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A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.

%I A172035 #10 Jul 22 2025 07:32:28
%S A172035 5,0,2,2,9,3,2,5,3,2,7,2,4,5,2,2,5,2,3,6,2,2,2,2,4,8,4,2,2,4,2,8,2,3,
%T A172035 2,2,4,4,6,2,4,2,10,3,4,2,3,2,4,3,5,6,3,4,4,2,2,2,2,2,3,4,3,3,3,5,3,3,
%U A172035 8,2,3,12,2,3,2,5,2,3,8,16,8,3,4,2,3,2,4,2,2,5,7,4,3,8,3,2,6,2,3,6,2,2,10
%N A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.
%C A172035 k = 1 is the "trivial" case: sod(prime(n)) = prime(m)
%C A172035 n = 2, prime(2) = 3: 3^k is for k > 1 a multiple of 3^2.
%D A172035 M. Fujiwara, Y. Ogawa: Introduction to truly beautiful Mathematics, Chikuma Shobo, Tokyo 2005.
%D A172035 Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
%D A172035 Hans Schubart: Einfuehrung in die klassische und moderne Zahlentheorie, Vieweg, Braunschweig 1974.
%e A172035 sod(2^5)=5, sod(5^2)=7, sod(7^2)=13, sod(11^9)=53, sod(13^3)=19, sod(17^2)=19, sod(19^5)=37, sod(23^3)=17, sod(29^2)=13, sod(31^7)=31, sod(37^2)=19, sod(41^4)=31, sod(43^5)=31, sod(47^2)=13, sod(53^2)=19, sod(59^5)=47, sod(61^2)=13, sod(67^3)=19, sod(71^6)=37, sod(73^2)=19, sod(79^2)=13, sod(83^2)=31, sod(89^2)=19, sod(97^4)=43, sod(101^8)=67, sod(103^4)=31, sod(107^2)=19, sod(109^2)=19, sod(113^4)=31, sod(127^2)=19, sod(131^8)=61, sod(137^2)=31, sod(139^3)=37, sod(149^2)=7, sod(151^2)=13, sod(157^4)=31, sod(163^4)=37, sod(167^6)=73, sod(173^2)=31, sod(179^4)=37, sod(181^2)=19, sod(191^10)=97, sod(193^3)=37, sod(197^4)=37, sod(199^2)=19, sod(211^3)=37, sod(223^2)=31, sod(227^4)=43, sod(229^3)=37.
%o A172035 (Magma) S:=[ 5, 0 ]; for n in [3..103] do j:=2; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S; // _Klaus Brockhaus_, Jan 29 2010
%Y A172035 Cf. A046704, A007605
%Y A172035 Cf. A172216. - _Klaus Brockhaus_, Jan 29 2010
%K A172035 easy,nonn,base
%O A172035 1,1
%A A172035 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 23 2010
%E A172035 More terms from _Klaus Brockhaus_, Jan 29 2010
%E A172035 Edited by _Charles R Greathouse IV_, Aug 02 2010