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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A082087 a(n) is the fixed point if function A008472 (= sum of prime factors with no repetition) is iterated when started at initial value n!.

Original entry on oeis.org

2, 5, 5, 7, 7, 17, 17, 17, 17, 3, 3, 41, 41, 41, 41, 31, 31, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 197, 197, 197, 197, 2, 2, 281, 281, 281, 281, 43, 43, 43, 43, 43, 43, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 73, 73, 73, 73, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13
Offset: 1

Views

Author

Labos Elemer, Apr 09 2003

Keywords

Examples

			n=73: iteration list=
{73!=61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000,639,74,39,16,2,2}

Crossrefs

Cf. A007504, A008472, A034387, A075860.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[FixedPoint[sopf, w! ], {w, 2, 100}]

Formula

a(n) = A075860(A000142(n)).

A082088 a(n) is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at initial value prime[n]!.

Original entry on oeis.org

2, 5, 7, 17, 3, 41, 31, 5, 7, 5, 7, 197, 2, 281, 43, 7, 5, 5, 73, 2, 7, 7, 13, 5, 7, 5, 3, 7, 13, 3, 7, 7, 7, 7, 571, 7, 7, 5, 7, 7, 5, 7, 5, 7, 2, 7, 19, 3, 3, 67, 5, 2, 71, 43, 7, 71, 239, 7, 7, 7699, 2, 5, 313, 8893, 2, 3, 199, 5, 5, 2, 5, 2, 3, 7, 6361, 3, 97, 5, 19, 97, 7, 2593, 5, 5
Offset: 1

Views

Author

Labos Elemer, Apr 09 2003

Keywords

Examples

			n=100,p(100)=541,start at 541! and get iteration list=
{541!,24133} ended immediately in a(100)=24133;
n=99,p(99)-523,start at 523! and get a list of
{523!,23592,988,34,19}, a(99)=19.

Crossrefs

Cf. A008472, A034387, A007504, A075860, A082087.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[FixedPoint[sopf, Prime[w]! ], {w, 2, 100}]

Formula

a(n)=A082087[A000142(p[n])].

A101256 Sum of composites <= n.

Original entry on oeis.org

0, 0, 0, 4, 4, 10, 10, 18, 27, 37, 37, 49, 49, 63, 78, 94, 94, 112, 112, 132, 153, 175, 175, 199, 224, 250, 277, 305, 305, 335, 335, 367, 400, 434, 469, 505, 505, 543, 582, 622, 622, 664, 664, 708, 753, 799, 799, 847, 896, 946, 997, 1049, 1049, 1103, 1158, 1214
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 23 2005

Keywords

Crossrefs

Cf. A065855, A002808.
Cf. A000217, A034387, A101203.

Programs

  • Mathematica
    Accumulate[Table[If[n<2,0,If[PrimeQ[n], 0, n]], {n, 1, 100}]] (* James C. McMahon, Jan 07 2024 *)

Formula

a(n) = A000217(n) - A034387(n) - 1 = A101203(n) - 1.

A228102 Numbers n such that sum of all primes <=n is prime.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 37, 38, 39, 40, 43, 44, 45, 46, 281, 282, 311, 312, 503, 504, 505, 506, 507, 508, 541, 542, 543, 544, 545, 546, 557, 558, 559, 560, 561, 562, 593, 594, 595, 596, 597, 598, 619, 620, 621, 622, 623, 624, 625
Offset: 1

Views

Author

Jayanta Basu, Aug 10 2013

Keywords

Comments

Numbers n such that A034387(n) is prime.

Examples

			8 is in the sequence since 2+3+5+7=17 is prime.

Links

Crossrefs

Cf. A034387.

Programs

  • Magma
    [n: n in [1..700] | IsPrime(s) where s is &+PrimesUpTo(n)]; // Bruno Berselli, Aug 10 2013
  • Mathematica
    t = {}; s = 0; Do[If[PrimeQ[n], s += n]; If[PrimeQ[s], AppendTo[t, n]], {n, 625}]; t
Position[Accumulate[Table[If[PrimeQ[n],n,0],{n,700}]],?PrimeQ]//Flatten (* _Harvey P. Dale, Apr 02 2024 *)

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

Original entry on oeis.org

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

Examples

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

Links

Crossrefs

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Programs

  • Mathematica
    Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]
  • PARI
    a(n) = my(f=factor(n)); vecsum(f[,2])^vecsum(f[,1]); \\ Michel Marcus, Apr 21 2018

Formula

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(k).

A352753 a(n) = (pi(2n-1) - pi(n-1)) * Sum_{p <= n, p prime} p.

Original entry on oeis.org

0, 4, 10, 10, 20, 20, 51, 34, 51, 68, 112, 112, 164, 123, 164, 205, 290, 232, 385, 308, 385, 462, 600, 600, 600, 600, 700, 700, 903, 903, 1280, 1120, 1120, 1280, 1280, 1440, 1970, 1773, 1773, 1970, 2380, 2380, 2810, 2529, 2810, 2810, 3280, 2952, 3280, 3280, 3608
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 01 2022

Keywords

Comments

Sum of the primes p from the ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

Examples

			a(5) = 20; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7). The sum of the corresponding prime parts p gives 2+2+3+3+5+5 = 20.

Crossrefs

Cf. A000720 (pi), A034387, A035250, A352749, A352754, A352775, A352777.

Programs

  • Mathematica
    Table[(PrimePi[2 n - 1] - PrimePi[n - 1]) Sum[k (PrimePi[k] - PrimePi[k - 1]), {k, n}], {n, 100}]

Formula

a(n) = A035250(n) * A034387(n). - Bernard Schott, Apr 02 2022
a(n) = A352775(n) - A352754(n).

A084409 Main diagonal of triangle A084408.

Original entry on oeis.org

1, 3, 11, 9, 29, 18, 59, 28, 40, 54, 107, 69, 179, 87, 106, 125, 271, 147, 389, 172, 200, 224, 541, 254, 286, 316, 346, 381, 727, 416, 941, 455, 495, 534, 575, 620, 1201, 665, 710, 756, 1493, 804, 1823, 856, 909, 962, 2203, 1018, 1077, 1136, 1194, 1254, 2621
Offset: 1

Views

Author

Amarnath Murthy, Jun 01 2003

Keywords

Comments

a(n) is prime iff n is prime.

Crossrefs

Cf. A084408, A084410, A084411.

Formula

For n prime, a(n) = p(A034387(n)). For n composite, a(n) = A002808([n*(n+1)/2]-A034387(n)-1).

Extensions

Edited and extended by David Wasserman, Dec 27 2004

A084410 First column of triangle A084408.

Original entry on oeis.org

1, 2, 5, 4, 13, 10, 31, 20, 30, 42, 61, 55, 109, 70, 88, 108, 181, 126, 277, 148, 174, 201, 397, 225, 255, 287, 318, 348, 547, 382, 733, 417, 456, 496, 535, 576, 947, 621, 666, 711, 1213, 758, 1499, 805, 858, 910, 1831, 963, 1020, 1078, 1137, 1195, 2207, 1255
Offset: 1

Views

Author

Amarnath Murthy, Jun 01 2003

Keywords

Comments

a(n) is prime iff n is prime.

Crossrefs

Cf. A084408, A084409, A084411.

Formula

For n prime, a(n) = p(A034387(n)-n+1). For n composite, a(n) = A002808([n*(n+1)/2]-A034387(n)-n).

Extensions

Edited and extended by David Wasserman, Dec 27 2004

A139562 Sum of primes < n^2.

Original entry on oeis.org

0, 0, 5, 17, 41, 100, 160, 328, 501, 791, 1060, 1593, 2127, 2914, 3831, 4661, 6081, 7982, 9523, 11599, 13887, 16840, 20059, 23592, 26940, 32353, 37561, 42468, 48494, 55837, 62797, 70241, 80189, 89672, 100838, 111587, 124211, 136114, 148827
Offset: 0

Views

Author

Cino Hilliard, Jun 11 2008

Keywords

Comments

This is also the sum of primes <= n^2.
Pi(x) is the prime counting function or the number of primes <= x.
SumP(n) is the sum of primes <= n.
SumP(n) ~ Pi(n^2).
For large n, a(n) is closely approximated by Pi(n^4). E.g., for n = 55, SumP(55^2) = 605877 and Pi(55^4) = 611827 with error = 0.0098...
For n = 10^5, SumP(10) = 2220822432581729238 and Pi(10^20) = 2220819602560918840 with error = 0.0000012...

Examples

			For n = 3, n^2 = 9, the sum of primes <= 9 is 2+3+5+7 = 17 = a(3).

Links

Crossrefs

Cf. A000720, A000290, A006880, A007504, A034387, A046731, A130739.
First differences: A108314.

Programs

  • Mathematica
    Array[Sum[p,{p,Prime@Range@PrimePi[#^2-1]}]&,51,0]
(* or *)
Table[Total@Select[Range[n^2-1],PrimeQ],{n,0,50}] (* Giorgos Kalogeropoulos, Jul 27 2021 *)
  • PARI
    a(n) = sum(k=1, n^2, k*isprime(k)); \\ Michel Marcus, Jul 27 2021
  • Python
    from sympy import primerange
def a(n): return sum(p for p in primerange(1, n*n))
print([a(n) for n in range(39)]) # Michael S. Branicky, Jul 29 2021

Formula

a(n) = A034387(n^2) for n >= 1. - Alois P. Heinz, Jul 30 2021

Extensions

a(16) corrected by Michael S. Branicky, Jul 29 2021

A063955 Sum of the unitary prime divisors of n!.

Original entry on oeis.org

0, 2, 5, 3, 8, 5, 12, 12, 12, 7, 18, 18, 31, 24, 24, 24, 41, 41, 60, 60, 60, 49, 72, 72, 72, 59, 59, 59, 88, 88, 119, 119, 119, 102, 102, 102, 139, 120, 120, 120, 161, 161, 204, 204, 204, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 281, 252, 311, 311
Offset: 1

Views

Author

Labos Elemer, Sep 04 2001

Keywords

Examples

			Prime divisors of 20! which have exponent 1 (i.e., unitary prime divisors) are {11, 13, 17, 19}, so a(20) = 11 + 13 + 17 + 19= 60. (The sum of all its prime divisors (unitary and non-unitary) is A034387(20).)

Links

Crossrefs

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063956.

Programs

  • Maple
    a:= n-> add(`if`(i[2]=1, i[1], 0), i=ifactors(n!)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 24 2018
  • Mathematica
    a[n_] := Select[FactorInteger[n!], #[[2]] == 1&][[All, 1]] // Total;
Array[a, 60] (* Jean-François Alcover, Jan 01 2022 *)
  • PARI
    a(n) = my(f=factor(n!)~); sum(i=1, length(f), if (f[2, i]==1, f[1, i])); \\ Harry J. Smith, Sep 04 2009

Formula

a(n) = Sum_{k=floor(n/2)+1..n} k*c(k), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Dec 23 2023
a(n) = A063956(n!). - Amiram Eldar, Jul 24 2024
Previous Showing 21-30 of 54 results. Next

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