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%I A364961 #22 Sep 03 2023 08:43:26
%S A364961 1,3,5,25,45,49,40131,50575,79625,1486485,1872507,3403125
%N A364961 Odd numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.
%C A364961 Odd numbers k such that A252464(k) is equal to A364569(k).
%C A364961 Apparently, A364960 without the even terms of A029747.
%C A364961 Note that 1, 25, 45, 49 are so far the only known integers x for which A005940(x) = A003961(x).
%e A364961 Term (and its factorization) A005940(term) (and its factorization)
%e A364961 1 -> 1
%e A364961 3 -> 3
%e A364961 5 -> 5
%e A364961 25 = 5^2 -> 49 = 7^2
%e A364961 45 = 3^2 * 5 -> 175 = 5^2 * 7
%e A364961 49 = 7^2 -> 121 = 11^2
%e A364961 40131 = 3^2 * 7^3 * 13 -> 100847877 = 3 * 13^2 * 19^3 * 29
%e A364961 50575 = 5^2 * 7 * 17^2 -> 22467159 = 3^3 * 11^2 * 13 * 23^2
%e A364961 79625 = 5^3 * 7^2 * 13 -> 787365187 = 7 * 19^3 * 23^2 * 31
%e A364961 1486485 = 3^3 * 5 * 7 * 11^2 * 13 -> 25468143451205
%e A364961 = 5 * 7 * 13 * 17^3 * 19 * 23 * 29^2 * 31
%e A364961 1872507 = 3 * 7 * 13 * 19^3 -> 240245795625
%e A364961 = 3 * 5^4 * 11 * 17 * 23 * 31^3,
%e A364961 3403125 = 3^2 * 5^5 * 11^2 -> 2394659631669305
%e A364961 = 5 * 7^3 * 11 * 13^2 * 17^5 * 23^2.
%e A364961 See also examples in A364959.
%o A364961 (PARI)
%o A364961 Abincompreflen(n, m) = { my(x=binary(n), y=binary(m), u=min(#x, #y)); for(i=1, u, if(x[i]!=y[i], return(i-1))); (u); };
%o A364961 A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
%o A364961 A364569(n) = Abincompreflen(A156552(n), (n-1));
%o A364961 A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
%o A364961 A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
%o A364961 isA364961(n) = ((n%2)&&(A252464(n)==A364569(n)));
%o A364961 (PARI)
%o A364961 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A364961 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A364961 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A364961 isA364961(n) = if(!(n%2),0,my(k=A005940(n)); while(k>n, k = A252463(k)); (k==n));
%Y A364961 Odd terms in A364960.
%Y A364961 Cf. A003961, A005940, A029747, A156552, A252464, A364569, A364570.
%Y A364961 Cf. also A364956 (odd terms there), A364959, A364962.
%K A364961 nonn,hard,more
%O A364961 1,2
%A A364961 _Antti Karttunen_, Aug 14 2023