A267892 -id:A267892 - cp's OEIS frontend

A266531 Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.

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

Offset: 1

Omar E. Pol, Apr 02 2016

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

			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, ...
  ...

Index entries for sequences that are permutations of the natural numbers

Row 1 is A038547.
Column 1-10: A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, A267892, A267893.
Cf. A001227, A182469, A236104, A237591, A237593, A240062, A261697, A261698, A261699, A279387, A286000, A286001, A296508.

A230577 Positive integers that have exactly 6 odd divisors.

Original entry on oeis.org

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

Offset: 1

Philippe Beaudoin, Oct 23 2013

nonn

Numbers that can be formed in exactly 5 ways by summing sequences of 2 or more consecutive integers.
Column 6 of A266531. - Omar E. Pol, Apr 03 2016
Numbers n such that the symmetric representation of sigma(n) has 6 subparts. - Omar E. Pol, Dec 28 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000203, A001227, A077646, A237593, A279387, A266531.

Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, this sequence, A267697, A267891, A267892, A267893.

Select[Range[500],Count[Divisors[#],?OddQ]==6&] (* _Harvey P. Dale, Mar 25 2015 *)

is(n)=numdiv(n>>valuation(n,2))==6 \\ Charles R Greathouse IV, Oct 28 2013

A267696 Numbers with 5 odd divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Column 5 of A266531.
Cf. A001227, A030514, A038547, A085964, A236104, A237593, A279387.
Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, this sequence, A230577, A267697, A267891, A267892, A267893.

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

isok(n) = sumdiv(n, d, (d%2)) == 5; \\ Michel Marcus, Apr 03 2016

A001227(a(n)) = 5.

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

More terms from Michel Marcus, Apr 03 2016

A267697 Numbers with 7 odd divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 03 2016

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

David A. Corneth, Table of n, a(n) for n = 1..10000

Column 7 of A266531.
Cf. A001227, A030516, A038547, A085966, A236104, A237593, A279387.
Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, this sequence, A267891, A267892, A267893.

isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016

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

from sympy import integer_log, primerange, integer_nthroot
def A267697(n):
    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)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A001227(a(n)) = 7.

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

More terms from Michel Marcus, Apr 03 2016

A267891 Numbers with 8 odd divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

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 n such that A000265(n) has prime signature {7} or {3,1} or {1,1,1}, i.e., is in A092759 or A065036 or A007304. - Robert Israel, Mar 15 2018
Numbers that can be formed in exactly 7 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Column 8 of A266531.
Cf. A000265, A001227, A007304, A038547, A065036, A092759, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, this sequence, A267892, A267893.

[n: n in [1..1000] | #[d: d in Divisors(n) | IsOdd(d)] eq 8]; // Bruno Berselli, Apr 04 2016

filter:= proc(n) local r;
  r:= n/2^padic:-ordp(n,2);
  numtheory:-tau(r)=8
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 15 2018

Select[Range@ 840, Length@ Select[Divisors@ #, OddQ] == 8 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 8; \\ after Michel Marcus

A001227(a(n)) = 8.

A267893 Numbers with 10 odd divisors.

Original entry on oeis.org

405, 567, 810, 891, 1053, 1134, 1377, 1539, 1620, 1782, 1863, 1875, 2106, 2268, 2349, 2511, 2754, 2997, 3078, 3240, 3321, 3483, 3564, 3726, 3750, 3807, 4212, 4293, 4375, 4536, 4698, 4779, 4941, 5022, 5427, 5508, 5751, 5913, 5994, 6156, 6399, 6480, 6642, 6723, 6875, 6966, 7128, 7203, 7209, 7452, 7500, 7614, 7857, 8125

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly 10 odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 10 subparts. - Omar E. Pol, Dec 29 2016
Numbers that can be formed in exactly 9 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Column 10 of A266531.
Cf. A001227, A038547, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, A267892, this sequence.

A:=List([1..10000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
a:=Filtered([1..Length(B)],i->Length(B[i])=10); # Muniru A Asiru, Aug 14 2018

Select[Range@ 8125, Length@ Select[Divisors@ #, OddQ] == 10 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 10; \\ after Michel Marcus

A001227(a(n)) = 10.

A267894 Numbers whose number of odd divisors is nonprime.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 04 2016

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A001227, A018252, A028982, A028983, A131651, A230577, A266531, A267891, A267892, A267893, A267895.

Select[Range[150], !PrimeQ[DivisorSigma[0, #/2^IntegerExponent[#, 2]]] &] (* Amiram Eldar, Dec 03 2020 *)

isok(n) = ! isprime(sumdiv(n, d, (d%2))); \\ Michel Marcus, Apr 04 2016

cp's OEIS Frontend

A266531 Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.

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A230577 Positive integers that have exactly 6 odd divisors.

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A267696 Numbers with 5 odd divisors.

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A267697 Numbers with 7 odd divisors.

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A267891 Numbers with 8 odd divisors.

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A267893 Numbers with 10 odd divisors.

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GAP

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A267894 Numbers whose number of odd divisors is nonprime.

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PARI