A334572 -id:A334572 - cp's OEIS frontend

A334573 Partial sums of A334572.

1, 2, 4, 6, 7, 8, 11, 14, 16, 17, 19, 21, 22, 23, 27, 31, 33, 35, 37, 39, 40, 41, 44, 47, 49, 52, 55, 57, 58, 59, 64, 69, 70, 71, 73, 75, 76, 77, 80, 83, 84, 85, 87, 89, 91, 92, 96, 100, 102, 104, 106, 108, 111, 114, 117, 120, 121, 122, 124, 126, 127, 129, 135, 141

Offset: 2

Michel Marcus, May 06 2020

nonn

a(n) = L_infinite(n) = Sum_{m=2..n} d_infinite(m, m-1) as defined in Kolossváry link.

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
István B. Kolossváry and István T. Kolossváry, Distance between natural numbers based on their prime signature, Journal of Number Theory, Vol. 234 (2022), pp. 120-139; arXiv preprint, arXiv:2005.02027 [math.NT], 2020-2021.

Cf. A051903, A334572, A334574.

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= n-> max(map(abs, {coeffs(f(n)-f(n-1))})):
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)) end:
seq(a(n), n=2..80);  # Alois P. Heinz, May 06 2020

f[n_] := Sum[{p, e} = pe; e x^p, {pe, FactorInteger[n]}];
b[n_] :=  CoefficientList[f[n] - f[n-1], x] // Abs // Max;
b /@ Range[2, 80] // Accumulate (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)
Accumulate[Max @@@ Partition[Join[{0}, Table[Max[FactorInteger[n][[;; , 2]]], {n, 2, 100}]], 2, 1]] (* Amiram Eldar, Jan 05 2024 *)

d(n) = {my(f=factor(n/(n-1))[,2]~); vecmax(apply(x->abs(x), f));}
a(n) = sum(k=2, n, d(k));

first(n)=my(v=vector(n-1),o,t,s); forfactored(k=2,n, t=vecmax(k[2][,2]); v[k[1]-1]=s+=max(o,t); o=t); v \\ Charles R Greathouse IV, Feb 01 2022

a(n) = Sum_{m=2..n} A334572(n).
a(n) = Sum_{m=2..n} max(A051903(n), A051903(n-1)).
a(n) ~ c * n, where c = 2.2883695... (A334574). - Amiram Eldar, Jan 05 2024

A176166 a(n) is the maximum exponent in the prime factorization of the n-th triangular number.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 3, 3, 1, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 5, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

From Amiram Eldar, Mar 28 2025: (Start)
The asymptotic density of the occurrences of terms < k is f(k) = Product_{p prime} (1 - 2/p^k) * (1+1/(2*(2^(k-1)-1))), for k >= 2.
The asymptotic density of the occurrences of k is d(1) = f(2) for k = 1, and d(k) = f(k+1) - f(k) for k >= 2.
The asymptotic mean of this sequence is lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k*d(k) = 1 + Sum_{k>=1} (1 - Product_{p prime} (1 - 2/p^(k+1)) * (1+1/(2*(2^k-1)))) = 1.89137712344735606085... . (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A051903, A135939, A334572.

f1[n_] := Max[Last/@FactorInteger[n]]; f2[n_]:=n*(n+1)/2; Join[{0}, Table[f1[f2[n]], {n, 2, 4*5!}]]
Max[FactorInteger[#][[;;,2]]]&/@Accumulate[Range[110]] // ReplacePart[1 -> 0] (* Harvey P. Dale, Oct 23 2024 *)

a(n) = if(n == 1, 0, vecmax(factor(n*(n+1)/2)[, 2])); \\ Amiram Eldar, Mar 28 2025

From Amiram Eldar, Mar 28 2025: (Start)
a(n) = A051903(A000217(n)).
a(n) = max(A051903(n), A051903((n+1)/2)) if n is odd, and max(A051903(n/2), A051903(n+1)) if n is even. (End)

a(1) inserted by Amiram Eldar, Mar 28 2025

A334574 Decimal expansion of limit when n->oo A334573(n)/n.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, May 06 2020

			2.2883695...

István B. Kolossváry and István T. Kolossváry, Distance between natural numbers based on their prime signature, Journal of Number Theory, Vol. 234 (2022), pp. 120-139; arXiv preprint, arXiv:2005.02027 [math.NT], 2020-2021.

Cf. A334572, A334573.

1 + sumpos(k = 2, 1 - prodeulerrat(1 - 2/p^k)) \\ Amiram Eldar, Jan 05 2024

Equals 1 + Sum_{k>=2} (1 - Product_{p prime} (1 - 2/p^k)). - Amiram Eldar, Jan 05 2024

More terms from Amiram Eldar, Jan 05 2024

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A334573 Partial sums of A334572.

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A176166 a(n) is the maximum exponent in the prime factorization of the n-th triangular number.

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A334574 Decimal expansion of limit when n->oo A334573(n)/n.

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