A038547 -id:A038547 - cp's OEIS frontend

A122819 Array read by rows in which the n-th row contains smallest odd numbers in increasing order of all possible prime signatures with n divisors.

1, 3, 9, 15, 27, 81, 45, 243, 729, 105, 135, 2187, 225, 6561, 405, 19683, 59049, 315, 675, 1215, 177147, 531441, 3645, 1594323, 2025, 4782969, 945, 1155, 3375, 10935, 14348907, 43046721, 1575, 6075, 32805, 129140163, 387420489, 2835, 10125, 98415

Offset: 1

Ray Chandler, Sep 22 2006

n-th row contains A001055(n) terms.

First item of each row gives A038547.

			Table begins:
  1,
  3,
  9,
  15, 27,
  81,
  45, 243,
  729,
  105, 135, 2187,
  ...

Amiram Eldar, Table of n, a(n) for n = 1..7314 (rows n=1..1000, flattened)

Cf. A001055, A038547, A077569, A162247.

row[n_] := Module[{e = f[n] - 1}, Sort[Times @@ (Prime[Range[2, Length[#]+1]]^Reverse[#]) & /@ e]]; Table[row[n], {n, 1, 25}] // Flatten (* Amiram Eldar, Jun 28 2025 using the function f by T. D. Noe at A162247 *)

A340232 a(n) is the least number with exactly 2*n bi-unitary divisors.

Original entry on oeis.org

2, 6, 32, 24, 512, 96, 8192, 120, 131072, 1536, 2097152, 480, 33554432, 24576, 536870912, 840, 8589934592, 7776, 137438953472, 7680, 2199023255552, 6291456, 35184372088832, 3360, 562949953421312, 100663296, 9007199254740992, 122880, 144115188075855872, 124416

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

Every integer except 1 has an even number of bi-unitary divisors.

			a(1) = 2 since 2 is the least number with 2*1 = 2 bi-unitary divisors, 1 and 2.
a(2) = 6 since 6 is the least number with 2*2 = 4 bi-unitary divisors, 1, 2, 3 and 6.

Amiram Eldar, Table of n, a(n) for n = 1..62

Subsequence of A025487.
Cf. A222266, A286324, A293185.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340233 (exponential).

f[p_, e_] := If[OddQ[e], e + 1, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 10; s = Table[0, {max}]; c = 0; n = 2;  While[c < max, i = d[n]/2; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

A286324(a(n)) = 2*n and A286324(k) != 2*n for all k < a(n).

A187941 Least number with exactly n even divisors.

Original entry on oeis.org

1, 2, 4, 8, 12, 32, 24, 128, 48, 72, 96, 2048, 120, 8192, 384, 288, 240, 131072, 360, 524288, 480, 1152, 6144, 8388608, 720, 2592, 24576, 1800, 1920, 536870912, 1440, 2147483648, 1680, 18432, 393216, 10368, 2520, 137438953472, 1572864, 73728, 3360, 2199023255552, 5760, 8796093022208

Offset: 0

Juri-Stepan Gerasimov, Mar 16 2011

nonn

The only odd term in the sequence is 1, having zero even divisors. All larger odd numbers also have zero even divisors.
Conjecture: a(n) = 2^n only if n is prime or if n = 1.
If the prime factorization of a number is 2^k p1^e1...pr^er, then the number of even divisors is k*(e1+1)...(er+1). Hence, to find the least number having n even divisors, factor n and determine k, e1,..., er such that n = k*(e1+1)...(er+1). Then a(n) will have the form 2^k 3^e1 5^e2.... It is obvious that if n is prime, then a(n) = 2^n. Similarly, if n is twice an odd prime p, then a(n) = 2^p * 3. - T. D. Noe, Mar 16 2011

Cf. A000005, A005179, A038547, A183063.

evenDivSigma[n_Integer] := Length[Select[Divisors[n], EvenQ]]; Flatten[Table[Take[Select[Range[2, 10^6, 2], evenDivSigma[#] == n &], 1], {n, 20}]] (* Alonso del Arte, Mar 16 2011 *)

a(n) = 2 * A005179(n) for n > 0.

A334034 a(n) is the least integer that can be expressed as the difference of two pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 22, 70, 715, 1330, 4025, 6370, 14014, 17290, 25025, 45815, 73150, 121030, 95095, 85085, 256025, 350350, 432250, 1179178, 425425, 575575, 734825, 950950, 1926925, 3751930, 2187185, 1616615, 1956955, 3148145, 3658655, 4029025, 2977975, 4352425, 6656650, 13918450

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 3 in exactly n ways.

Index of first occurrence of n in A333815.

Chai Wah Wu, Table of n, a(n) for n = 1..72
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A016777, A038547, A068314, A333815, A334008, A334035, A334036, A334037.

More terms from Jinyuan Wang, Apr 13 2020

A334035 a(n) is the least integer that can be expressed as the difference of two hexagonal numbers in exactly n ways.

Original entry on oeis.org

1, 45, 225, 585, 2415, 4725, 9945, 10395, 31185, 28665, 45045, 58905, 143325, 257985, 135135, 225225, 329175, 487305, 405405, 831285, 1091475, 675675, 1396395, 1576575, 2927925, 3132675, 2436525, 2027025, 2567565, 2297295, 6235515, 5360355, 4729725, 3828825, 10503675

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 4 in exactly n ways.

Index of first occurrence of n in A333816.

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A016813, A038547, A068314, A333816, A334010, A334034, A334036, A334037.

nmax = 10000; A333816 = Rest[CoefficientList[Series[Sum[x^(k*(2*k - 1))/(1 - x^(4*k)), {k, 1, 1 + Sqrt[nmax/2]}], {x, 0, nmax}], x]]; Flatten[Table[FirstPosition[A333816, k], {k, 1, Max[A333816]}]] (* Vaclav Kotesovec, Apr 19 2020 *)

More terms from Jinyuan Wang, Apr 13 2020

A334037 a(n) is the least integer that can be expressed as the difference of two octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 133, 560, 1729, 4160, 10640, 14560, 22400, 44800, 58240, 138320, 98560, 123200, 203840, 246400, 394240, 320320, 492800, 800800, 640640, 1047200, 1823360, 1724800, 1281280, 2094400, 1601600, 2475200, 2722720, 4484480, 3203200, 5532800, 6697600, 5445440, 7958720

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 6 in exactly n ways.

Index of first occurrence of n in A333818.

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000567, A016921, A038547, A068314, A333818, A334012, A334034, A334035, A334036.

More terms from Jinyuan Wang, Apr 13 2020

A340233 a(n) is the least number with exactly n exponential divisors.

Original entry on oeis.org

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.

See the link for more values of this sequence.

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Amiram Eldar, Table of n, a(n) for n = 1..12
Amiram Eldar, Table of n, a(n) for n = 1..100 (given by prime factorizations)

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

A357450 a(n) is the smallest integer having exactly n odd square divisors (A298735).

Original entry on oeis.org

1, 9, 81, 225, 6561, 2025, 531441, 11025, 50625, 164025, 3486784401, 99225, 282429536481, 13286025, 4100625, 893025, 1853020188851841, 2480625, 150094635296999121, 8037225, 332150625, 87169610025, 984770902183611232881, 12006225, 2562890625, 7060738412025, 121550625

Offset: 1

Bernard Schott, Sep 29 2022

nonn

All terms are odd and squares (A016754).

			2025 has 6 divisors that are odd squares: {1, 9, 25, 81, 225, 2025}; also, 2025 is the smallest integer that has 6 odd squares divisors, hence a(6) = 2025.

Cf. A000290, A016754, A038547, A130279, A147516, A298735.

f(n) = factorback(apply(e->e\2+1, factor(n/2^valuation(n, 2))[, 2])); \\ A298735
a(n) = my(k=1); while (f(k)!=n, k++); k; \\ Michel Marcus, Sep 29 2022

a(n) = A038547(n)^2. - Thomas Scheuerle, Sep 30 2022
Proof: Suppose a(n) = Product p_i^(2*e_i), where the p_i are odd primes. Then the n odd square divisors are all of the form d = Product p_i^(2*k_i) with 0 <= k_i <= e_i. As a(n) = Product (p_i^e_i)^2 = (Product (p_i^e_i))^2, we get that sqrt(a(n)) = Product (p_i^e_i). This is the prime decomposition of sqrt(a(n)). As there is a bijection between prime factors p_i^(2*k_i) and (p_i^k_i), there is also bijection between odd square divisors of a(n) and odd divisors of sqrt(a(n)). We conclude that sqrt(a(n)) is the smallest integer that has exactly n odd divisors. - Bernard Schott, Oct 01 2022
a(p) = 3^(2*(p-1)) for primes p. - Bernard Schott, Oct 03 2022

a(7)-a(10) from Michel Marcus, Sep 29 2022

More terms from Amiram Eldar, Sep 29 2022

A122811 Number of prime factors (counted with multiplicity) of the smallest odd number with exactly n divisors.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 5, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 6, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 7, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41

Offset: 1

Ray Chandler, Sep 22 2006

nonn

a(p) = p-1 for prime p.

This sequence first differs from A059975(n) at n=16, because 3^3.5.7 is less than 3.5.7.11. - Hugo van der Sanden, May 21 2010

Amiram Eldar, Table of n, a(n) for n = 1..2000 (calculated from the b-file at A038547)

Cf. A001222, A038547, A122376, A059975.

a(n) = Omega(A038547(n)), where Omega(n) = A001222(n).

A334036 a(n) is the least integer that can be expressed as the difference of two heptagonal numbers in exactly n ways.

Original entry on oeis.org

1, 81, 468, 1911, 6237, 11781, 21021, 51051, 81081, 121737, 261261, 318087, 513513, 671517, 1145529, 1072071, 1582581, 1378377, 3216213, 2513511, 4135131, 4700619, 5666661, 11792781, 8729721, 11810799, 15444891, 19270251, 15162147, 24657633, 28945917, 26189163

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 5 in exactly n ways.

Index of first occurrence of n in A333817.

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A016861, A038547, A068314, A333817, A334011, A334034, A334035, A334037.

More terms from Jinyuan Wang, Apr 13 2020

cp's OEIS Frontend

A122819 Array read by rows in which the n-th row contains smallest odd numbers in increasing order of all possible prime signatures with n divisors.

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A340232 a(n) is the least number with exactly 2*n bi-unitary divisors.

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A187941 Least number with exactly n even divisors.

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A334034 a(n) is the least integer that can be expressed as the difference of two pentagonal numbers in exactly n ways.

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A334035 a(n) is the least integer that can be expressed as the difference of two hexagonal numbers in exactly n ways.

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A334037 a(n) is the least integer that can be expressed as the difference of two octagonal numbers in exactly n ways.

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A340233 a(n) is the least number with exactly n exponential divisors.

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A357450 a(n) is the smallest integer having exactly n odd square divisors (A298735).

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A122811 Number of prime factors (counted with multiplicity) of the smallest odd number with exactly n divisors.