A364961 -id:A364961 - cp's OEIS frontend

A364959 Odd numbers k such that A348717(k) = A348717(A005940(k)).

1, 3, 5, 17, 25, 45, 49, 133, 257, 65537

Offset: 1

Antti Karttunen, Sep 02 2023

In contrast to condition A348717(n) = A348717(A163511(n)), which seems to admit only Mersenne primes for odd n (see A364297), here we also have some extra terms in addition to Fermat primes, A019434.

			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.)

Cf. A005940, A019434 (subsequence), A348717, A364961, A364962.

Cf. also A364297.

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))));

A364960 Numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 25, 32, 40, 45, 48, 49, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40131, 40960, 49152, 50575, 65536, 79625, 81920, 98304, 131072, 163840, 196608, 262144, 327680

Offset: 1

Antti Karttunen, Aug 14 2023

nonn

Numbers k such that A252464(k) is equal to A364569(k).

Positions of 0's in A364570.
Cf. A252464, A364569.
Subsequences: A029747, A364961 (odd terms).

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));
isA364960(n) = { my(k=A005940(n)); while(k>n, k = A252463(k)); (k==n); };

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));
isA364960(n) = (A252464(n)==A364569(n));

A364962 Odd numbers k such that A005941(k) is either k itself or its descendant in Doudna-tree, A005940.

Original entry on oeis.org

1, 3, 5, 11, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 127

Offset: 1

Antti Karttunen, Aug 14 2023

Questions: Is 85 the only composite in this sequence? (See also A364565). Are there any more terms after 127, or is the sequence finite?

			85 = 5*17 is a term, because A005941(85) = 133 = 7*19 = A003961(85), thus 133 is a left hand side child of 85 in the tree depicted in A005940, and therefore 85 is included in this sequence. (See also the last example in A364959).

Cf. A005940, A005941, A029747, A156552, A252464.

Cf. also A364959, A364961.

A005941(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]); (1+res) }; \\ (After David A. Corneth's program for A156552)
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));
isA364962(n) = if(!(n%2),0,my(k=A005941(n)); while(k>n, k = A252463(k)); (k==n));

cp's OEIS Frontend

A364959 Odd numbers k such that A348717(k) = A348717(A005940(k)).

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A364960 Numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.

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A364962 Odd numbers k such that A005941(k) is either k itself or its descendant in Doudna-tree, A005940.

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PARI