A339812 -id:A339812 - cp's OEIS frontend

A339906 Numbers k for which A339812(2k) >= A339902(k).

1, 2, 4, 5, 8, 9, 10, 14, 16, 18, 32, 64, 65, 72, 84, 128, 129, 132, 136, 141, 145, 170, 256, 258, 261, 385, 448, 512, 516, 578, 642, 912, 1024, 1040, 1049, 1160, 1352, 2048, 4096, 4097, 4100, 4111, 4160, 4652, 4675, 4864, 5124, 5280, 8192, 8193, 8194, 8195, 8196, 8200, 8214, 8216, 8258, 8320, 8329, 8468, 8704

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

Terms of (1/2)*A048675(A339907(i)), for i >= 1, sorted into ascending order.
The first term not present in A339816 is 10, the second is 642; the first term of A339816 not present here is 12, the second is 21.
First terms with binary weights (A000120) w = 1..9 are: 1, 5, 14, 141, 4111, 25676, 41674, 1094530, 423297.

			10 ("1010" in binary) is present, because it encodes an odd squarefree number 5*11, for which phi(55) = 4*10 = 40, and bigomega(55-1) = 4 >= 4 = bigomega(40).
12 ("1100" in binary) is NOT present, because it encodes an odd squarefree number 7*11, for which phi(77) = 6*10 = 60, and bigomega(77-1) = 3 < 4 = bigomega(60).

Antti Karttunen, Table of n, a(n) for n = 1..514

Cf. A000010, A001222, A019565, A048675, A339812, A339902, A339907.
Cf. A000079 (a subsequence).
Cf. also A339816.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339812(n) = bigomega(A019565(n)-1);
A339902(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += bigomega(p-1)); n >>= 1); (s); };
isA339906(n) = (A339812(2*n) >= A339902(n));

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339906(n) = { my(x=A019565(2*n)); (bigomega(eulerphi(x))<=bigomega(x-1)); };

A339809 a(n) = A019565(n) - 1.

Original entry on oeis.org

0, 1, 2, 5, 4, 9, 14, 29, 6, 13, 20, 41, 34, 69, 104, 209, 10, 21, 32, 65, 54, 109, 164, 329, 76, 153, 230, 461, 384, 769, 1154, 2309, 12, 25, 38, 77, 64, 129, 194, 389, 90, 181, 272, 545, 454, 909, 1364, 2729, 142, 285, 428, 857, 714, 1429, 2144, 4289, 1000, 2001, 3002, 6005, 5004, 10009, 15014, 30029, 16, 33, 50

Offset: 0

Antti Karttunen, Dec 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A019565, A339810, A339811, A339812.

Cf. also A339820, A339821.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339809(n) = (A019565(n)-1);

A339810 a(n) = A046523(A019565(n) - 1).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 2, 6, 2, 12, 2, 6, 6, 24, 6, 6, 6, 32, 6, 24, 2, 12, 6, 12, 12, 30, 2, 384, 2, 6, 2, 12, 4, 6, 6, 64, 6, 6, 2, 60, 2, 48, 6, 6, 12, 60, 2, 6, 30, 12, 2, 210, 2, 96, 2, 216, 30, 30, 6, 180, 2, 6, 2, 16, 6, 12, 2, 60, 4, 6, 2, 6, 6, 12, 6, 120, 6, 24, 6, 30, 2, 240, 6, 6, 30, 12, 6, 60, 2, 30, 2, 48

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A019565, A046523, A339809, A339811 (rgs-transform), A339812.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A339810(n) = A046523(A019565(n)-1);

a(n) = A046523(A339809(n)) = A046523(A019565(n) - 1).

A339811 Lexicographically earliest infinite sequence such that a(i) = a(j) => A339810(i) = A339810(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 2, 5, 2, 4, 4, 6, 4, 4, 4, 7, 4, 6, 2, 5, 4, 5, 5, 8, 2, 9, 2, 4, 2, 5, 3, 4, 4, 10, 4, 4, 2, 11, 2, 12, 4, 4, 5, 11, 2, 4, 8, 5, 2, 13, 2, 14, 2, 15, 8, 8, 4, 16, 2, 4, 2, 17, 4, 5, 2, 11, 3, 4, 2, 4, 4, 5, 4, 18, 4, 6, 4, 8, 2, 19, 4, 4, 8, 5, 4, 11, 2, 8, 2, 12, 4, 4, 4, 11, 20, 4, 5, 19, 3, 4, 4, 4, 4

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

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

Cf. A019565, A046523, A339809, A339810, A339812.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A339810(n) = A046523(A019565(n)-1);
v339811 = rgs_transform(vector(up_to, n, A339810(n)));
A339811(n) = v339811[n];

cp's OEIS Frontend

A339906 Numbers k for which A339812(2k) >= A339902(k).

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A339809 a(n) = A019565(n) - 1.

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A339810 a(n) = A046523(A019565(n) - 1).

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A339811 Lexicographically earliest infinite sequence such that a(i) = a(j) => A339810(i) = A339810(j), for all i, j >= 1.

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