A201825
G.f.: exp( Sum_{n>=1} A119616(n)*x^n/n ) where A119616(n) = (sigma(n)^2 - sigma(n,2))/2.
Original entry on oeis.org
1, 0, 1, 1, 4, 2, 12, 6, 26, 21, 61, 46, 155, 114, 317, 293, 704, 627, 1539, 1400, 3149, 3124, 6469, 6463, 13341, 13467, 26271, 27889, 51970, 55513, 101961, 110261, 195586, 217238, 373391, 418299, 708816, 800833, 1323842, 1521372, 2461081, 2846722, 4543987
Offset: 0
G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 2*x^5 + 12*x^6 + 6*x^7 + 26*x^8 +...
where
log(A(x)) = 2*x^2/2 + 3*x^3/3 + 14*x^4/4 + 5*x^5/5 + 47*x^6/6 + 7*x^7/7 + 70*x^8/8 + 39*x^9/9 + 97*x^10/10 + 11*x^11/11 +...+ A119616(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n,(sigma(m)^2-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)}
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{a(n)=if(n==0,1,(1/n)*sum(k=1,n,(sigma(k)^2 - sigma(k,2))/2 * a(n-k)))}
A224381
Table of coefficients in the expansion of product((1+d_i*x), d_i|n).
Original entry on oeis.org
1, 1, 1, 1, 3, 2, 1, 4, 3, 1, 7, 14, 8, 1, 6, 5, 1, 12, 47, 72, 36, 1, 8, 7, 1, 15, 70, 120, 64, 1, 13, 39, 27, 1, 18, 97, 180, 100, 1, 12, 11, 1, 28, 287, 1400, 3444, 4032, 1728, 1, 14, 13, 1, 24, 163, 336, 196, 1, 24, 158, 360, 225, 1, 31, 310, 1240, 1984, 1024
Offset: 0
Row n = 6 : 1, 12, 47, 72, 36 because (1+x)*(1+2x)*(1+3x)*(1+6x) = 1 + 12*x + 47*x^2 + 72*x^3 + 36*x^4.
Table begins :
1;
1, 1;
1, 3, 2;
1, 4, 3;
1, 7, 14, 8;
1, 6, 5;
1, 12, 47, 72, 36;
1, 8, 7;
1, 15, 70, 120, 64;
1, 13, 39, 27;
1, 18, 97, 180, 100;
1, 12, 11;
1, 28, 287, 1400, 3444, 4032, 1728;
1, 14, 13;
1, 24, 163, 336, 196;
1, 24, 158, 360, 225;
1, 31, 310, 1240, 1984, 1024;
...
Last elements of rows give:
A007955.
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with(numtheory):
T:= proc(n) local p;
p:= mul(1+d*x, d=divisors(n));
seq(coeff(p, x, k), k=0..degree(p))
end:
seq(T(n), n=0..30); # Alois P. Heinz, Apr 05 2013
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T[n_] := CoefficientList[Product[1+d*x, {d, Divisors[n]}], x]; T[0] = {1};
Array[T, 20, 0] // Flatten (* Jean-François Alcover, Mar 27 2017 *)
A119623
Composite numbers for which the second elementary symmetric function of divisors (s2) is prime.
Original entry on oeis.org
6, 10, 14, 26, 34, 62, 82, 122, 142, 146, 202, 206, 226, 254, 334, 346, 362, 394, 446, 542, 562, 566, 586, 734, 766, 794, 842, 926, 934, 982, 1046, 1126, 1286, 1294, 1346, 1382, 1514, 1546, 1594, 1622, 1654, 1706, 1766, 1906, 1934
Offset: 1
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dv:=Divisors[n];le:=Length[dv];re=Reap[Do[If[ !PrimeQ[n],su=Sum[dv[[i]]*dv[[i+j]],{i,1,le-1},{j,1,le-i}];If[PrimeQ[su],Sow[{n,su}]]],{n,2,2000}]][[2,1]]
A337329
Sum of the products of all pairs of divisors of n, (d1,d2), such that d1|n, d2|n, d1|d2 and d1
Original entry on oeis.org
0, 2, 3, 14, 5, 41, 7, 70, 39, 87, 11, 245, 13, 149, 143, 310, 17, 455, 19, 539, 241, 321, 23, 1165, 155, 431, 390, 945, 29, 1521, 31, 1302, 509, 699, 467, 2639, 37, 857, 679, 2595, 41, 2687, 43, 2093, 1664, 1221, 47, 5053, 399, 2387, 1091, 2835, 53, 4370, 951, 4585, 1333, 1887, 59
Offset: 1
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Table[Sum[Sum[(i*k) (1 - Ceiling[k/i] + Floor[k/i])*(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 80}]
Showing 1-4 of 4 results.
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