Original entry on oeis.org
1, 3, 5, 17, 25, 45, 49, 133, 257, 65537
Offset: 1
For n = 17, A005940(17) = 11, and A348717(11) = A348717(17) = 2, therefore 17 is included in this sequence. Moreover, any prime in this sequence must be one of the Fermat primes (A019434), because the primes are located at positions 2^k + 1 in the offset-1 variant of Doudna-tree, A005940.
For n = 25 and n = 49, A005940(25) = 49 and A005940(49) = 121, which are all squares of primes, and thus have the same A348717-value (4), therefore both 25 and 49 are terms.
For n = 45 = 3^2 * 5, A005940(45) = 175 = 5^2 * 7 [= A003961(45)], with A348717(45) = A348717(175) = 12, therefore 45 is included as a term. (See also examples in A364961).
For n = 133 = 7*19, A005940(133) = 85 = 5*17 [= A064989(133)], with A348717(133) = A348717(85) = 22, therefore 133 is included as a term. (See also example in A364962.)
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A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
isA364959(n) = ((n%2)&&(A348717(n)==A348717(A005940(n))));
A364961
Odd numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.
Original entry on oeis.org
1, 3, 5, 25, 45, 49, 40131, 50575, 79625, 1486485, 1872507, 3403125
Offset: 1
Term (and its factorization) A005940(term) (and its factorization)
1 -> 1
3 -> 3
5 -> 5
25 = 5^2 -> 49 = 7^2
45 = 3^2 * 5 -> 175 = 5^2 * 7
49 = 7^2 -> 121 = 11^2
40131 = 3^2 * 7^3 * 13 -> 100847877 = 3 * 13^2 * 19^3 * 29
50575 = 5^2 * 7 * 17^2 -> 22467159 = 3^3 * 11^2 * 13 * 23^2
79625 = 5^3 * 7^2 * 13 -> 787365187 = 7 * 19^3 * 23^2 * 31
1486485 = 3^3 * 5 * 7 * 11^2 * 13 -> 25468143451205
= 5 * 7 * 13 * 17^3 * 19 * 23 * 29^2 * 31
1872507 = 3 * 7 * 13 * 19^3 -> 240245795625
= 3 * 5^4 * 11 * 17 * 23 * 31^3,
3403125 = 3^2 * 5^5 * 11^2 -> 2394659631669305
= 5 * 7^3 * 11 * 13^2 * 17^5 * 23^2.
See also examples in A364959.
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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); };
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
A364569(n) = Abincompreflen(A156552(n), (n-1));
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
isA364961(n) = ((n%2)&&(A252464(n)==A364569(n)));
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A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA364961(n) = if(!(n%2),0,my(k=A005940(n)); while(k>n, k = A252463(k)); (k==n));
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