A088821 -id:A088821 - cp's OEIS frontend

A088822 a(n) is the sum of largest prime factors of numbers from 1 to n.

0, 2, 5, 7, 12, 15, 22, 24, 27, 32, 43, 46, 59, 66, 71, 73, 90, 93, 112, 117, 124, 135, 158, 161, 166, 179, 182, 189, 218, 223, 254, 256, 267, 284, 291, 294, 331, 350, 363, 368, 409, 416, 459, 470, 475, 498, 545, 548, 555, 560, 577, 590, 643, 646, 657, 664, 683

Offset: 1

Labos Elemer, Oct 22 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A046670, A006530, A088821, A088823, A088824, A088825.

P:=List(List([2..60],n->Reversed(Factors(n))),i->i[1]);;
 a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Nov 29 2018

-1 + Accumulate@ Array[FactorInteger[#][[-1, 1]] &, 57] (* Michael De Vlieger, Jul 23 2017 *)

gpf(n)=if(n<4, n, n=factor(n)[, 1]; n[#n])
a(n)=sum(k=2, n, gpf(k)) \\ Charles R Greathouse IV, Feb 19 2014

a(n) = Pi^2/12 * n^2/log n + O(n^2/log^2 n). - Charles R Greathouse IV, Feb 19 2014

a(n) ~ zeta(2) * A088821(n), where zeta(2) = Pi^2/6. - Thomas Ordowski, Nov 29 2018

A046669 Partial sums of A020639.

Original entry on oeis.org

1, 3, 6, 8, 13, 15, 22, 24, 27, 29, 40, 42, 55, 57, 60, 62, 79, 81, 100, 102, 105, 107, 130, 132, 137, 139, 142, 144, 173, 175, 206, 208, 211, 213, 218, 220, 257, 259, 262, 264, 305, 307, 350, 352, 355, 357, 404, 406, 413, 415, 418, 420, 473

Offset: 1

N. J. A. Sloane

M. Kalecki, On certain sums extended over primes or prime factors (in Polish), Prace Mat., Vol. 8 (1963/64), pp. 121-129.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section IV.1, p. 121.

T. D. Noe, Table of n, a(n) for n = 1..1000
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]

Cf. A020639, A046670, A072486, A088821.

a046669 n = a046669_list !! (n-1)
A046669_list = scanl1 (+) a020639_list -- Reinhard Zumkeller, Jun 15 2013

Accumulate[Array[FactorInteger[#][[1,1]]&,60]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = A088821(n) + 1.
From Amiram Eldar, Mar 04 2021: (Start)
a(n) ~ ((1 + o(1))/2)* n^2/log(n) (Kalecki, 1963/64).
a(n) = (1/2) * n^2/log(n) + O(n^2/log(n)^2) (Brouwer, 1974). (End)

A088824 Numbers n such that the sum of smallest prime factors of numbers from 1 to n is divisible by n.

Original entry on oeis.org

1, 2, 7, 14, 78, 113, 153, 439, 462, 1215, 2294, 8363, 11102, 12302, 36382, 38370, 60398, 199953, 224090, 421399, 427131, 1947938, 2467022, 2571633, 62395623, 462027217, 2140648015, 6418011931, 43074345625, 52714450814, 71229445182, 90719472005, 105685014433

Offset: 1

Labos Elemer, Oct 22 2003

nonn

Giovanni Resta, Table of n, a(n) for n = 1..38 (terms < 7.5*10^12)

Cf. A088821-A088825.

smp[n_] := If[n==1, 0, FactorInteger[n][[1, 1]]]; s = Accumulate @ Array[ smp, 10^5]; Select[Range[10^5], Mod[s[[#]], #] == 0 &] (* Giovanni Resta, Apr 27 2017 *)

Solutions to Mod[A088821[x], x]=0.

More terms from Ray Chandler, Oct 31 2003
a(27)-a(31) from Donovan Johnson, Jul 09 2010
a(32)-a(33) from Giovanni Resta, Apr 27 2017

A088825 Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n.

Original entry on oeis.org

1, 2, 8, 9, 32, 62, 558, 993, 995, 1947, 2150, 4343, 9944, 10977, 43054658, 202275890, 2291937393, 2459073795, 2836929091, 3737529738, 21382629569, 248208997602, 389691028017, 838566394212, 1019000924619

Offset: 1

Labos Elemer, Oct 22 2003

nonn

a(21) > 15*10^9. - Donovan Johnson, Nov 01 2009
a(22) > 10^11. - Donovan Johnson, Jul 09 2010
a(26) > 5*10^12. - Giovanni Resta, Apr 25 2017

Cf. A088821-A088824.

gp[n_] := If[n==1, 0, FactorInteger[n][[-1, 1]]]; Flatten@ Position[ Accumulate[ gp /@ Range[10^5]] / Range[10^5], Integer] (* _Giovanni Resta, Apr 25 2017 *)

Solutions to Mod[A088822[x], x]=0.

More terms from Ray Chandler, Oct 31 2003
a(17)-a(20) from Donovan Johnson, Nov 01 2009
a(21) from Donovan Johnson, Jul 09 2010
a(22)-a(25) from Giovanni Resta, Apr 25 2017

A088823 a(n) is the GCD of the sum of largest prime factors of numbers from 1 to n and of the sum of smallest prime factors of numbers from 1 to n.

Original entry on oeis.org

0, 2, 5, 7, 12, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 10, 1, 1, 1, 2, 29, 1, 1, 2, 1, 7, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 5, 1, 14, 1, 1

Offset: 1

Labos Elemer, Oct 22 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A088821, A088822, A088824, A088825.

map(igcd@op, ListTools:-PartialSums([[0,0],seq([min,max](numtheory:-factorset(n)),n=2..N)])); # Robert Israel, Dec 16 2015

GCD @@@ Transpose[{Accumulate[Prepend[First /@ #, 0]], Accumulate[Prepend[Last /@ #, 0]]}] &@ Map[First /@ FactorInteger@ # &, Range[2, 103]] (* Michael De Vlieger, Dec 15 2015 *)

a(n) = gcd(A088821(n), A088822(n)).

A355441 Numbers k such that the sum of the least prime factors of i=2..k is prime.

Original entry on oeis.org

2, 3, 4, 8, 12, 15, 16, 20, 24, 40, 43, 52, 55, 60, 63, 68, 72, 79, 87, 95, 96, 108, 111, 120, 123, 136, 140, 148, 151, 160, 184, 211, 215, 216, 227, 232, 235, 239, 252, 255, 256, 260, 264, 280, 283, 288, 299, 307, 323, 324, 327, 332, 360, 363, 371, 372, 375, 379

Offset: 1

Jean-Marc Rebert, Jul 02 2022

nonn

			8 is a term since the least prime factors of 2..8 are 2, 3, 2, 5, 2, 7, 2 and their sum 23 is prime.

Jean-Marc Rebert, Table of n, a(n) for n = 1..5676

Cf. A088821.

Position[Accumulate[Join[{0}, Table[FactorInteger[k][[1, 1]], {k, 2, 400}]]], ?PrimeQ] // Flatten (* _Amiram Eldar, Jul 02 2022 *)

isok(k) = isprime(sum(i=2, k, factor(i)[1,1])); \\ Michel Marcus, Jul 04 2022

from sympy import isprime, factorint
from itertools import accumulate, count, islice
def agen(): yield from (k for k, sk in enumerate(accumulate(min(factorint(i)) for i in count(2)), 2) if isprime(sk))
print(list(islice(agen(), 75))) # Michael S. Branicky, Jul 02 2022

A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 7, 4, 5, 12, 4, 6, 14, 7, 6, 7, 21, 7, 6, 8, 23, 11, 14, 9, 26, 20, 14, 10, 28, 25, 24, 11, 39, 25, 24, 12, 41, 28, 28, 6, 12, 13, 54, 28, 28, 6, 12, 14, 56, 35, 42, 6, 12, 15, 59, 40, 57, 6, 12, 16, 61, 44, 65, 22, 12, 17, 78, 44, 65, 22, 12, 18, 80, 47, 71, 31, 30, 19, 99, 47, 71, 31, 30

Offset: 1

Omar E. Pol, Feb 07 2023

Also, looking at all the partitions into equal-sized parts of all positive integers <= n, T(n,k) is the total number of parts in the partitions with the k-th largest parts.
Column k lists the partial sums of the column k of A027750.
The rows where the length row increases to a record gives A002182.

			Triangle begins:
   1;
   2,   2;
   3,   5;
   4,   7,  4;
   5,  12,  4;
   6,  14,  7,   6;
   7,  21,  7,   6;
   8,  23, 11,  14;
   9,  26, 20,  14;
  10,  28, 25,  24;
  11,  39, 25,  24;
  12,  41, 28,  28,  6, 12;
  ...
For n = 6 the divisors, in increasing order, of all positive integers <= 6 are as follows:
  -----------------------------
  n\k |    1     2     3     4
  -----------------------------
  1   |    1
  2   |    1,    2
  3   |    1,    3
  4   |    1,    2,    4
  5   |    1,    5
  6   |    1,    2,    3,    6
.
The sum of the first divisors (k = 1) is equal to 1+1+1+1+1+1 = 6, so T(6,1) = 6.
The sum of the second divisors (k = 2) is equal to 2+3+2+5+2 = 14, so T(6,2) = 14.
The sum of the third divisors (k = 3) is equal to 4+3 = 7, so T(6,3) = 7.
The sum of the fourth divisors (k = 4) is equal to 6, so T(6,4) = 6.
So the 6th row of the triangle is [6, 14, 7, 6].
Also, for n = 6 the partitions into equal parts, with the sizes of the parts in decreasing order, of all positive integers <= 6 are as follows:
  ----------------------------------------------------
  n\k |     1      2              3           4
  ----------------------------------------------------
  1   |    [1]
  2   |    [2],  [1,1]
  3   |    [3],  [1,1,1]
  4   |    [4],  [2,2],       [1,1,1,1]
  5   |    [5],  [1,1,1,1,1]
  6   |    [6],  [3,3],       [2,2,2],   [1,1,1,1,1,1]
.
The total number of parts in the 1st partitions (k = 1) is 6, so T(6,1) = 6.
The total number of parts in the 2nd partitions (k = 2) is 14, so T(6,2) = 14.
The total number of parts in the 3rd partitions (k = 3) is 7, so T(6,3) = 7.
The total number of parts in the 4th partitions (k = 4) is 6, so T(6,4) = 6.
So the 6th row of the triangle is [6, 14, 7, 6].

Paolo Xausa, Table of n, a(n) for n = 1..10175 (rows 1..550 of triangle, flattened)
Michael De Vlieger, Log log scatterplot of row n, n = 1..5040, showing T(n, k) with trajectory of k in a color function where black indicates k = 1, red k = 2, ..., magenta k = 60, ignoring zeros.

Row sums give A024916.
Row lengths give A070319.
Column 1 gives A000027.
Column 2 gives A088821.
The sum of the first n rows gives A175254.
Main sequences: A027750 and A244051.
Cf. A000005, A002182.

nn = 20; s[] = 0; m[0] = 0; Do[Set[m[n], Max[m[n - 1], DivisorSigma[0, n]]], {n, nn}]; Do[MapIndexed[(s[First[#2]] += #1; Set[t[n, First[#2]], s[First[#2]]]) &, PadRight[Divisors[n], m[n]]], {n, nn}]; Table[t[n, k], {n, nn}, {k, m[n]}] // Flatten (* _Michael De Vlieger, Mar 04 2023 *)
A360461[rowmax_]:=DeleteCases[Accumulate[PadRight[Divisors[Range[rowmax]]]],0,{2}];
A360461[20] (* Generates 20 rows *) (* Paolo Xausa, Mar 05 2023 *)

rowlen(n) = vecmax(vector(n, k, numdiv(k))); \\ A070319
row(n) = my(vd=vector(n, i, divisors(i)), nb=rowlen(n)); vector(nb, k, sum(i=1, #vd, if (#(vd[i]) >= k, vd[i][k]))); \\ Michel Marcus, Mar 06 2023

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A088822 a(n) is the sum of largest prime factors of numbers from 1 to n.

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A046669 Partial sums of A020639.

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A088824 Numbers n such that the sum of smallest prime factors of numbers from 1 to n is divisible by n.

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A088825 Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n.

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A088823 a(n) is the GCD of the sum of largest prime factors of numbers from 1 to n and of the sum of smallest prime factors of numbers from 1 to n.

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A355441 Numbers k such that the sum of the least prime factors of i=2..k is prime.

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A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).

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