A340149 -id:A340149 - cp's OEIS frontend

A340150 Positions of ones in A340149.

1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 26, 29, 31, 37, 39, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 78, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 138, 139, 149, 151, 157, 163, 167, 173, 179, 181, 182, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 259, 263, 269, 271, 277, 281, 283

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Differs from its subsequence A340076 for the first time at n=98, where occurs the first term 445, which is not present in A340076. See A340151.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A340076, A340149, A340151 (for terms not present in A340076).

isA340150(n) = (1==A340149(n)); \\ Needs code from A340149 also.

A340075 The odd part of A340072(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 5, 3, 1, 3, 1, 5, 3, 27, 1, 5, 1, 9, 5, 3, 1, 9, 7, 1, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 1, 27, 1, 5, 1, 9, 5, 7, 1, 27, 11, 21, 9, 3, 1, 25, 1, 45, 11, 15, 1, 9, 1, 9, 25, 243, 3, 3, 1, 27, 7, 3, 1, 45, 1, 5, 21, 33, 15, 1, 1, 81, 125, 21, 1, 15, 9, 23, 15, 27, 1, 15, 5, 21, 9, 13, 33, 81

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Each term a(n) is a multiple of A340149(n), therefore, as both sequences have only positive terms, it follows that if a(n) = 1 then A340149(n) = 1 also, but not necessarily vice versa.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A019565, A339901, A339904, A340072, A340074, A340076 (positions of ones), A340149 (differs from the first time at n=85).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };
A340075(n) = A000265(A340072(n));

a(n) = A000265(A340072(n)).
a(n) = A339904(n) / A340074(n) = A339904(n) / gcd(A003961(n)-1, A339904(n)).
For all n >= 0, a(A019565(n)) = A339901(n).

A340091 Odd numbers k such that A064989(k) is in A340151.

Original entry on oeis.org

679, 703, 1387, 1729, 1891, 2047, 2509, 2701, 2821, 3277, 3367, 5551, 7471, 7735, 8119, 8827, 9997, 10963, 11305, 12403, 13021, 13747, 13981, 14491, 14701, 15841, 16471, 17563, 19951, 21349, 21907, 21931, 22015, 23959, 24727, 25669, 26281, 27511, 28939, 29341, 31417, 32407, 38503, 39091, 39831, 39865, 40501, 41041

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Sequence A003961(A340151(i)), for i >= 1, sorted into ascending order.

By definition, this has no common terms with A340077 nor any of its subsequences like A339869 or A339880.

Antti Karttunen, Table of n, a(n) for n = 1..3290

Cf. A002997, A003961, A339869, A339880, A340077, A340151.

Cf. A340092 (Carmichael numbers in this sequence).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339904(n) = A000265(eulerphi(A003961(n)));
A340075(n) = { my(u=A339904(n)); u/gcd(A003961(n)-1, u); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340149(n) = A000265(A247074(A003961(n)));
isA340151(n) = ((1!=A340075(n))&&(1==A340149(n)));
A064989(n) = { my(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) };
isA340091(n) = ((n%2)&&isA340151(A064989(n)));

A340147 a(n) = A247074(A003961(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 9, 5, 3, 1, 3, 1, 5, 6, 27, 1, 10, 1, 9, 10, 6, 1, 18, 7, 8, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 4, 27, 1, 5, 1, 9, 10, 14, 1, 27, 11, 21, 18, 6, 1, 50, 2, 45, 22, 15, 1, 18, 1, 18, 50, 243, 24, 12, 1, 27, 7, 3, 1, 90, 1, 20, 21, 33, 30, 16, 1, 81, 125, 21, 1, 30, 3, 23, 30, 54, 1, 15, 40, 21

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Prime shifted analog of A247074.

Each term a(n) is a divisor of A340072(n).

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A003972, A247074, A340072, A340148, A340149 (odd part).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340147(n) = A247074(A003961(n));

a(n) = A247074(A003961(n)).

a(n) = A003972(n) / A340148(n).

A340151 Setwise difference A340150 \ A340076.

Original entry on oeis.org

445, 527, 935, 1207, 1577, 1595, 1705, 1711, 2101, 2145, 2201, 2507, 3245, 3315, 4895, 6045, 6631, 6931, 7511, 8371, 9707, 9845, 10189, 10295, 10505, 11023, 11895, 12137, 12194, 13145, 13571, 13845, 14101, 14245, 15042, 15281, 16235, 16643, 17355, 17701, 18559, 19567, 20119, 20865, 22703, 23347, 25123, 26581, 27101, 27695

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Positions k where A340149(k) = 1, but A340075(k) > 1.

Setwise difference A340150 \ A340076.
Cf. A340091 (gives the same terms prime shifted once and sorted into ascending order).
Cf. A340075, A340149.

isA340151(n) = ((1!=A340075(n))&&(1==A340149(n))); \\ Uses code from both A340075 and A340149

cp's OEIS Frontend

A340150 Positions of ones in A340149.

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A340075 The odd part of A340072(n).

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A340091 Odd numbers k such that A064989(k) is in A340151.

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A340147 a(n) = A247074(A003961(n)).

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A340151 Setwise difference A340150 \ A340076.

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