A074373 -id:A074373 - cp's OEIS frontend

A362584 Integers k > 1 such that k >= the square of the sum of their prime factors (A074373(k)).

243, 256, 270, 288, 300, 320, 324, 336, 360, 375, 378, 384, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 495, 500, 504, 512, 525, 528, 540, 550, 560, 567, 576, 585, 588, 594, 600, 616, 624, 625, 630, 640, 648, 650, 660, 672, 675, 686, 693, 700, 702, 704

Offset: 1

Simon R Blow, Jun 23 2023

nonn

			243 >= A001414(243)^2 = (3+3+3+3+3=15)^2 = 225 so 243 is a term.
800 >= A001414(800)^2 = (2+2+2+2+2+5+5=20)^2 = 400 so 800 is a term.

Simon R Blow, Table of n, a(n) for n = 1..5000

Cf. A001414, A074373.

q:= n-> n>=add(i[1]*i[2], i=ifactors(n)[2])^2:
select(q, [$2..800])[];  # Alois P. Heinz, Jun 23 2023

Select[Range[2, 700], # >= (Plus @@ Times @@@ FactorInteger[#])^2 &] (* Amiram Eldar, Jun 24 2023 *)

isok(n) = ((p=factor(n))[, 1]~*p[, 2])^2 <= n \\ Thomas Scheuerle, Jun 23 2023

A074375 s(s+3)/2 where s is the sum of the prime factors of n (with repetition).

Original entry on oeis.org

0, 5, 9, 14, 20, 20, 35, 27, 27, 35, 77, 35, 104, 54, 44, 44, 170, 44, 209, 54, 65, 104, 299, 54, 65, 135, 54, 77, 464, 65, 527, 65, 119, 209, 90, 65, 740, 252, 152, 77, 902, 90, 989, 135, 77, 350, 1175, 77, 119, 90, 230, 170, 1484, 77, 152, 104, 275, 527, 1829, 90

Offset: 1

W. Neville Holmes, Aug 29 2002

			a(20) = 9(9+3)/2 = 54 because 9 = 2+2+5 and 20 = 2*2*5.

Neville Holmes, Integer Sequence Combinations

Applies A000096 to A001414. Cf. A074373, A074374.

spf[n_]:=Module[{c=Total[Times@@@FactorInteger[n]]},(c(c+3))/2]; Join[ {0}, Rest[Array[spf,60]]] (* Harvey P. Dale, Aug 16 2011 *)

A074376 s(3s-1)/2 where s is the sum of the prime factors of n (with repetition).

Original entry on oeis.org

0, 5, 12, 22, 35, 35, 70, 51, 51, 70, 176, 70, 247, 117, 92, 92, 425, 92, 532, 117, 145, 247, 782, 117, 145, 330, 117, 176, 1247, 145, 1426, 145, 287, 532, 210, 145, 2035, 651, 376, 176, 2501, 210, 2752, 330, 176, 925, 3290, 176, 287, 210, 590, 425, 4187

Offset: 1

W. Neville Holmes, Aug 29 2002

			a(20) = 9(3*9-1)/2 = 117 because 9 = 2+2+5 and 20 = 2*2*5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Neville Holmes, Integer Sequence Combinations [Broken link?]

Applies A000326 to A001414. Cf. A074373, A074374, A074375.

spf[n_]:=Module[{t=Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]]},(t(3t-1))/2]; Join[{0},Array[spf,60,2]] (* Harvey P. Dale, Sep 23 2016 *)

sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2])
fn(n) = my(s=sopfr(n)); s*(3*s-1)/2 \\ Michel Marcus, Jul 11 2013

A255384 a(n) = sopfr(n)^2 - 2n, where sopfr(n) is the sum of the prime factors of n with multiplicity.

Original entry on oeis.org

-2, 0, 3, 8, 15, 13, 35, 20, 18, 29, 99, 25, 143, 53, 34, 32, 255, 28, 323, 41, 58, 125, 483, 33, 50, 173, 27, 65, 783, 40, 899, 36, 130, 293, 74, 28, 1295, 365, 178, 41, 1599, 60, 1763, 137, 31, 533, 2115, 25, 98, 44, 298, 185, 2703, 13, 146, 57, 370, 845

Offset: 1

Wesley Ivan Hurt, May 05 2015

sign

If n is prime, then a(n) = n*(n-2). If n is semiprime, then a(n) gives the sum of the squares of the prime factors of n (with multiplicity).

a(n) is negative for n = 1, 81, 90, 96, 100, 108, 120, 125, 126, 128, 135, .... - Charles R Greathouse IV, May 06 2015

			a(6) = sopfr(6)^2 - 2(6) = (2+3)^2 - 12 = 25 - 12 = 13.
a(8) = sopfr(8)^2 - 2(8) = (2+2+2)^2 - 16 = 36 - 16 = 20.

Cf. A074373 (sopfr^2), A001414 (sopfr).

sopfr[n_] := Plus @@ Times @@@ FactorInteger@n; f[1] = 0; Table[sopfr[n]^2 - 2 n, {n, 100}]

sopfr(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]*f[i,2])
a(n)=sopfr(n)^2 - 2*n \\ Charles R Greathouse IV, May 06 2015

a(n) = A074373(n) - A005843(n).

cp's OEIS Frontend

A362584 Integers k > 1 such that k >= the square of the sum of their prime factors (A074373(k)).

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A074375 s(s+3)/2 where s is the sum of the prime factors of n (with repetition).

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A074376 s(3s-1)/2 where s is the sum of the prime factors of n (with repetition).

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A255384 a(n) = sopfr(n)^2 - 2n, where sopfr(n) is the sum of the prime factors of n with multiplicity.

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