cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A355331 Numbers k that divide A020696(k).

Original entry on oeis.org

1, 2, 6, 12, 20, 24, 42, 60, 72, 84, 90, 120, 126, 140, 144, 156, 168, 180, 210, 216, 220, 240, 252, 280, 312, 336, 342, 360, 420, 432, 440, 462, 468, 480, 504, 540, 560, 600, 624, 630, 660, 672, 684, 700, 720, 770, 780, 816, 840, 864, 880, 900, 924, 936, 945, 960, 990
Offset: 1

Views

Author

Amiram Eldar, Jun 29 2022

Keywords

Comments

If k and m are coprime terms then k*m is also a term.
The least odd term above 1 is a(55) = 945, the least term above 1 that is coprime to 6 is a(378) = 10465, least term above 1 that is coprime to 30 is a(3122) = 151487, and the least term above 1 that is coprime to 210 is a(6858) = 414713.

Examples

			2 is a term since A020696(2) = 6 is divisible by 2.

Links

Crossrefs

Cf. A020696.
A355332 is a subsequence.

Programs

  • Mathematica
    v[n_] := Times @@ (Divisors[n] + 1); Select[Range[1000], Divisible[v[#], #] &]
  • PARI
    f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696
isok(k) = !(f(k) % k); \\ Michel Marcus, Jun 30 2022
  • Python
    from itertools import count, islice
from functools import reduce
from sympy import divisors
def A355331_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:reduce(lambda a,b:a*b%n,(d+1 for d in divisors(n,generator=True)))%n==0,count(max(startvalue,1)))
A355331_list = list(islice(A355331_gen(),30)) # Chai Wah Wu, Jun 30 2022

A355332 Numbers k such that k | A020696(k) and (k+1) | A020696(k+1).

Original entry on oeis.org

1, 201824, 227799, 313599, 395199, 544824, 638000, 654975, 799799, 862784, 1056159, 1204280, 1269729, 1447550, 1890944, 2276351, 2460975, 2481115, 2781999, 2821272, 3348224, 3382379, 3403700, 3832191, 3864575, 3956120, 5142500, 5961950, 6116175, 6401024, 7050120
Offset: 1

Views

Author

Amiram Eldar, Jun 29 2022

Keywords

Comments

Numbers k such that k and k+1 are both in A355331.
Are there 3 consecutive integers in A355331?
There are no such 3 consecutive integers below 10^10. - Amiram Eldar, Oct 12 2023

Examples

			1 is a term since A020696(1) = 2 is divisible by 1 and A020696(2) = 6 is divisible 2.

Links

Crossrefs

Cf. A020696, A355331.

Programs

  • Mathematica
    q[n_] := Divisible[Times @@ (Divisors[n] + 1), n]; Select[Range[10^6], q[#] && q[#+1] &]
  • PARI
    f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696
isok(k) = !(f(k) % k) && !(f(k+1) % (k+1)); \\ Michel Marcus, Jun 30 2022

A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 2, 3, 16, 36, 2, 30, 4, 6, 16, 45, 4, 80, 2, 108, 16, 6, 2, 210, 24, 12, 32, 18, 4, 1008, 2, 45, 64, 12, 48, 8400, 4, 18, 16, 2268, 4, 240, 2, 90, 512, 18, 2, 3150, 32, 216, 64, 540, 4, 160, 144, 2430, 32, 12, 2, 166320, 4, 6, 1280, 405, 48, 1344
Offset: 1

Views

Author

Amiram Eldar, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A000079, A002144, A002145, A020696, A377484, A378056.

Programs

  • Mathematica
    a[n_] := GCD[Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@ d - 1)]; Array[a, 70]
  • PARI
    a(n) = if(n == 1, 1, my(d = divisors(n)); gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)));

Formula

a(n) = 2 if and only if n = 6 or n is a prime of the form 4*k+3 (A002145).
a(n) = 4 if and only if n is a prime of the form 4*k+1 (A002144).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).

A299436 G.f.: exp( Sum_{n>=1} A020696(n) * x^n/n ), where A020696(n) = Product_{d|n} (d + 1).

Original entry on oeis.org

1, 2, 5, 10, 24, 44, 109, 198, 423, 766, 1555, 2730, 6269, 11090, 22127, 39246, 77541, 134242, 270348, 467004, 895797, 1546922, 2905899, 4943126, 9666435, 16471506, 30604583, 52206218, 96412319, 162467222, 303289098, 510436808, 929735638, 1564811464, 2818065892, 4700325864, 8619686709, 14378564170, 25693238857, 42876196186, 76267527522, 126317457712
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2018

Keywords

Comments

Self-convolution of A299437.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 10*x^3 + 24*x^4 + 44*x^5 + 109*x^6 + 198*x^7 + 423*x^8 + 766*x^9 + 1555*x^10 + 2730*x^11 + 6269*x^12 + 11090*x^13 + ...
such that
log(A(x)) = 2*x + 6*x^2/2 + 8*x^3/3 + 30*x^4/4 + 12*x^5/5 + 168*x^6/6 + 16*x^7/7 + 270*x^8/8 + 80*x^9/9 + 396*x^10/10 + 24*x^11/11 + 10920*x^12/12 + 28*x^13/13 + 720*x^14/14 + 768*x^15/15 + ... + A020696(n)*x^n/n + ...

Links

Crossrefs

Cf. A299437 (sqrt(A(x))), A020696.

Programs

  • PARI
    A020696(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1)); } \\ after Michel Marcus
{a(n) = my(A = exp( sum(m=1,n, A020696(m)*x^m/m ) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

A299437 G.f.: exp( Sum_{n>=1} A020696(n)/2 * x^n/n ), where A020696(n) = Product_{d|n} (d + 1).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 27, 33, 73, 100, 203, 269, 987, 1163, 2283, 3234, 6706, 8812, 21455, 27211, 55718, 76055, 147048, 196483, 533149, 659549, 1262531, 1759301, 3462333, 4593487, 10261739, 13213278, 25944342, 35397849, 66694451, 89412873, 209286231, 266115126, 499426529, 689936238, 1311854563, 1750578063, 3676669661, 4787587399, 9114353938, 12427479022, 22925519170
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2018

Keywords

Comments

Self-convolution equals A299436.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 27*x^6 + 33*x^7 + 73*x^8 + 100*x^9 + 203*x^10 + 269*x^11 + 987*x^12 + 1163*x^13 + 2283*x^14 + ...
such that
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 84*x^6/6 + 8*x^7/7 + 135*x^8/8 + 40*x^9/9 + 198*x^10/10 + 12*x^11/11 + 5460*x^12/12 + 14*x^13/13 + 360*x^14/14 + 384*x^15/15 + ... + A020696(n)/2*x^n/n + ...

Links

Crossrefs

Cf. A299436 (A(x)^2), A020696.

Programs

  • PARI
    A020696(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1)); } \\ after Michel Marcus
{a(n) = my(A = exp( sum(m=1,n, A020696(m)/2*x^m/m ) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

A355330 Numbers k such that A020696(2^k-1) < A020696(2^k+1).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 41, 45, 46, 47, 49, 51, 53, 57, 59, 61, 62, 65, 67, 69, 71, 73, 77, 78, 81, 83, 85, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 118, 121, 122, 123, 125
Offset: 1

Views

Author

Amiram Eldar, Jun 29 2022

Keywords

Comments

Sándor (2021) showed that all the Mersenne exponents (A000043) are in this sequence and conjectured that both this sequence and its complement are infinite.

Examples

			2 is a term since A020696(2^2-1) = A020696(3) = 8 and A020696(2^2+1) = A020696(5) = 12 > 8.

Links

Crossrefs

Cf. A000043, A000668, A020696.

Programs

  • Mathematica
    v[n_] := Times @@ (Divisors[n] + 1); Select[Range[150], v[2^# - 1] < v[2^# + 1] &]
  • PARI
    f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696
isok(k) = f(2^k-1) < f(2^k+1); \\ Michel Marcus, Jun 30 2022

A081845 Decimal expansion of Product_{k>=0} (1 + 1/2^k).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Apr 09 2003

Keywords

Comments

Twice the product in A079555.

Examples

			4.76846205806274344829979857....

Links

Crossrefs

Cf. A006125, A020696, A028361, A079555, A083864.
Cf. A000593, A048651, A100220, A098844, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132269, A132270.

Programs

  • Mathematica
    digits = 105; NProduct[1 + 1/2^k, {k, 0, Infinity}, WorkingPrecision -> digits+5, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013 *)
N[QPochhammer[-1, 1/2], 100] (* Vaclav Kotesovec, Dec 13 2015 *)
2*N[QPochhammer[-1/2, 1/2], 200] (* G. C. Greubel, Dec 20 2015 *)
  • PARI
    prodinf(k=0,1/2^k,1) \\ Hugo Pfoertner, Feb 21 2020

Formula

lim sup Product_{k=0..floor(log_2(n))} (1 + 1/floor(n/2^k)) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132369(n)/A098844(n) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
2*exp(Sum_{n>0} 2^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*2^n)). - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n+1)/A132269(n) = 4.76846205806274344... for n-->oo. - Hieronymus Fischer, Aug 20 2007
Sum_{k>=1} (-1)^(k+1) * 2^k / (k*(2^k-1)) = log(A081845) = 1.562023833218500307570359922772014353168080202860122... . - Vaclav Kotesovec, Dec 13 2015
Equals 2*(-1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals 1 + Sum_{n>=1} 2^n/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 21 2020
From Peter Bala, Jan 18 2021: (Start)
Constant C = 3*Sum_{n >= 0} (1/2)^n/Product_{k = 1..n} (2^k - 1).
Faster converging series:
C = (2*3*5)/(2^3)*Sum_{n >= 0} (1/4)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9)/(2^6)*Sum_{n >= 0} (1/8)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9*17)/(2^10)*Sum_{n >= 0} (1/16)^n/Product_{k = 1..n} (2^k - 1), and so on. The sequence [2,3,5,9,17,...] is A000051. (End)
From Amiram Eldar, Mar 20 2022: (Start)
Equals sqrt(2) * exp(log(2)/24 + Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(2))) (McIntosh, 1995).
Equals 1/A083864. (End)
Equals lim_{n->oo} A020696(2^n)/A006125(n+1) (Sándor, 2021). - Amiram Eldar, Jun 29 2022

A057643 Least common multiple of all (k+1)'s, where the k's are the positive divisors of n.

Original entry on oeis.org

2, 6, 4, 30, 6, 84, 8, 90, 20, 66, 12, 5460, 14, 120, 48, 1530, 18, 7980, 20, 2310, 88, 276, 24, 81900, 78, 378, 140, 3480, 30, 114576, 32, 16830, 204, 630, 72, 3838380, 38, 780, 280, 284130, 42, 397320, 44, 4140, 5520, 1128, 48, 9746100, 200, 14586, 468
Offset: 1

Views

Author

Leroy Quet, Oct 11 2000

Keywords

Comments

a(n) is a divisor of A020696(n). - Ivan Neretin, May 27 2015

Examples

			Since the positive divisors of 6 are 1, 2, 3 and 6, a(6) = LCM(1+1,2+1,3+1,6+1) = LCM(2,3,4,7) = 84.

Links

Crossrefs

Cf. A119250.
Cf. A020696 (product instead of LCM).

Programs

  • Maple
    f:= n -> ilcm(op(map(`+`,numtheory:-divisors(n),1)));
seq(f(n),n=1..100); # Robert Israel, Jul 24 2014
  • Mathematica
    a057643[n_Integer] := Apply[LCM, Map[# + 1 &, Divisors[n]]]; Table[a057643[n], {n, 10000}] (* Michael De Vlieger, Jul 19 2014 *)
  • PARI
    a(n)=lcm(apply(d->d+1,divisors(n))) \\ Charles R Greathouse IV, Feb 14 2013
  • Python
    from math import lcm
from sympy import divisors
def A057643(n): return lcm(*(d+1 for d in divisors(n,generator=True))) # Chai Wah Wu, Jun 30 2022

A377484 a(n) = Product_{d|n, d>1} (d - 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 6, 21, 16, 36, 10, 330, 12, 78, 112, 315, 16, 1360, 18, 2052, 240, 210, 22, 53130, 96, 300, 416, 6318, 28, 146160, 30, 9765, 640, 528, 816, 1570800, 36, 666, 912, 560196, 40, 639600, 42, 27090, 39424, 990, 46, 37456650, 288, 42336, 1600, 45900, 52, 1874080, 2160
Offset: 1

Views

Author

Ridouane Oudra, Oct 29 2024

Keywords

Examples

			a(12) = (2-1)*(3-1)*(4-1)*(6-1)*(12-1) = 1*2*3*5*11 = 330.

Links

Crossrefs

Cf. A007955, A020696.
Cf. A000079, A005329, A027871, A027872, A027875, A027879, A175787.

Programs

  • Maple
    with(numtheory): seq(mul(d-1, d in divisors(n) minus {1}), n=1..80);
  • Mathematica
    a[n_] := Times @@ (Rest@ Divisors[n] - 1); Array[a, 60] (* Amiram Eldar, Nov 01 2024 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=2, #d, d[k]-1); \\ Michel Marcus, Oct 30 2024

Formula

a(n) = Product_{k=2..A000005(n)} (A027750(n,k) - 1).
a(p^n) = Product_{k=1..n} (p^k - 1), where p is prime, and n an integer.
a(2^n) = A005329(n).
a(3^n) = A027871(n).
a(5^n) = A027872(n).
a(7^n) = A027875(n).
a(11^n) = A027879(n).
From Amiram Eldar, Nov 02 2024: (Start)
a(n) = n-1 if and only if n is in A175787 (i.e., n = 4 or n is prime).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079). (End)

A229337 Sum of products of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 5, 7, 29, 11, 167, 15, 269, 79, 395, 23, 10919, 27, 719, 767, 4589, 35, 31919, 39, 41579, 1407, 1655, 47, 2456999, 311, 2267, 2239, 104399, 59, 5499647, 63, 151469, 3263, 3779, 3455, 76767599, 75, 4679, 4479, 15343019, 83, 19071359, 87, 372599, 353279, 6767
Offset: 1

Views

Author

Jaroslav Krizek, Sep 20 2013

Keywords

Comments

Number of nonempty subsets of divisors of n = A100587(n).

Examples

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of products of elements of subsets = 1 + 2 + 4 + 2 + 4 + 8 + 8 = 29 = (1+1) * (2+1) * (4+1) - 1.

Crossrefs

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n), A229336 (product of sums of elements of nonempty subsets of divisors of n), A229338 (product of products of elements of nonempty subsets of divisors of n).

Formula

Let a, b, c, ..., k be all divisors of n; a(n) = (a+1) * (b+1) * ... * (k+1) - 1.
a(p) = 2p+1, a(p^2) = 2(p+1)(p^2+1) - 1.
a(n) = A020696(n) - 1.
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