, 10 terms are actually sufficient; is there a better way? - Michel Marcus, Dec 07 2013

a(12) corrected and more terms from Michel Marcus, Dec 07 2013

A114648 Number of occurrences of 6 as a denominator in Bernoulli(k) for 0<=k<=10^n.

Original entry on oeis.org

0, 1, 10, 87, 834, 7992, 77696, 758582, 7436825, 73129588, 720726912

Offset: 0

Eric W. Weisstein, Dec 21 2005

Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A002445, A114649.

Cf. A051222, A045979. - Enrique Pérez Herrero, Aug 16 2010

DenominatorBernoulliBQ[n_Integer, denom_Integer] := If[EvenQ[n], Times @@ Select[Divisors[n] + 1, PrimeQ] == denom, 1 + KroneckerDelta[n, 1] == denom]; A114648[n_] := A114648[n] = Length[Select[Range[10^n], DenominatorBernoulliBQ[#, 6] &]] (* Enrique Pérez Herrero, Aug 01 2010 *)

More terms added by Enrique Pérez Herrero, Aug 01 2010
More terms from Enrique Pérez Herrero, Aug 16 2010
a(10) from Arkadiusz Wesolowski, Dec 02 2012

A090863 Numbers n such that F(n+1)F(n-1)B(2n) is an integer, where F(k)=k-th Fibonacci number and B(2k)=2k-th Bernoulli number.

Original entry on oeis.org

1, 7, 13, 17, 19, 31, 37, 43, 47, 49, 59, 61, 67, 71, 73, 79, 91, 97, 101, 103, 107, 109, 121, 127, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 181, 193, 197, 199, 211, 217, 223, 227, 229, 241, 247, 257, 259, 263, 269, 271, 277, 283, 289

Offset: 1

Benoit Cloitre, Feb 12 2004

nonn

Differs from A045979 for some terms. Contains A053176.

A272383 Numbers n such that Bernoulli number B_{n} has denominator 3318.

Original entry on oeis.org

78, 1014, 2418, 3354, 7566, 8502, 10842, 11622, 12246, 12714, 13026, 15054, 15366, 15522, 16458, 17394, 23946, 26286, 27222, 27534, 29562, 29874, 30342, 31434, 31902, 33774, 34242, 35646, 36114, 40794, 42198, 43602, 44538, 47814, 48126, 48282, 49218, 50154, 52494, 55302, 57174, 57642, 59046, 59982

Offset: 1

Paolo P. Lava, Apr 28 2016

3318 = 2 * 3 * 7 * 79.
All terms are multiples of a(1) = 78.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 37.

			Bernoulli B_{78} is 414846365575400828295179035549542073492199375372400483487/3318, hence 78 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,3318);

Select[78 Range@ 800, Denominator@ BernoulliB@ # == 3318 &] (* Michael De Vlieger, Apr 28 2016 *)

lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 3318, print1(n, ", "))); \\ Altug Alkan, Apr 28 2016

from sympy import divisors, isprime
A272383_list = []
for i in range(78, 10**6, 78):
    for d in divisors(i):
        if d not in (1,2,6,78) and isprime(d+1):
            break
    else:
        A272383_list.append(i) # Chai Wah Wu, May 02 2016

a(9)-a(22) from Altug Alkan, Apr 28 2016

More terms from Michael De Vlieger, Apr 28 2016

A275770 Primes p == 5 (mod 6) that are not Sophie Germain primes.

Original entry on oeis.org

17, 47, 59, 71, 101, 107, 137, 149, 167, 197, 227, 257, 263, 269, 311, 317, 347, 353, 383, 389, 401, 449, 461, 467, 479, 503, 521, 557, 563, 569, 587, 599, 617, 647, 677, 701, 773, 797, 821, 827, 839, 857, 863, 881, 887, 929, 941, 947, 971, 977, 983, 1061, 1091, 1097

Offset: 1

Hilko Koning, Aug 08 2016

nonn

Cf. A005384, A007528, A045979, A053176, A156543.

Select[Prime[Range[150]], Mod[#, 6] == 5 && \[Not] PrimeQ[2 # + 1] &]

forprime(p=1, 900, if(Mod(p, 6)==5 && !ispseudoprime(2*p+1), print1(p, ", "))) \\ Felix Fröhlich, Aug 08 2016

Subset of A156543 and subset of A045979

Intersection of A007528 and A053176. - Felix Fröhlich, Aug 08 2016

A295587 Numbers k such that Bernoulli number B_{k} has denominator 13530.

Original entry on oeis.org

40, 6680, 7880, 8920, 9080, 10280, 12520, 12680, 14120, 15320, 15560, 18280, 20840, 21640, 22760, 23480, 25720, 26440, 28040, 30040, 30280, 31880, 33080, 33560, 34520, 35240, 35480, 36280, 38680, 39640, 42040, 43880, 44360, 46120, 46520, 46840, 47240, 47720, 48520

Offset: 1

Paolo P. Lava, Nov 24 2017

13530 = 2*3*5*11*41.
All terms are multiples of a(1) = 40.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 11519.

			Bernoulli B_{40} is -261082718496449122051/13530, hence 40 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186, A272369.

with(numtheory): P:=proc(q, h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,13530);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 11, 41}:
select(filter, [seq(i, i=1..10^5)]);

Select[Range[50000],Denominator[BernoulliB[#]]==13530&] (* Harvey P. Dale, Jul 29 2025 *)

A295588 Numbers k such that Bernoulli number B_{k} has denominator 14322.

Original entry on oeis.org

30, 1770, 3810, 4170, 4470, 4890, 5910, 5970, 6810, 8070, 9210, 10590, 11370, 11670, 12030, 12990, 13470, 13890, 14370, 14970, 15630, 16890, 17070, 17610, 18510, 18570, 19290, 19410, 20190, 20310, 21270, 22710, 24810, 25710, 26310, 27570, 27870, 29010, 29490, 29730

Offset: 1

Paolo P. Lava, Nov 24 2017

14322 = 2*3*7*11*31.
All terms are multiples of a(1) = 30.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 12899.

			Bernoulli B_{30} is 8615841276005/14322, hence 30 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186, A272369.

with(numtheory): P:=proc(q, h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,14322);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 7, 11, 31}:
select(filter, [seq(i, i=1..10^5)]);

cp's OEIS Frontend

A156543 Multiplicative closure of primes that are not Sophie Germain primes (A053176).

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A202318 Let (n)_p denote the exponent of prime p in the prime power factorization of n. Then a(n) is defined by the formulas a(1)=1; for n >= 2, (a(n))_2 = (n)_2, (a(n))_3 = (n)_3 and, for p >= 5, (a(n))_p = 1 + ((2n)/(p-1))_p if p-1|2*n, and (a(n))_p = 0 otherwise.

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A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

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A081863 Largest integer m such that m divides (sigma_(2n+1)(2k-1)-sigma(2k-1)) for all k>=1.

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A114648 Number of occurrences of 6 as a denominator in Bernoulli(k) for 0<=k<=10^n.

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Mathematica

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A090863 Numbers n such that F(n+1)*F(n-1)*B(2n) is an integer, where F(k)=k-th Fibonacci number and B(2k)=2k-th Bernoulli number.

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A272383 Numbers n such that Bernoulli number B_{n} has denominator 3318.

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A275770 Primes p == 5 (mod 6) that are not Sophie Germain primes.

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A295587 Numbers k such that Bernoulli number B_{k} has denominator 13530.

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A090863 Numbers n such that F(n+1)F(n-1)B(2n) is an integer, where F(k)=k-th Fibonacci number and B(2k)=2k-th Bernoulli number.