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5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 135, 136, 137, 140, 143, 146, 148, 149
COMMENTS
Absolute values of the negative terms of A209662. Sequence related to an infinite series which is equal to pi, due to Leonhard Euler.
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 25, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 50, 54, 56, 57, 59, 62, 63, 64, 65, 66, 67, 69, 71, 72, 75, 76, 77, 79, 81, 83, 84, 85, 86, 88, 92, 93, 94, 96, 98, 99, 100, 103, 107, 108, 112, 114
COMMENTS
Sequence related to an infinite series which is equal to pi, due to Leonhard Euler.
1, 2, 3, 4, -5, 6, 7, 8, 9, -10, 11, 12, -13, 14, -15, 16, -17, 18, 19, -20, 21, 22, 23, 24, 25, -26, 27, 28, -29, -30, 31, 32, 33, -34, -35, 36, -37, 38, -39, -40, -41, 42, 43, 44, -45, 46, 47, 48, 49, 50, -51, -52, -53, 54, -55, 56, 57, -58, 59, -60, -61, 62, 63, 64, 65
COMMENTS
Also denominators of an infinite series which is equal to pi, if the numerators are 1, 1, 1,..., for example: pi = 1/1 + 1/2 + 1/3 + 1/4 + 1/(-5) + 1/6 + 1/7 + 1/8 + 1/9 + 1/(-10) + 1/11 + 1/12 + 1/(-13) + 1/14 ... = 3.14159263... This arises from an infinite series due to Leonhard Euler which is given by: Pi = 1/1 + 1/2 + 1/3 + 1/4 - 1/5 + 1/6 + 1/7 + 1/8 + 1/9 - 1/10 + 1/11 + 1/12 - 1/13 + 1/14 ... = 3.14159263... For another version see A209661. a(n) = -n if n has an odd number of prime factors of the form 4k+1 (counted with multiplicity), else a(n) = n. - M. F. Hasler, Apr 15 2012
REFERENCES
Leonhard Euler, Introductio in analysin infinitorum, 1748.
EXAMPLE
For n = 10 we have that the 10th row of triangle A207338 is [2, -5] therefore a(10) = 2*(-5) = -10.
MATHEMATICA
f[p_, e_] := If[Mod[p, 4] == 1, (-1)^e, 1]; a[n_] := n * Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)
PROG
(PARI) a(n)={my(f=factor(n)); n*prod(i=1, #f~, my([p, e]=f[i, ]); if(p%4==1, -1, 1)^e)} \\ Andrew Howroyd, Aug 04 2018
Triangle read by rows in which row n lists the prime factors of n with repetition, with a(1) = 1, but with the primes of the form 4k + 1 multiplied by -1.
+10
2
1, 2, 3, 2, 2, -5, 2, 3, 7, 2, 2, 2, 3, 3, 2, -5, 11, 2, 2, 3, -13, 2, 7, 3, -5, 2, 2, 2, 2, -17, 2, 3, 3, 19, 2, 2, -5, 3, 7, 2, 11, 23, 2, 2, 2, 3, -5, -5, 2, -13, 3, 3, 3, 2, 2, 7, -29, 2, 3, -5, 31, 2, 2, 2, 2, 2, 3, 11, 2, -17, -5, 7, 2, 2, 3, 3, -37
COMMENTS
The row products of triangle give A209662. Also the row products of triangle divided by n give A209661. The mentioned sequences are related to an infinite series which is equal to pi, due to Leonhard Euler.
EXAMPLE
Written as a triangle begins:
1;
2;
3;
2, 2;
-5;
2, 3;
7;
2, 2, 2;
3, 3;
2, -5;
11;
2, 2, 3;
-13;
2, 7;
3, -5;
2, 2, 2, 2;
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