A339906 -id:A339906 - cp's OEIS frontend

A339907 Odd squarefree numbers k > 1 for which the bigomega(phi(k)) <= bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 33, 37, 41, 43, 47, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 129, 131, 137, 139, 141, 145, 149, 151, 157, 161, 163, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201, 209, 211, 217, 223, 227, 229, 233, 235, 239, 241, 249, 251, 253, 257

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

Terms of A003961(A019565(A339906(i))) [or equally, of A019565(2*A339906(i))], for i = 1.., sorted into ascending order.

Natural numbers n > 2 that satisfy equation k * phi(n) = n - 1 (for some integer k) all occur in this sequence. Lehmer conjectured that there are no composite solutions.

Antti Karttunen, Table of n, a(n) for n = 1..18526
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.

Cf. A339906.
Cf. A065091, A339908 (subsequences).
Cf. also A339817.
Apart from initial 3, a subsequence of A339910.

isA339907(n) = ((n>1)&&(n%2)&&issquarefree(n)&&(bigomega(eulerphi(n))<=bigomega(n-1)));

A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 16, 20, 32, 33, 34, 35, 38, 41, 50, 56, 64, 128, 176, 256, 259, 290, 512, 1024, 2048, 2056, 2081, 2089, 2096, 2180, 4096, 4130, 8192, 9218, 16384, 18436, 32768, 65536, 131072, 131140, 262144, 279552, 524288, 524308, 524546, 1048576, 1048736, 2097152, 4194304, 4194352, 4194420, 4196656, 4202499, 8388608

Offset: 1

Antti Karttunen, Dec 26 2020

nonn

Numbers k such that A339971(k) divides A339809(2k).

Union of {0}, A000079 and the terms of (1/2)*A048675(A339880(i)), for i >= 1, sorted into ascending order.

Antti Karttunen, Table of n, a(n) for n = 1..94

Positions of zeros in A339898, and of ones in A339901.
Cf. A000010, A000265, A048675, A339821, A339880, A339809, A339971, A339974.
Cf. A000079 (subsequence).
Cf. also A339816, A339906.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339971(n) = { my(x=A019565(2*n)); !((x-1)%A000265(eulerphi(x))); };

A339902 Number of prime divisors of A339821(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 5, 6, 7, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 4, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 7, 8, 9, 10, 9, 10, 11, 12, 9, 10, 11, 12, 11, 12, 13, 14, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 6, 7, 8, 9, 8, 9, 10, 11, 8, 9, 10, 11, 10, 11, 12, 13, 7

Offset: 0

Antti Karttunen, Dec 21 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000010, A001222, A003972, A019565, A023508, A339821, A339822, A339906.

A339902(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += bigomega(p-1)); n >>= 1); (s); };

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A023508(e1) + A023508(e2) + ... + A023508(ek).
a(n) = A001222(A339821(n)).
a(n) = A001222(A003972(A019565(n))) = A001222(A000010(A019565(2n))).
a(n) >= A339822(n).

cp's OEIS Frontend

A339907 Odd squarefree numbers k > 1 for which the bigomega(phi(k)) <= bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

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A339902 Number of prime divisors of A339821(n), counted with multiplicity.

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