Interprimes

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The $\scriptstyle n \,$ ^th interprime number is the mid-point between the $\scriptstyle n \,$ ^th odd prime and the $\scriptstyle (n+1) \,$ ^th odd prime.

$i_n = \frac{p_{n+1} + p_{n+2}}{2},\quad n \ge 1, \,$

where $\scriptstyle p_n \,$ is the $\scriptstyle n \,$ ^th prime.

The interprime numbers are obviously all composite numbers. An interprime between twin primes (except for the (3, 5) and (5, 7) twin primes) is always even, congruent to 0 (mod 6) and is congruent to 0, 2 or 8 (mod 10). An interprime between cousin primes (except for the (3, 7) cousin primes) is always odd, congruent to 3 (mod 6) and is congruent to 1, 5 or 9 (mod 10). An interprime between the second (ending in 3 in base 10) and third (ending in 7 in base 10) member of a prime quadruplet (except for the (5, 7, 11, 13) prime quadruplet) will always be odd and congruent to 5 (mod 10).

The following sum

$\sum_{n=1}^{\infty} (-1)^n ~ e^{-(x - p_{n})^2} \,$

where $\scriptstyle p_n \,$ is the $\scriptstyle n \,$ ^th prime, has zeros at almost integer approximations of the interprimes, and also the non integer "interprime" $\scriptstyle \frac{5}{2} \,$ , mid-point of the even prime 2 and first odd prime 3.^[1]

**?????**
Now, does the following sum $\sum_{n=0}^{\infty} (-1)^n ~ e^{-(x - i_{n})^2} \,$ where $\scriptstyle i_0 \,$ is the non integer "interprime" $\scriptstyle \frac{5}{2} \,$ (mid-point of the even prime 2 and first odd prime 3,) $\scriptstyle i_n,\, n \,\ge\, 1, \,$ is the $\scriptstyle n \,$ ^th interprime, have zeros at almost integer approximations of the primes? — Daniel Forgues 22:32, 2 August 2011 (UTC)

Sequences

Interprimes: mid-point between consecutive odd primes (average of two consecutive odd primes.) (Cf. A024675)

{4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279, 282, 288, 300, ...}

Even interprimes. (Cf. A072568)

{4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, 64, 72, 76, 86, 102, 108, 120, 134, 138, 144, 150, 154, 160, 170, 176, 180, 186, 192, 198, 228, 236, 240, 246, 254, 260, 266, 270, 274, 282, 288, 300, 312, 324, 334, 342, 348, 356, 370, 376, 386, 414, 420, 426, 432, ...}

Odd interprimes. (Cf. A072569)

{9, 15, 21, 39, 45, 69, 81, 93, 99, 105, 111, 129, 165, 195, 205, 217, 225, 231, 279, 309, 315, 351, 363, 381, 393, 399, 405, 441, 453, 459, 465, 473, 483, 489, 495, 501, 515, 615, 625, 645, 667, 675, 687, 705, 723, 741, 747, 759, 765, 771, 803, 825, 855, 861, ...}

Odd interprimes divisible by 3. (Cf. A072572)

{9, 15, 21, 39, 45, 69, 81, 93, 99, 105, 111, 129, 165, 195, 225, 231, 279, 309, 315, 351, 363, 381, 393, 399, 405, 441, 453, 459, 465, 483, 489, 495, 501, 615, 645, 675, 687, 705, 723, 741, 747, 759, 765, 771, 825, 855, 861, 879, 885, 897, 909, 915, 933, ...}

Odd interprimes not divisible by 3. (Cf. A072573)

{205, 217, 473, 515, 625, 667, 803, 1003, 1207, 1243, 1313, 1465, 1505, 1517, 1537, 1681, 1715, 1795, 1817, 1895, 2075, 2105, 2191, 2303, 2405, 2453, 2585, 2627, 2783, 2933, 3055, 3073, 3175, 3197, 3265, 3337, 3353, 3505, 3565, 3665, 3937, 3995, 4085, ...}

A126228 Odd interprimes divisible by 5.

A126229 Odd interprimes divisible by 7.

A126230 Odd interprimes divisible by 11.

A124619 Odd interprimes divisible by 13.

A124620 Odd interprimes divisible by 17.

A126231 Odd interprimes divisible by 19.

A124622 Smallest odd interprime divisible by n-th odd prime.

Numbers n such that n^2 is an interprime = average of two successive primes. (Cf. A075190)

{2, 3, 8, 9, 12, 15, 18, 21, 25, 33, 41, 51, 60, 64, 72, 78, 92, 112, 117, 129, 138, 140, 159, 165, 168, 172, 192, 195, 198, 213, 216, 218, 228, 237, 273, 295, 298, 303, 304, 309, 322, 327, 330, 338, 342, 356, 360, 366, 387, 393, 408, 416, 429, 432, 441, 447, 456, ...}

Numbers n such that n^3 is an interprime = average of two successive primes. (Cf. A075191)

{4, 12, 16, 26, 28, 36, 48, 58, 66, 68, 74, 78, 102, 106, 112, 117, 124, 126, 129, 130, 148, 152, 170, 174, 184, 189, 190, 192, 224, 273, 280, 297, 321, 324, 369, 372, 373, 399, 408, 410, 421, 426, 429, 435, 447, 449, 450, 470, 475, 496, 504, 507, 531, 537, ...}

Numbers n such that n^4 is an interprime = average of two successive primes. (Cf. A075192)

{3, 5, 8, 21, 55, 66, 87, 99, 104, 105, 110, 120, 129, 135, 141, 144, 152, 168, 172, 186, 187, 192, 211, 222, 243, 279, 283, 295, 297, 321, 342, 385, 395, 398, 408, 425, 426, 460, 520, 541, 559, 597, 626, 627, 638, 642, 657, 666, 673, 680, 713, 755, 759, 765, ...}

Numbers n such that n^5 is an interprime = average of two successive primes. (Cf. A075228)

{20, 42, 77, 81, 186, 198, 200, 220, 248, 266, 270, 294, 300, 387, 411, 477, 498, 537, 630, 678, 682, 696, 728, 741, 774, 819, 872, 985, 987, 1001, 1014, 1037, 1060, 1083, 1084, 1087, 1098, 1140, 1155, 1162, 1232, 1245, 1278, 1316, 1370, 1392, 1397, 1402, ...}

Numbers n such that n^6 is an interprime = average of two successive primes. (Cf. A075229)

{2, 4, 6, 18, 24, 27, 30, 53, 96, 122, 175, 195, 213, 231, 265, 300, 408, 420, 426, 450, 492, 532, 570, 614, 618, 657, 682, 705, 774, 777, 822, 858, 915, 946, 948, 1001, 1008, 1061, 1073, 1107, 1145, 1186, 1233, 1269, 1323, 1352, 1374, 1406, 1413, 1440, 1480, ...}

Numbers n such that n^7 is an interprime = average of two successive primes. (Cf. A075230)

{20, 33, 41, 71, 82, 99, 151, 165, 254, 267, 283, 316, 345, 462, 486, 496, 516, 630, 657, 668, 676, 681, 687, 724, 760, 945, 1004, 1050, 1085, 1167, 1305, 1314, 1316, 1326, 1335, 1389, 1414, 1420, 1454, 1456, 1512, 1638, 1644, 1726, 1803, 1874, 1905, 1963, ...}

Numbers n such that n^8 is an interprime = average of two successive primes. (Cf. A075231)

{12, 111, 116, 175, 183, 205, 246, 305, 313, 406, 438, 593, 594, 620, 696, 714, 788, 824, 844, 969, 1014, 1023, 1061, 1080, 1153, 1176, 1204, 1288, 1367, 1456, 1470, 1515, 1533, 1538, 1572, 1626, 1659, 1689, 1692, 1695, 1734, 1759, 1788, 1860, 1928, ...}

Numbers n such that n^9 is an interprime = average of two successive primes. (Cf. A075232)

{9, 74, 110, 141, 340, 370, 411, 423, 546, 687, 720, 723, 725, 744, 813, 834, 966, 1033, 1054, 1137, 1178, 1233, 1264, 1284, 1287, 1320, 1335, 1460, 1636, 1642, 1768, 1934, 2046, 2053, 2064, 2103, 2214, 2397, 2447, 2465, 2486, 2496, 2510, 2716, 2741, 2775, ...}

Numbers n such that n^10 is an interprime = average of two successive primes. (Cf. A075233)

{9, 42, 87, 105, 108, 141, 144, 166, 215, 250, 381, 387, 482, 490, 500, 645, 748, 792, 831, 860, 876, 968, 990, 1377, 1448, 1468, 1526, 1769, 1780, 1922, 1968, 2084, 2118, 2228, 2245, 2252, 2373, 2381, 2478, 2565, 2672, 2781, 2883, 2915, 2972, 2988, 3008, ...}

Least k such that k^n is the smallest interprime which is an n-th power. (Cf. A075234)

{4, 2, 4, 3, 20, 2, 20, 12, 9, 9, 24, 2, 23, 26, 20, 66, 10, 3, 16, 3, 92, 13, 18, 48, 230, 129, 78, 181, 315, 33, 231, 19, 14, 152, 78, 39, 39, 4, 144, 9, 143, 55, 106, 25, 10, 91, 17, 7, 107, 91, 35, 44, 426, 81, 380, 97, 265, 237, 611, 1034, 122, 1072, 298, 1213, 18, 51, ...}

Notes

↑ Weisstein, Eric W., Interprime, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Interprime.html]

[0] Weisstein, Eric W., Interprime, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Interprime.html]

[1]

Interprimes

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