Displaying 1-10 of 11 results found.
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(3+I*4)/5.
+10
13
1, 9, 12, 16, 19, 24, 28, 34, 39, 45, 51, 57, 64, 71, 78, 85, 93, 102, 110, 119, 127, 137, 146, 156, 166, 176, 187, 197, 208, 219, 231, 243, 254, 267, 279, 291, 304, 317, 330, 344, 358, 371, 386, 400, 414, 429, 444, 459, 474, 490, 506, 522, 538, 554, 570, 587
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
Sequences A084799 - A084804 are related to zeros of the Riemann zeta function. The least integers that satisfy sum(n>0, 1/a(n)^z ) = 0, where a(1)=1, a(n+1)>a(n) and z = unit complex numbers using Pythagorean triples: (3+I*4)/5, (4+I*3)/5, (12+I*5)/13, (24+I*7)/25, (40+I*9)/41; these z produce a special pattern to the sequences.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(60+I*11)/61.
+10
3
1, 6072, 6078, 6084, 6091, 6097, 6104, 6110, 6116, 6123, 6129, 6136, 6142, 6148, 6155, 6161, 6168, 6174, 6180, 6187, 6193, 6200, 6206, 6212, 6219, 6225, 6232, 6238, 6244, 6251, 6257, 6264, 6270, 6276, 6283, 6289, 6296, 6302, 6308, 6315, 6321, 6328
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(1+I*sqrt(3))/2.
+10
3
1, 8, 12, 16, 21, 27, 33, 41, 49, 58, 67, 78, 89, 102, 114, 128, 143, 158, 174, 191, 208, 227, 246, 266, 286, 308, 330, 353, 377, 402, 427, 454, 481, 508, 537, 566, 597, 627, 659, 692, 725, 759, 794, 829, 866, 903, 941, 980, 1019, 1060, 1101, 1143, 1185, 1229
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(4+I*3)/5.
+10
2
1, 16, 18, 21, 25, 28, 31, 34, 38, 41, 44, 48, 52, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 104, 108, 112, 116, 121, 125, 130, 134, 139, 143, 148, 152, 157, 162, 166, 171, 176, 180, 185, 190, 195, 200, 205, 209, 214, 219, 224, 229, 234, 239, 244, 249, 255, 260
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(12+I*5)/13.
+10
2
1, 62, 65, 69, 72, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 113, 116, 120, 124, 128, 131, 135, 139, 143, 147, 150, 154, 158, 162, 166, 169, 173, 177, 181, 185, 189, 193, 197, 201, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(24+i*7)/25.
+10
2
1, 276, 280, 285, 289, 294, 298, 303, 307, 312, 316, 321, 325, 330, 334, 339, 343, 348, 352, 357, 361, 366, 370, 375, 379, 384, 388, 393, 397, 402, 406, 411, 415, 420, 425, 429, 434, 438, 443, 447, 452, 456, 461, 466, 470, 475, 479, 484, 488, 493, 498, 502
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(40+I*9)/41.
+10
2
1, 1285, 1290, 1296, 1301, 1306, 1312, 1317, 1323, 1328, 1334, 1339, 1344, 1350, 1355, 1361, 1366, 1372, 1377, 1382, 1388, 1393, 1399, 1404, 1410, 1415, 1421, 1426, 1431, 1437, 1442, 1448, 1453, 1459, 1464, 1469, 1475, 1480, 1486, 1491, 1497, 1502
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(1+I)/sqrt(2).
+10
2
1, 11, 14, 17, 20, 24, 27, 31, 35, 39, 44, 48, 52, 57, 62, 67, 72, 77, 82, 87, 93, 98, 104, 109, 115, 121, 127, 133, 139, 145, 151, 158, 164, 170, 177, 184, 190, 197, 204, 211, 218, 225, 232, 239, 246, 254, 261, 268, 276, 283, 291, 299, 306, 314, 322, 330, 338
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=1/2 + I.
+10
2
1, 6, 9, 12, 16, 20, 25, 31, 37, 43, 50, 58, 66, 75, 84, 94, 104, 115, 127, 139, 152, 165, 179, 193, 208, 223, 239, 256, 273, 290, 309, 327, 347, 367, 387, 408, 430, 452, 475, 498, 522, 546, 571, 596, 622, 649, 676, 704, 732, 761, 790, 820, 851, 882, 913, 945
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
Least positive integers, all distinct, that satisfy sum(n>0,1/a(n)^z)=0, where z=(1+I)/2.
+10
2
1, 28, 40, 54, 69, 87, 107, 128, 152, 177, 205, 235, 267, 300, 336, 374, 414, 455, 499, 545, 593, 643, 695, 749, 805, 863, 923, 985, 1048, 1115, 1183, 1253, 1325, 1399, 1475, 1553, 1633, 1715, 1799, 1886, 1974, 2064, 2156, 2250, 2347, 2445, 2545, 2648, 2752
COMMENTS
Sequence satisfies sum(n>0,1/a(n)^z)=0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude for the given z.
PROG
(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))
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