A356414
Number k such that k and k+1 both have an equal sum of even and odd exponents in their prime factorization (A356413).
Original entry on oeis.org
819, 1035, 1196, 1274, 1275, 1449, 1665, 1924, 1925, 1988, 2324, 2331, 2540, 3068, 3195, 3324, 3339, 3549, 3555, 3626, 3717, 4164, 4220, 4235, 4556, 4598, 4635, 4675, 4796, 5084, 5525, 5634, 5660, 6003, 6027, 6068, 6164, 6363, 6740, 6867, 6908, 7028, 7227, 7275
Offset: 1
819 is a term since A350386(819) = A350387(819) = 2 and A350386(820) = A350387(820) = 2.
-
f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[10^4], q[#] && q[# + 1] &]
-
is(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};
is1 = is(1); for(k = 2, 10^4, is2 = is(k); if(is1 && is2, print1(k-1,", ")); is1 = is2);
A356416
a(n) is the least start of exactly n consecutive numbers that have an equal sum of even and odd exponents in their prime factorization (A356413), or -1 if no such run of consecutive numbers exists.
Original entry on oeis.org
1, 819, 1274, 19940, 204323, 149228720, 3144583275
Offset: 1
a(2) = 819 since 819 = 3^2 * 7 * 13 and 820 = 2^2 * 5 * 41 both have an equal sum of even and odd exponents (2) in their prime factorization, 818 and 821 have no even exponent, and 819 is the least number with this property.
-
f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; seq[len_, nmax_] := Module[{s = Table[0, {len}], v = {1}, n = 2, c = 0, m}, While[c <= len && n <= nmax, If[q[n], v = Join[v, {n}], m = Length[v]; v = {}; If[0 <= m <= len && s[[m]] == 0, c++; s[[m]] = n - m]]; n++]; s]; seq[4, 2*10^4]
A371600
Numbers of least prime signature (A025487) whose prime factorization has equal sum of even and odd exponents.
Original entry on oeis.org
1, 60, 2160, 12600, 18480, 77760, 180180, 216000, 453600, 665280, 2646000, 2799360, 3880800, 7776000, 10810800, 16329600, 16336320, 23950080, 32016600, 45360000, 66528000, 95256000, 100776960, 139708800, 214414200, 232792560, 279936000, 389188800, 555660000, 587865600
Offset: 1
The prime signatures of the first 12 terms are:
n a(n) signature A350386(a(n)) = A350387(a(n))
-- ------- ------------ ------------- -------------
1 1 {} 0 0
2 60 {1,1,2} 2 1+1=2
3 2160 {1,3,4} 4 1+3=4
4 12600 {1,2,2,3} 2+2=4 1+3=4
5 18480 {1,1,1,1,4} 4 1+1+1+1=4
6 77760 {1,5,6} 6 1+5=6
7 180180 {1,1,1,1,2,2} 2+2=4 1+1+1+1=4
8 216000 {3,3,6} 6 3+3=6
9 453600 {1,2,4,5} 2+4=6 1+5=6
10 665280 {1,1,1,3,6} 6 1+1+1+3=6
11 2646000 {2,3,3,4} 2+4=6 3+3=6
12 2799360 {1,7,8} 8 1+7=8
-
fun[p_, e_] := (-1)^e * e; q[n_] := Module[{f = FactorInteger[n]}, n == 1 || (f[[-1, 1]] == Prime[Length[f]] && Plus @@ fun @@@ f == 0 && Max@ Differences[f[[;; , 2]]] < 1)]; Select[Range[4*10^6], q]
-
is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); n == 1 || (sum(i = 1, #e, (-1)^e[i] * e[i]) == 0 && e == vecsort(e, , 4) && primepi(p[#p]) == #p);}
Showing 1-3 of 3 results.
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