A088825 -id:A088825 - cp's OEIS frontend

A088821 a(n) is the sum of smallest prime factors of numbers from 1 to n.

0, 2, 5, 7, 12, 14, 21, 23, 26, 28, 39, 41, 54, 56, 59, 61, 78, 80, 99, 101, 104, 106, 129, 131, 136, 138, 141, 143, 172, 174, 205, 207, 210, 212, 217, 219, 256, 258, 261, 263, 304, 306, 349, 351, 354, 356, 403, 405, 412, 414, 417, 419, 472, 474, 479, 481, 484

Offset: 1

Labos Elemer, Oct 22 2003

nonn

M. Kalecki, On certain sums extended over primes or prime factors, Prace Mat, Vol. 8 (1963), pp. 121-127.
J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, Chapter IV, p. 121.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Project Euler, Problem 521 - Smallest prime factor

Cf. A020639, A046669, A088822, A088823, A088824, A088825.

P:=List(List([2..60],n->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

Prepend[Accumulate[Rest[Table[FactorInteger[i][[1,1]],{i,60}]]],0] (* Harvey P. Dale, Jan 09 2011 *)

a(n) = sum(k=2, n, factor(k)[1,1]); \\ Michel Marcus, May 15 2017

a(n) ~ n^2/(2 log n) [Kalecki]. - Thomas Ordowski, Nov 29 2018

a(n) = Sum_{prime p} n(p)*p, where n(p) is the number of integers in [1,n] with smallest prime factor spf(.) = A020639(.) = p, decreasing from n(2) = floor(n/2) to n(p) = 1 for p >= sqrt(n), possibly earlier, and n(p) = 0 for p > n. One has n(p) ~ D(p)*n where D(p) = (Product_{primes q < p} 1-1/q)/p = A038110/A038111 is the density of numbers having p as smallest prime factor. - M. F. Hasler, Dec 05 2018

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

Original entry on oeis.org

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

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

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)).

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A088821 a(n) is the sum of smallest prime factors of numbers from 1 to n.

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

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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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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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