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A302755 Strongly prime-additive numbers: numbers n with at least 2 distinct prime factors that can be represented as n = Sum_{p|n} p^e_p, with e_p > 0 and p^e_p < n < p^(e_p+1). 2

%I #12 Aug 23 2018 17:29:00

%S 228,3115,190233,1090918,11352818,32647591,98437335,670402467,

%T 809609110,869424040,1008317892,8911808588,37104085671,243006777356,

%U 297252832082

%N Strongly prime-additive numbers: numbers n with at least 2 distinct prime factors that can be represented as n = Sum_{p|n} p^e_p, with e_p > 0 and p^e_p < n < p^(e_p+1).

%C The first 3 terms were given in the paper by Erdős & Hegyvári. They were found by P. Massias.

%C Subsequence of A302753.

%C a(16) > 5.4*10^11. - _Giovanni Resta_, Aug 23 2018

%H Paul Erdős and Norbert Hegyvári, <a href="http://real-j.mtak.hu/5469/1/StudScientMath_27.pdf">On prime-additive numbers</a>, Studia Sci. Math. Hungar., Vol. 27, No. 1-2 (1992), pp. 207-212. <a href="https://www.emis.de/classics/Erdos/cit/68810043.htm">Review</a>.

%e 228 = 2^2 * 3 * 19 = 2^7 + 3^4 + 19.

%e 3115 = 5 * 7 * 89 = 5^4 + 7^4 + 89.

%e 190233 = 3^2 * 23 * 919 = 3^11 + 23^3 + 919.

%e 1090918 = 2 * 199 * 2741 = 2^20 + 199^2 + 2741.

%e 11352818 = 2 * 41 * 138449 = 2^23 + 41 ^4 + 138449.

%e 32647591 = 29 * 59 * 19081 = 29^5 + 59^4 + 19081.

%e 98437335 = 3 * 5 * 6562489 = 3^16 + 5^11 + 6562489.

%e 670402467 = 3^3 * 7^2 * 506729 = 3^18 + 7^10 + 506729.

%e 809609110 = 2 * 5 * 211 * 257 * 1493 = 2^29 + 5^12 + 211^3 + 257^3 + 1493^2.

%e 869424040 = 2^3 * 5 * 19 * 197 * 5807 = 2^29 + 5^12 + 19^6 + 197^3 + 5807^2.

%e 1008317892 = 2^2 * 3 * 84026491 = 2^29 + 3^18 + 84026491.

%e 8911808588 = 2^2 * 683 * 3262009 = 2^33 + 683^3 + 3262009.

%e 37104085671 = 3 * 89 * 138966613 = 3^22 + 89^5 + 138966613.

%e 243006777356 = 2^2 * 7 * 8678813477 = 2^37 + 7^13 + 8678813477.

%e 297252832082 = 2 * 29 * 5125048829 = 2^38 + 29^7 + 5125048829.

%t p[n_] := First[Transpose[FactorInteger[n]]]; powerMax[p_, n_] :=

%t Module[{k = 0, nn = n}, While[nn > 1, nn /= p; k++]; p^(k - 1)]; a[n_] := Module[{primes = p[n]}, np = Length[primes]; s = 0; If[np > 1, Do[s += powerMax[primes[[k]], n], {k, 1, np}]]; s]; aQ[n_] := a[n] == n; seq={}; Do[If[aQ[n], AppendTo[seq, n]], {n,2,100000}]; seq

%Y Cf. A296717, A302753.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Apr 12 2018

%E a(9)-a(15) from _Giovanni Resta_, Aug 23 2018

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)