A082299 -id:A082299 - cp's OEIS frontend

A373365 a(n) = gcd(A001414(n), A064097(n)), where A001414 is the sum of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 5, 7, 1, 1, 2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 6, 2, 8, 1, 7, 1, 2, 1, 1, 1, 1, 1, 1, 9, 2, 1, 1, 4, 1, 1, 2, 2, 9, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

As A001414 and A064097 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001414, A064097.

Cf. also A082299, A373362, A373363, A373364, A373366.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A373365(n) = gcd(A001414(n), A064097(n));

A082300 Numbers relatively prime to the sum of their prime factors (with repetition).

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

In other words, numbers n such that n and sopfr(n) are relatively prime, where sopfr(n) (A001414) is the sum of the primes (with repetition) dividing n.
Conjecture: a(n) ~ (Pi^2/6)n. - Charles R Greathouse IV, Aug 04 2016
No term is prime since for prime p, p and 2p are not coprime; similarly no term is a prime power. A050703 is a subsequence because then n+sopfr(n) is prime, and so coprime to n. - David James Sycamore, Mar 04 2018

			gcd(2*2*5,2+2+5) = gcd(2*2*5,3*3)=1, therefore 20 is a term;
gcd(3*11,3+11) = gcd(3*11,2*7)=1, therefore 33 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001414, A275665, A050703.

A082299(a(n)) = 1.

Select[Range@ 106, CoprimeQ[#, Total@ Flatten@ Map[Table[#1, {#2}] & @@ # &, FactorInteger[#]]] &] (* Michael De Vlieger, Aug 06 2016 *)

sopfr(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]*f[i,2])
is(n)=gcd(sopfr(n),n)==1 \\ Charles R Greathouse IV, Aug 04 2016

Revised definition from Lior Manor, Apr 14 2004

A333980 Partial sums of A337490.

Original entry on oeis.org

1, 3, 7, 14, 25, 30, 36, 49, 70, 100, 110, 131, 164, 210, 224, 253, 298, 360, 378, 415, 472, 550, 572, 617, 686, 780, 806, 859, 940, 1050, 1190, 1361, 1564, 1800, 2070, 2375, 2716, 3094, 3510, 3549, 3628, 3748, 3910, 4115, 4364, 4658, 4998, 5385, 5388, 5440, 5542, 5559, 5628, 5750

Offset: 0

Scott R. Shannon, Sep 04 2020

nonn

See A337490 for an explanation of the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 4, 13, 23, 34, 46, 59, 73, 88, 8, 25, 43, 62, 10, 31, 53, 76, 100, 125, 151, 178, 206, 235, 15, 46, 78, 111, 145, 5, 41, 78, 116, 155, 195, 236, 278, 321, 365, 410, 456, 503, 551, 600, 50, 101, 153, 206, 260, 315, 371, 428, 486, 545, 605, 666, 728, 791, 855, 920, 986, 1053

Offset: 1

Scott R. Shannon, Sep 03 2020

nonn

This is a variation of A337490; here we start with an offset of 1, so a(1) = 1. See that sequence for further details.

In the first 4212 terms the sequence decreases 69 times while 45 terms are repeated, the first being 13 at n=9 and the last 399876 at n=4212. After n(4166)=84 the sequence does not decrease again for n up to at least 100 million. The lowest numbers that have not appeared in that range are 30,37,47,48,49,51. The 100 millionth term is 4999999941527298.

			a(2) = 3 as the sum of all previous terms is a(1) = 1, and the GCD of 1 and 2 is 1. However 1 has already appeared so a(2) = a(1) + n = 1 + 2 = 3.
a(4) = 2 as the sum of all previous terms is a(1)+a(2)+a(3) = 10, and the GCD of 10 and 4 is 2, and as 2 has not previous appeared a(4) = 2.
a(8) = 4 as the sum of all previous terms is a(1)+...+a(7) = 52, and the GCD of 52 and 8 is 4, and as 4 has not previous appeared a(8) = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Graph of the terms for n=1..4500. This includes the last known decrease in the sequence, n(4166)=84.

Cf. A337490 (same sequence rules but starting a(0)=1), A333980, A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A333716 a(0)=1; for n>0, a(n) is the greatest common divisor (GCD) of n and the sum of the previous terms back to the last GCD term, if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 2, 4, 7, 11, 5, 11, 18, 26, 3, 13, 24, 36, 49, 63, 78, 94, 111, 129, 148, 168, 189, 211, 234, 258, 283, 309, 336, 364, 393, 423, 454, 486, 519, 553, 588, 624, 661, 699, 738, 778, 819, 861, 43, 87, 132, 178, 225, 273, 322, 10, 61, 113, 166, 220, 275, 331, 388, 446, 505, 565

Offset: 0

Scott R. Shannon, Sep 03 2020

nonn

This sequence is similar to A337490 except that here when a GCD term is added to the sequence the sum of previous terms is reset to the value of that GCD. Subsequent terms calculate the sum of previous terms back to this last GCD value. See the examples below.

Unlike A337490 this sequence shows numerous drops in value as the sum used when calculating the GCD with n is constantly being reset back to a smaller value anytime a unique GCD value greater than 1 is found. In the first one million terms the sequence drops in value 1516 times, the largest drop occurring from a(738133) = 45463489818 to a(738134) = 738134.

			a(4) = 11 as the sum of the previous terms is a(0)+...+a(3) = 14, and the GCD of 14 and 4 is 2. However 2 has already appeared so a(4) = a(3) + n = 7 + 4 = 11.
a(5) = 5 as the sum of all previous terms is a(0)+...+a(4) = 25, and the GCD of 25 and 5 is 5, and as 5 has not previously appeared a(5) = 5. As this term adds a GCD value to the sequence, the running sum of previous terms is now set to 5.
a(6) = 11 as the sum of previous terms is now just a(5) = 5, and as the GCD of 5 and 6 is 1, which already appears in the sequence, a(6) = a(5) + 6 = 5 + 6 = 11.
a(9) = 3 as the sum of previous terms back to the last GCD term is a(5)+...+a(8) = 60, and the GCD of 60 and 9 is 3, and as 3 has not previously appeared, a(9) = 3. As this term adds a GCD value to the sequence, the running sum of previous terms is now set to 3.

Scott R. Shannon, Graph of the terms for n=0 to n=1000000.

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

Block[{k = 0}, Nest[Append[#, If[FreeQ[#1, #3], Set[k, #2]; #3, #1[[-1]] + #2]] & @@ {#1, #2, GCD[Total@ #1[[k + 1 ;; #2]], #2]} & @@ {#, Length@ #} &, {1}, 60]] (* Michael De Vlieger, Sep 20 2020 *)

A381249 Indices of records in k/A001414(k), k>=2.

Original entry on oeis.org

2, 6, 8, 9, 12, 15, 16, 18, 24, 27, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 81, 90, 96, 108, 120, 128, 135, 144, 160, 162, 180, 192, 216, 240, 243, 270, 288, 320, 324, 360, 384, 405, 432, 480, 486, 540, 576, 640, 648, 720, 729, 810, 864, 960, 972, 1024, 1080, 1152

Offset: 1

Clark Kimberling, Apr 19 2025

nonn

Except for initial 2, this is a subsequence of A381972.

			f(2) = 1 < f(6) = 6/5 < f(8) = 4/3 < f(9) = 3/2 < f(12) = 12/7, where f(k) = k/A001414(k).

Cf. A001414, A082299, A082343, A082344, A381972.

z = 800; g[n_] := FactorInteger[n];
f[n_] := Map[First, g[n]] . Map[Last, g[n]];
mx = -1; k = 2; u = {}; While[k < z, a = k/f[k]; If[a > mx, mx = a; AppendTo[u, k]]; k++]; u

sopfr(n) = (n=factor(n))[, 1]~*n[, 2];
lista(nn) = my(r=oo, list=List()); for (n=2, nn, my(x=sopfr(n)/n); if (x < r, listput(list, n); r = x)); Vec(list); \\ Michel Marcus, Apr 27 2025

More terms from Michel Marcus, Apr 27 2025

A381972 Numbers k>=3 such that k/A001414(k) > (k-1)/A001414(k-1).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 24, 27, 30, 32, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 77, 78, 80, 81, 84, 87, 88, 90, 95, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 117, 119, 120, 123, 124, 125, 126

Offset: 1

Clark Kimberling, Mar 16 2025

nonn

1 <= a(n+1) - a(n) <= 10 for n = 2..3000000.

			f(2) = 1 < f(6) = 6/5 < f(8) = 4/3 < f(9) = 3/2 < f(12) = 12/7, where f(k) = k/A001414(k).

Cf. A001414, A082299, A082343, A082344, A381249.

z = 200; f[n_] := FactorInteger[n];
g[n_] := Map[First, f[n]] . Map[Last, f[n]];
h[n_] := If[n/g[n] > (n - 1)/g[n - 1], n, 0];
Rest[Union[Table[h[n], {n, 2, z}]]]

Definition corrected by Clark Kimberling, May 08 2025

cp's OEIS Frontend

A373365 a(n) = gcd(A001414(n), A064097(n)), where A001414 is the sum of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

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A082300 Numbers relatively prime to the sum of their prime factors (with repetition).

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A333980 Partial sums of A337490.

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A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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A333716 a(0)=1; for n>0, a(n) is the greatest common divisor (GCD) of n and the sum of the previous terms back to the last GCD term, if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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A381249 Indices of records in k/A001414(k), k>=2.

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A381972 Numbers k>=3 such that k/A001414(k) > (k-1)/A001414(k-1).

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