A233999 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 7 raised to an odd power.

1, 2, 4, 8, 9, 11, 15, 16, 18, 22, 23, 25, 29, 30, 32, 36, 37, 39, 43, 44, 46, 49, 50, 51, 53, 57, 58, 60, 64, 65, 67, 71, 72, 74, 78, 79, 81, 85, 86, 88, 92, 93, 95, 98, 99, 100, 102, 106, 107, 109, 113, 114, 116, 120, 121, 123, 127, 128, 130, 134, 135, 137, 141, 142, 144, 148, 149

Offset

1,2

Comments

Equivalently, numbers of the form 49^n*(7m+1), 49^n*(7m+2), or 49^n*(7m+4). [Corrected by Charles R Greathouse IV, Jan 12 2017]

From Peter Munn, Feb 08 2024: (Start)

Numbers whose squarefree part is congruent to a (nonzero) quadratic residue modulo 7.

The integers in a subgroup of the positive rationals under multiplication. As such the sequence is closed under multiplication and - where the result is an integer - under division. The subgroup has index 4 and is generated by the primes congruent to a quadratic residue (1, 2 or 4) modulo 7, the square of 7, and 3 times the other primes; that is a generator corresponding to each prime: 2, 3*3, 3*5, 7^2, 11, 3*13, 3*17, 3*19, 23, 29, 3*31, ... .

(End)

Links

Table of n, a(n) for n=1..67.

Eric Weisstein's World of Mathematics, Squarefree Part.

Formula

a(n) = 16n/7 + O(log n). - Charles R Greathouse IV, Jan 12 2017

Prog

(PARI) is(n)=n/=49^valuation(n, 49); n%7==1||n%7==2||n%7==4 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013
(PARI) is_A233999(n)=bittest(22, n/49^valuation(n, 49)%7) \\ - M. F. Hasler, Jan 02 2014
(PARI) list(lim)=my(v=List(), t, u); forstep(k=1, lim\=1, [1, 2, 4], listput(v, k)); for(e=1, logint(lim, 49), u=49^e; for(i=1, #v, t=u*v[i]; if(t>lim, break); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Jan 12 2017

Crossrefs

Cf. A055046, A233998.

Numbers whose squarefree part is congruent to a coprime quadratic residue modulo k: A003159 (k=2), A055047 (k=3), A277549 (k=4), A352272 (k=6), A234000 (k=8), A334832 (k=24).

First differs from A047350 by including 49.

Sequence in context: A034036 A010410 A010459 * A047350 A319742 A190723

Adjacent sequences: A233996 A233997 A233998 * A234000 A234001 A234002

Keyword

nonn,easy

Author

V. Raman, Dec 18 2013

Status

approved