A088822 -id:A088822 - 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

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)).

A133133 a(n) is the largest prime factor of the sum of the largest prime factors of numbers from 2 to n.

Original entry on oeis.org

2, 5, 7, 3, 5, 11, 3, 3, 2, 43, 23, 59, 11, 71, 73, 5, 31, 7, 13, 31, 5, 79, 23, 83, 179, 13, 7, 109, 223, 127, 2, 89, 71, 97, 7, 331, 7, 11, 23, 409, 13, 17, 47, 19, 83, 109, 137, 37, 7, 577, 59, 643, 19, 73, 83, 683, 89, 257, 97, 31, 31, 7, 877, 89, 53, 11, 197, 7, 29, 181, 11

Offset: 2

Paul Finley (pfinley(AT)touro.edu), Sep 21 2007

			a(11) = gpf(43) = 43;
a(12) = gpf(46) = 23.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A006530, A088822.

gpf:= proc(n) max(numtheory:-factorset(n)) end proc:
map(gpf, ListTools:-PartialSums(map(gpf, [$2..100])));

FactorInteger[#][[-1,1]]&/@Accumulate[Table[FactorInteger[n][[-1,1]],{n,2,80}]] (* Harvey P. Dale, Feb 01 2025 *)

gpf(n) = vecmax(factor(n)[,1]);
a(n) = gpf(sum(k=2, n, gpf(k))); \\ Michel Marcus, Jun 02 2021

A303659 Numbers k such that the average of greatest prime factors of all positive integers <= k is an integer.

Original entry on oeis.org

2, 5, 6, 24, 27, 28, 79, 1171, 1901, 1932, 7845, 19152, 360698, 649358, 1731441, 1979944, 30658517, 34337898, 400114748, 412626117, 3738454462, 7725156081, 97366414806, 129951504625, 290788509485

Offset: 1

Ilya Gutkovskiy, Apr 28 2018

Numbers k such that (k - 1) | (A046670(k) - 1).
Numbers k such that (k - 1) | A088822(k).
a(26) > 10^12. - Giovanni Resta, May 05 2018

			6 is in the sequence because (gpf(2) + gpf(3) + gpf(4) + gpf(5) + gpf(6))/5 = (2 + 3 + 2 + 5 + 3)/5 = 3 is an integer.

Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A046670, A088822, A226647, A284755, A303480, A303482.

isok(n) = (n>1) && !(sum(k=2, n, vecmax(factor(k)[,1])) % (n-1)); \\ Michel Marcus, Apr 29 2018

lista(nn) = {s = 0; for (n=2, nn, s += vecmax(factor(n)[,1]); if (! (s % (n-1)), print1(n, ", ")););} \\ Michel Marcus, Apr 29 2018

a(13)-a(18) from Michel Marcus, Apr 29 2018

a(19)-a(25) from Giovanni Resta, May 05 2018

cp's OEIS Frontend

A088821 a(n) is the sum of smallest prime factors of numbers from 1 to 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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A133133 a(n) is the largest prime factor of the sum of the largest prime factors of numbers from 2 to n.

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Maple

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PARI

A303659 Numbers k such that the average of greatest prime factors of all positive integers <= k is an integer.

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PARI

PARI

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