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Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.
+10
13
1, 2, 3, 4, 5, 9, 8, 6, 18, 15, 16, 7, 25, 21, 81, 32, 10, 36, 27, 162, 45, 64, 11, 49, 30, 324, 63, 729, 128, 12, 50, 33, 625, 75, 1458, 105, 256, 13, 72, 35, 648, 90, 2916, 135, 225, 512, 14, 98, 39, 1250, 99, 5832, 165, 441, 405, 1024, 17, 100, 42, 1296, 117, 11664, 189, 450, 567, 59049, 2048, 19, 121, 51, 2401, 126, 15625
COMMENTS
T(n,k) is the n-th positive integer with exactly k odd divisors.
This is a permutation of the natural numbers.
T(n,k) is also the n-th number j with the property that the symmetric representation of sigma(j) has k subparts (cf. A279387). - Omar E. Pol, Dec 27 2016 T(n,k) is also the n-th positive integer with exactly k partitions into consecutive parts. - Omar E. Pol, Aug 16 2018
EXAMPLE
The corner of the square array begins:
1, 3, 9, 15, 81, 45, 729, 105, 225, 405, ...
2, 5, 18, 21, 162, 63, 1458, 135, 441, 567, ...
4, 6, 25, 27, 324, 75, 2916, 165, 450, 810, ...
8, 7, 36, 30, 625, 90, 5832, 189, 882, 891, ...
16, 10, 49, 33, 648, 99, 11664, 195, 900, 1053, ...
32, 11, 50, 35, 1250, 117, 15625, 210, 1089, 1134, ...
64, 12, 72, 39, 1296, 126, 23328, 231, 1225, 1377, ...
128, 13, 98, 42, 2401, 147, 31250, 255, 1521, 1539, ...
...
CROSSREFS
Cf. A001227, A182469, A236104, A237591, A237593, A240062, A261697, A261698, A261699, A279387, A286000, A286001, A296508.
Positive integers that have exactly 6 odd divisors.
+10
8
45, 63, 75, 90, 99, 117, 126, 147, 150, 153, 171, 175, 180, 198, 207, 234, 243, 245, 252, 261, 275, 279, 294, 300, 306, 325, 333, 342, 350, 360, 363, 369, 387, 396, 414, 423, 425, 468, 475, 477, 486, 490
COMMENTS
Numbers that can be formed in exactly 5 ways by summing sequences of 2 or more consecutive integers.
Numbers n such that the symmetric representation of sigma(n) has 6 subparts. - Omar E. Pol, Dec 28 2016
MATHEMATICA
Select[Range[500], Count[Divisors[#], _?OddQ]==6&] (* Harvey P. Dale, Mar 25 2015 *)
Numbers with 5 odd divisors.
+10
8
81, 162, 324, 625, 648, 1250, 1296, 2401, 2500, 2592, 4802, 5000, 5184, 9604, 10000, 10368, 14641, 19208, 20000, 20736, 28561, 29282, 38416, 40000, 41472, 57122, 58564, 76832, 80000, 82944, 83521, 114244, 117128, 130321, 153664, 160000, 165888, 167042, 228488, 234256, 260642, 279841
COMMENTS
Positive integers that have exactly five odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 5 subparts. - Omar E. Pol, Dec 28 2016 Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 4 ways; e.g., 81 = 40+41 = 26+27+28 = 11+12+13+14+15+16 = 5+6+7+8+9+10+11+12+13. - Julie Jones, Aug 13 2018
FORMULA
Sum_{n>=1} 1/a(n) = 2 * P(4) - 1/8 = 0.00289017370127..., where P(4) is the value of the prime zeta function at 4 ( A085964). - Amiram Eldar, Sep 16 2024
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 5; \\ Michel Marcus, Apr 03 2016 (GAP) A:=List([1..700000], n->DivisorsInt(n));;
B:=List([1..Length(A)], i->Filtered(A[i], IsOddInt));;
a:=Filtered([1..Length(B)], i->Length(B[i])=5); # Muniru A Asiru, Aug 14 2018
Numbers with 7 odd divisors.
+10
8
729, 1458, 2916, 5832, 11664, 15625, 23328, 31250, 46656, 62500, 93312, 117649, 125000, 186624, 235298, 250000, 373248, 470596, 500000, 746496, 941192, 1000000, 1492992, 1771561, 1882384, 2000000, 2985984, 3543122, 3764768, 4000000, 4826809, 5971968, 7086244, 7529536, 8000000, 9653618
COMMENTS
Positive integers that have exactly seven odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - Omar E. Pol, Dec 28 2016 Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018 Numbers of the form p^6 * 2^k where p is an odd prime. - David A. Corneth, Aug 14 2018
FORMULA
Sum_{n>=1} 1/a(n) = 2 * P(6) - 1/32 = 0.00289017370127..., where P(6) is the value of the prime zeta function at 6 ( A085966). - Amiram Eldar, Sep 16 2024
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016 (PARI) upto(n) = {my(res = List()); forprime(p = 3, sqrtnint(n, 6), listput(res, p^6)); q = #res; for(i = 1, q, odd = res[i]; for(j = 1, logint(n \ odd, 2), listput(res, odd <<= 1))); listsort(res); res} \\ David A. Corneth, Aug 14 2018 (Python)
from sympy import integer_log, primerange, integer_nthroot
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x-sum(integer_log(x//p**6, 2)[0]+1 for p in primerange(3, integer_nthroot(x, 6)[0]+1)))
Numbers with 8 odd divisors.
+10
8
105, 135, 165, 189, 195, 210, 231, 255, 270, 273, 285, 297, 330, 345, 351, 357, 375, 378, 385, 390, 399, 420, 429, 435, 455, 459, 462, 465, 483, 510, 513, 540, 546, 555, 561, 570, 594, 595, 609, 615, 621, 627, 645, 651, 660, 663, 665, 690, 702, 705, 714, 715, 741, 750, 756, 759, 770, 777, 780, 783, 795, 798, 805, 837
COMMENTS
Positive integers that have exactly eight odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 8 subparts. - Omar E. Pol, Dec 29 2016 Numbers that can be formed in exactly 7 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018
MAPLE
filter:= proc(n) local r;
r:= n/2^padic:-ordp(n, 2);
numtheory:-tau(r)=8
end proc:
MATHEMATICA
Select[Range@ 840, Length@ Select[Divisors@ #, OddQ] == 8 &] (* Michael De Vlieger, Dec 30 2016 *)
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 8; \\ after Michel Marcus(Magma) [n: n in [1..1000] | #[d: d in Divisors(n) | IsOdd(d)] eq 8]; // Bruno Berselli, Apr 04 2016
Numbers with 9 odd divisors.
+10
8
225, 441, 450, 882, 900, 1089, 1225, 1521, 1764, 1800, 2178, 2450, 2601, 3025, 3042, 3249, 3528, 3600, 4225, 4356, 4761, 4900, 5202, 5929, 6050, 6084, 6498, 6561, 7056, 7200, 7225, 7569, 8281, 8450, 8649, 8712, 9025, 9522, 9800, 10404, 11858, 12100, 12168, 12321, 12996, 13122, 13225, 14112, 14161, 14400, 14450, 15129
COMMENTS
Positive integers that have exactly nine odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 9 subparts. - Omar E. Pol, Dec 29 2016 Numbers of the form 2^j*p^8 or 2^j*p^2*q^2 where p and q are distinct odd primes. (End)
Numbers that can be formed in exactly 8 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018
FORMULA
Sum_{n>=1} 1/a(n) = (P(2)-1/4)^2 - P(4) + 2*P(8) + 7/128 = 0.026721189882055998428..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 16 2024
MAPLE
N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i, i=3..floor(sqrt(N)/2), 2)]);
Aodd:= select(`<=`, map(t -> t^8, P) union {seq(seq(P[i]^2*P[j]^2, i=1..j-1), j=1..nops(P))}, N):
A:= map(t -> seq(2^j*t, j=0..ilog2(N/t)), Aodd):
MATHEMATICA
Select[Range[5^6], Length[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] == 9 &] (* Michael De Vlieger, Apr 04 2016 *) Select[Range[16000], Total[Boole[OddQ[Divisors[#]]]]==9&] (* Harvey P. Dale, May 12 2019 *)
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 9; \\ after Michel Marcus. (GAP) A:=List([1..16000], n->DivisorsInt(n));; B:=List([1..Length(A)], i->Filtered(A[i], IsOddInt));;
a:=Filtered([1..Length(B)], i->Length(B[i])=9); # Muniru A Asiru, Aug 14 2018
Numbers whose number of odd divisors is nonprime.
+10
2
1, 2, 4, 8, 15, 16, 21, 27, 30, 32, 33, 35, 39, 42, 45, 51, 54, 55, 57, 60, 63, 64, 65, 66, 69, 70, 75, 77, 78, 84, 85, 87, 90, 91, 93, 95, 99, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 123, 125, 126, 128, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150
EXAMPLE
The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. The odd divisors of 42 are 1, 3, 7, 21. There are 4 odd divisors of 42 and 4 is a nonprime number, so 42 is in the sequence.
MATHEMATICA
Select[Range[150], !PrimeQ[DivisorSigma[0, #/2^IntegerExponent[#, 2]]] &] (* Amiram Eldar, Dec 03 2020 *)
PROG
(PARI) isok(n) = ! isprime(sumdiv(n, d, (d%2))); \\ Michel Marcus, Apr 04 2016
CROSSREFS
Cf. A000079, A001227, A018252, A028982, A028983, A131651, A230577, A266531, A267891, A267892, A267893, A267895.
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