A339815 -id:A339815 - cp's OEIS frontend

A339822 The exponent of the highest power of 2 dividing A339821(n).

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

Offset: 0

Antti Karttunen, Dec 18 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, A003972, A007814, A019565, A023506, A053574, A339814, A339815, A339821.

A339822(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += valuation((p-1),2)); n >>= 1); (s); };

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A023506(e1) + A023506(e2) + ... + A023506(ek).
a(n) = A007814(A339821(n)) = A053574(A019565(2n)).
a(n) = A007814(A003972(A019565(n))) = A007814(A000010(A019565(2n))).

A339814 The exponent of the highest power of 2 dividing (A019565(2n) - 1).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 1, 5, 1, 2, 2, 1, 7, 1, 2, 1, 6, 1, 1, 4, 1, 2, 1, 2, 1, 5, 3, 1, 2, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 1, 4, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 3, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 6, 1, 2, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 5, 1

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

The 2-adic valuation of A339809(2n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Bisection of A339813.

Cf. A003961, A007814, A019565, A339809, A339815, A339822.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339814(n) = valuation((A019565(2*n)-1),2);

a(n) = A007814(A339809(2*n)) = A007814(A019565(2*n)-1).

a(n) = A007814(A003961(A019565(n))-1).

A339816 Numbers k for which A339814(k) >= A339822(k).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 12, 14, 16, 18, 21, 32, 64, 65, 68, 72, 84, 128, 129, 132, 136, 138, 141, 145, 159, 170, 192, 204, 208, 256, 258, 261, 324, 385, 448, 462, 512, 513, 515, 516, 520, 536, 576, 578, 581, 640, 705, 723, 776, 908, 912, 1024, 1036, 1040, 1049, 1160, 1172, 1280, 1352, 1537, 1600, 1609, 1666, 1732, 1795, 2048

Offset: 1

Antti Karttunen, Dec 19 2020

nonn

Terms of (1/2)*A048675(A339817(i)), for i = 2.., sorted into ascending order.

First occurrences of terms with binary weight (A000120) w = 1..19 are at n=1, 4, 8, 23, 50, 25, 125, 136, 176, 502, 749, 1142, 791, 1882, 2913, 4327, 17979, 16991, 12441. The terms themselves are: 1, 5, 14, 141, 908, 159, 8921, 9948, 18390, 175449, 400237, 1223389, 441805, 3234271, 28743379, 53892047, 1631024969, 1331412056, 725475951.

Antti Karttunen, Table of n, a(n) for n = 1..19468

Positions of zeros and negative terms in A339815.
Cf. A000010, A000120, A007814, A019565, A048675, A339814, A339817, A339822.
Cf. A000079 (a subsequence).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339816(n) = { my(x=A019565(2*n)); (valuation(eulerphi(x),2)<=valuation(x-1,2)); };

cp's OEIS Frontend

A339822 The exponent of the highest power of 2 dividing A339821(n).

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A339814 The exponent of the highest power of 2 dividing (A019565(2n) - 1).

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A339816 Numbers k for which A339814(k) >= A339822(k).

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