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.

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A309802 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+2} (prime(i)*x-1).

Original entry on oeis.org

1, 10, 101, 1358, 20581, 390238, 8130689, 201123530, 6166988769, 201097530280, 7754625545261, 329758834067168, 14671637258193181, 711027519310719868, 38706187989054920001, 2338431642812927422310, 145908145906128304198449, 9976861293427674211625032
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Links

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309803, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+2), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 18 2019

Formula

a(n) = [x^n] Product_{i=1..n+2} (prime(i)*x-1).
a(n) = abs(A070918(n+2,2)).
a(n) = abs(A238146(n+2,n)) for n>0.
a(n) = A260613(n+2,n).

A309803 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+3} (prime(i)*x-1).

Original entry on oeis.org

-1, -17, -288, -5102, -107315, -2429223, -64002818, -2057205252, -69940351581, -2788890538777, -122099137635118, -5580021752377242, -276932659619923555, -15388458479166668283, -946625238259888348698, -60082571176666116692888, -4171440414742758122621945
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Links

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309802, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+3), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019

Formula

a(n) = [x^n] Product_{i=1..n+3} (prime(i)*x-1).
a(n) = -abs(A070918(n+3,3)).
a(n) = -abs(A238146(n+3,n)) for n>0.
a(n) = -A260613(n+3,n).

A353534 a(n) is the least prime p such that the numerator of the sum of reciprocals of the 2*n+1 consecutive primes starting with p is a prime.

Original entry on oeis.org

2, 2, 5, 197, 7, 157, 29, 41, 2, 599, 3, 13, 293, 19, 181, 59, 7, 1489, 557, 43, 11, 23, 2, 227, 191, 349, 179, 2, 103, 5479, 2, 7, 131, 971, 37, 2, 6917, 23, 1279, 10903, 593, 311, 239, 2711, 6277, 1669, 257, 293, 503, 1861, 13613, 11, 569, 719, 619, 709, 4523, 3, 3, 2549, 1361, 383, 3, 10193
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, May 29 2022

Keywords

Comments

We use 2*n+1 consecutive primes rather than n because the numerator of the sum of reciprocals of an even number of odd primes is even.
The numerators are in A354221.

Examples

			a(3) = 5 because the sum of reciprocals of 2*3 + 1 = 7 primes starting with 5 is 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 = 24749279/37182145, and 24749279 is prime.

Links

Crossrefs

Cf. A024451, A244622, A354221.

Programs

  • Maple
    f:= proc(n) local i,k,v;
   for k from 1 do
     v:= numer(add(1/ithprime(i),i=k..k+2*n));
     if isprime(v) then return ithprime(k) fi
   od
end proc:
map(f, [$1..70]);

A356253 a(n) is the largest coefficient of P(x) := Product_{k} (x + p_k), where (p_k) are the primes dividing n listed with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 32, 17, 21, 19, 24, 21, 22, 23, 44, 25, 26, 27, 32, 29, 31, 31, 80, 33, 34, 35, 60, 37, 38, 39, 68, 41, 42, 43, 48, 45, 46, 47, 112, 49, 50, 51, 56, 53, 81, 55, 92, 57, 58, 59, 92, 61, 62, 63, 240, 65, 66, 67, 72
Offset: 1

Views

Author

Thomas Scheuerle, Jul 31 2022

Keywords

Comments

a(n) is the greatest number we may obtain by applying elementary symmetric functions onto the prime factors of n with multiplicity.
The record values of a(n)/n appear at powers of two.
If a(n) is greater than n, then it equals in most cases A003415(n), the first exception where a(n) > A003415(n) > n is at n = 64.
Conjectured: a(A002110(n)) = A024451(n), for n > 2.
Conjecture equality breaks down after n = 175, as a(A002110(176)) > A024451(176). - Antti Karttunen, Feb 08 2024

Links

Crossrefs

Cf. A002110, A003415, A024451, A070918, A083348, A109388, A260613, A369657 (difference between this sequence and A003415).
Cf. A065048 (same concept but uses numbers 1..n instead of prime factors of n).

Programs

  • PARI
    a(n) = vecmax(Vec(vecprod([(x+f[1])^f[2] | f<-factor(n)~]))) \\ Edited by M. F. Hasler, Feb 14 2024

Formula

a(n) = n iff n is not in A083348, otherwise a(n) > n.
a(2^n) = A109388(n) = binomial( n, floor(n/3) )*2^(n-floor(n/3)).
a(p^n) = binomial( n, floor(n/(p+1)) )*p^(n-floor(n/(p+1))), if p is prime.
a(p*n)/a(n) >= n and <= n+1 if p is prime.
a(p*q)/a(q) = p if p and q are prime. This is also true if q is a prime greater than 7 and p is a product of two primes greater than 4.
a(A002110(n)) >= A024451(n), for n > 2. The maximum of row n in A260613 a variant of A070918.

A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

Original entry on oeis.org

1, 6, 1, 14, 1, 1, 74, 38, 1, 1, 1551, 338, 1, 1, 1, 21084, 8631, 1330, 1, 1, 3550, 172655, 72938, 1970, 3410, 1, 1, 5822, 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610, 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431, 2156564414, 929540471, 68769335, 335525472, 4283242, 21900155, 348965439, 109820278, 185002, 32593310
Offset: 1

Views

Author

Antti Karttunen, Jun 21 2024

Keywords

Comments

Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.
Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.
See also comments in A373844, and in A373848.

Examples

			Triangle begins as:
        1,
        6,        1,
       14,        1,       1,
       74,       38,       1,       1,
     1551,      338,       1,       1,      1,
    21084,     8631,    1330,       1,      1,  3550,
   172655,    72938,    1970,    3410,      1,     1,     5822,
  3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
etc.

Links

Crossrefs

Cf. A002110, A003415, A370121, A373844, A373848.
Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373845(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A003415(1+(A002110(1+c)+x)); };

Formula

For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).

A096795 Numerator of sum of reciprocals of first n prime powers; denominator=A051451(n).

Original entry on oeis.org

1, 3, 11, 25, 137, 1019, 2143, 6709, 76319, 1019867, 2084779, 36161963, 699329537, 16317371911, 82657705331, 250947687593, 7357796373397, 230420777138107, 465354165304139, 17362507669146743, 717205745892079663
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 17 2004

Keywords

Examples

			n=6: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/7 =
(420+210+140+105+84+60)/420 = 1019/420 = a(6)/A051451(6).

Crossrefs

Cf. A000961, A024451.

A244621 (First arithmetic derivative of primorials) read mod 12.

Original entry on oeis.org

1, 5, 7, 7, 11, 5, 7, 7, 11, 1, 1, 7, 5, 5, 1, 11, 7, 1, 1, 5, 11, 11, 7, 5, 11, 1, 1, 5, 11, 1, 1, 5, 7, 7, 5, 5, 11, 11, 7, 5, 1, 7, 11, 5, 7, 7, 7, 7, 11, 5, 7, 11, 5, 1, 11, 7, 5, 5, 11, 1, 1, 11, 11, 7, 1, 11, 11, 5
Offset: 1

Views

Author

Freimut Marschner, Jul 02 2014

Keywords

Comments

A024451 as numerator of Sum_{i = 1..n} 1/prime(i) is the first arithmetic derivative of primorials prime(n)# of A002110. a(n) shows the distribution of A024451 over four residual classes.

Examples

			a(4) = [(prime(4)#)' = (4#)' = (210)' = 247] mod 12 = 7,
a(6) = [(prime(6)#)' = (13#)' = (30030)' = 40361] mod 12 = 5.

Crossrefs

Cf. A002110, A024451, A003415, A244622.

Programs

  • PARI
    a(n) = numerator(sum(i=1, n, 1/prime(i))) % 12; \\ Michel Marcus, Jul 07 2014

Formula

a(n) = (prime(n)#)' mod 12 or a(n) = A024451(n) mod 12.

A309804 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+4} (prime(i)*x-1).

Original entry on oeis.org

1, 28, 652, 16186, 414849, 11970750, 411154568, 14802996860, 617651235401, 28112591190218, 1330940558814492, 68134228016658366, 3888046744502816953, 244783216404832868510, 15878401438954693327808, 1123935467586630569656024, 83970858613393528568199649
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309802, A309803, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+4), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019
  • Mathematica
    a[n_] := CoefficientList[Series[Product[Prime[i]*x - 1, {i, 1, n+4}], {x, 0, 25}], x] [[n+1]]; Array[a, 17, 0] (* Amiram Eldar, Aug 24 2019 *)
  • PARI
    a(n) = polcoef(prod(i=1, n+4, prime(i)*x-1), n); \\ Michel Marcus, Aug 25 2019

Formula

a(n) = [x^n] Product_{i=1..n+4} (prime(i)*x-1).
a(n) = abs(A070918(n+4,4)).
a(n) = abs(A238146(n+4,n)) for n>0.
a(n) = A260613(n+4,n).

A348301 a(n) is the difference between the numerator and denominator of the (reduced) fraction Sum_{i = 1..n} 1/prime(i).

Original entry on oeis.org

-1, -1, 1, 37, 617, 10331, 205657, 4417993, 111313529, 3451185211, 113456434771, 4398448576657, 187757129777747, 8377806843970331, 406839682998275587, 22177392981497097521, 1341055344385518798469, 83727136357670859345679, 5727006517323354547143763
Offset: 1

Views

Author

Greg Tener, Oct 10 2021

Keywords

Examples

			a(1) = (p_1# / p_1) - p_1 = (2 / 2) - 2 = -1.
a(2) = (p_2# / p_1 + p_2# * p_2) - p_1 * p_2 = (6 / 2 + 6 / 3) - 2 * 3 = -1.
a(3) = 2*3*5/2 + 2*3*5/3 + 2*3*5/5 - 2*3*5 = 31 - 30 = 1.

Links

Crossrefs

Cf. A024451 (numerators), A002110 (denominators).

Programs

  • Mathematica
    Numerator[#]-Denominator[#]&/@Accumulate[1/Prime[Range[20]]] (* Harvey P. Dale, Feb 05 2023 *)
  • PARI
    a(n) = my(q=sum(i=1, n, 1/prime(i))); numerator(q)-denominator(q); \\ Michel Marcus, Oct 18 2021
  • Python
    from itertools import islice
from sympy import primorial, sieve
def a(n): return sum(primorial(n) // p for p in islice(sieve, n)) - primorial(n) # Greg Tener, Oct 18 2021

Formula

a(n) = (Sum_{i = 1..n} p_n# / p_i) - p_n# where p_n# is the primorial of the n-th prime.
a(n) = A024451(n) - A002110(n).

A353299 a(n) is the length of the continued fraction for the sum of the reciprocals of the first n primes.

Original entry on oeis.org

2, 3, 2, 5, 9, 10, 11, 16, 13, 20, 27, 27, 31, 43, 37, 41, 43, 47, 50, 58, 53, 57, 65, 83, 69, 62, 80, 84, 88, 93, 88, 110, 119, 117, 104, 111, 116, 126, 114, 140, 130, 164, 166, 132, 158, 154, 166, 168, 178, 178, 146, 176, 192, 188, 190, 203, 213, 191, 224, 236, 234, 238, 236, 236, 251
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 09 2022

Keywords

Examples

			Sum_{k=1..2} 1/prime(k) = 1/2 + 1/3 = 5/6 = 0 + 1/(1 + 1/5), so a(2) = 3.
Sum_{k=1..4} 1/prime(k) = 1/2 + 1/3 + 1/5 + 1/7 = 247/210 = 1 + 1/(5 + 1/(1 + 1/(2 + 1/12))), so a(4) = 5.

Links

Crossrefs

Row lengths of A260615.
Cf. A002110, A024451, A055573.

Programs

  • Mathematica
    Table[Length[ContinuedFraction[Sum[1/Prime[k], {k, 1, n}]]], {n, 1, 65}]
  • PARI
    a(n) = #contfrac(sum(k=1, n, 1/prime(k))); \\ Michel Marcus, Apr 10 2022
Previous Showing 51-60 of 69 results. Next

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