A100042 -id:A100042 - cp's OEIS frontend

A181794 Numbers n such that the number of even divisors of n is an even divisor of n.

4, 6, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206, 208, 212, 214, 216, 218, 226, 234, 236, 240, 244, 252, 254, 256, 262, 264, 268, 272, 274, 278

Offset: 1

Matthew Vandermast, Nov 14 2010

nonn

All terms are even, since odd numbers, even if they have an even count of divisors, don't have any even divisors.

Includes all numbers of the form A000040(m)*A001146(n).

			a(4)=12 has four even divisors (2, 4, 6, and 12), and 4 is one of those even divisors.
The number 21 is not in this sequence: it has four divisors (1, 3, 7, and 21), and 4 is not one of those divisors.

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

A100484 and A001749 are subsequences. A001146 and A100042 are also subsequences except for their initial terms.

A370493 Numbers k such that A006530(k) = A051903(k).

Original entry on oeis.org

4, 24, 27, 54, 72, 108, 160, 216, 480, 800, 896, 1215, 1440, 2400, 2430, 2688, 3125, 4000, 4320, 4480, 4860, 6075, 6250, 6272, 7200, 8064, 9375, 9720, 12000, 12150, 12500, 12960, 13440, 15309, 18750, 18816, 19440, 20000, 21600, 22400, 22528, 24192, 24300, 25000

Offset: 1

Amiram Eldar, Feb 20 2024

nonn

			72 = 2^3 * 3^2 is a term since A006530(72) = A051903(72) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..3000

Cf. A002110, A006530, A051903, A017665, A017666.

Subsequences: A051674, A100042, A370492.

q[n_] := Module[{f = FactorInteger[n]}, Max[f[[;; , 2]]] == f[[-1, 1]]]; Select[Range[2, 25000], q]

is(n)={my(f = factor(n), p = f[,1], e = f[,2]); n > 1 && p[#p] == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=1} ((Sum_{i=1..prime(k)-1} 1/p^i) * (s(p(k-1)^prime(k)) - s(p(k-1)^(prime(k)-1))) + s(p(k-1)^prime(k))/prime(k)^prime(k)) = 0.39239336056178266729..., where s(k) = sigma_{-1}(k) = A017665(k)/A017666(k), and p(k) = prime(k)# = A002110(k).

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

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A181794 Numbers n such that the number of even divisors of n is an even divisor of n.

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A370493 Numbers k such that A006530(k) = A051903(k).

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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