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%I #20 Sep 08 2022 08:45:07
%S 1,2,5,4,5,12,17,10,15,16,31,36,9,28,41,48,57,24,31,50,9,16,37,48,49,
%T 76,15,42,85,116,79,114,137,52,41,96,121,148,27,52,79,144,139,16,65,
%U 136,109,84,141,220,49,86,169,166,209,254,33,124,169,240,55,48,297,66
%N Let n^2 = q*prime(n) + r with 0 <= r < prime(n); then a(n) = q + r.
%C The digital sum (base the n-th prime) of n^2.
%C A lower bound is a(n) >= n^2/prime(n) ~ n/log(n log n). No term less than this can occur after index n, e.g., a(n) > 126 for n > 10^3 and a(n) > 954 for n > 10^4. "Late birds" (such that a(k) > a(n) for all k > n) are a(1) = 1, a(2) = 2, a(4) = 4, a(5) = 5, a(21) = 9, a(27) = 15, a(44) = 16, a(104) = 24, a(173) = 59, a(365) = 61, a(369) = 81, a(500) = 100, a(590) = 124, a(735) = 129, a(840) = 152, a(987) = 169, a(1564) = 196, a(1797) = 249, a(2415) = 305, a(3368) = 400, a(3545) = 425, a(4025) = 475, a(4466) = 520, a(5018) = 556, a(5477) = 565, a(6686) = 676, a(7239) = 771, a(8025) = 795, a(8182) = 904, a(9369) = 939, ... Values that occur not less often than any smaller one are: 1, 2, 4 (once), 5, 9, 15 (twice), 16, 48, 64, 86, 100 (three times), 144 (five times), 169 (seven times), ... Values that never occur are: 3, 6, 7, 8, 11, 13, 14, 18, 19, 20, 21, 22, 23, 25, 26, 29, 30, 32, 34, 35, 38, 39, 40, 43, 44, 45, 47, 51, 53, 54, 56, 58, 62, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 80, 82, 83, 87, 89, 90, 91, 92, 94, 97, 98, 99, ... - _M. F. Hasler_, Nov 25 2016
%H M. F. Hasler, <a href="/A075771/b075771.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Lawson-Perfect and D. Cushing, <a href="http://aperiodical.com/2016/08/integer-sequence-reviews-a075771-a032799-a002717/">Integer Sequence Reviews: A075771, A032799, A002717</a>, The Aperiodical, Aug. 6, 2016
%F a(n) = ds_prime(n)(n^2), where ds_prime(n) = digital sum base the n-th prime.
%F a(n) = n^2 - (prime(n)-1)*floor(n^2/prime(n)). For example, a(2) = ds_prime(2)(2^2) = ds_3(4) = 1 + 1 = 2; a(6) = ds_prime(6)(6^2) = ds_13(36) = 2 + 10 = 12.
%e 6^2/p(6) = 36/13 = 2+10/13; a(6) = 2+10 = 12.
%t Table[n^2 - (Prime[n] - 1) Floor[n^2 / Prime[n]], {n, 80}] (* _Vincenzo Librandi_, Feb 18 2015 *)
%o (PARI) a(n) = {my(sq = n^2); my(p = prime(n)); (sq % p) + sq\p;} \\ _Michel Marcus_, Feb 18 2015
%o (Magma) [n^2-(NthPrime(n)-1)*Floor(n^2/NthPrime(n)): n in [1..70]]; // _Vincenzo Librandi_, Feb 18 2015
%o (PARI) A075771(n)=[1,1]*divrem(n^2,prime(n)) \\ _M. F. Hasler_, Nov 25 2016
%Y Cf. A135101, A135102, A135103.
%K nonn
%O 1,2
%A _Werner D. Sand_, Oct 09 2002
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