A038547 -id:A038547 - cp's OEIS frontend

A338105 a(n) is the least integer that can be expressed as the difference of two n-gonal numbers in exactly n ways.

9, 96, 1330, 4725, 21021, 22400, 421515, 675675, 5370365, 576576, 10790325, 39255125, 51548805, 7286400, 978624647, 144729585, 649593945, 125245120, 1109593485, 4519064403, 13908638315, 253955520, 8860666815, 30587913125, 33144736086, 859541760, 147839441750

Offset: 3

Ilya Gutkovskiy, Oct 10 2020

nonn

a(17) <= 1340770739, a(18) = 144729585, a(19) <= 9381302307, a(20) <= 1257818848, a(21) <= 6299438145, a(22) <= 32911706919, a(23) <= 26720105555, a(24) <= 3141537984, a(25) <= 59558175105, a(26) <= 71119743695, a(27) <= 260207700831, a(28) <= 28582652736, a(29) <= 688883385190, a(30) <= 593086020813. - Chai Wah Wu, Oct 14 2020

			a(3) = 9 because 9 = 10 - 1 = 15 - 6 = 45 - 36 and this is the least integer that can be expressed as the difference of two triangular numbers in exactly 3 ways.

Martin Ehrenstein, Table of n, a(n) for n = 3..40
Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers

Cf. A038547, A068314, A072502, A257411, A334034, A334035, A334036, A334037.

a(11)-a(16) from Chai Wah Wu, Oct 13 2020

a(17) and a(19)-a(40) from Martin Ehrenstein, Oct 23 2020

A358252 a(n) is the least number with exactly n non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 13824, 10368, 20736, 28800, 41472, 64800, 279936, 115200, 331776, 345600, 663552, 259200, 1679616, 518400, 1620000, 1166400, 4860000, 1036800, 17915904, 2073600, 15552000, 6998400, 26873856, 4147200, 53747712, 8294400

Offset: 0

Amiram Eldar, Nov 05 2022

nonn

a(n) is the least number k such that A056626(k) = n.

Since A056626(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

			a(1) = 8 since 8 is the least number that has exactly one non-unitary square divisor, 4.

Amiram Eldar, Table of n, a(n) for n = 0..176

Cf. A056626, A025487, A358253.

Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[21, 10^6]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A364586 a(n) is the least number with exactly n divisors of the form 5*k+1.

Original entry on oeis.org

1, 6, 36, 66, 252, 336, 672, 1008, 3528, 2016, 4032, 3696, 9072, 7392, 13104, 11088, 36288, 38304, 26208, 22176, 68544, 44352, 91728, 66528, 154224, 99792, 209664, 96096, 301392, 144144, 222768, 188496, 487872, 399168, 471744, 421344, 1079568, 288288, 1097712, 432432

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A001876, A005179, A038547, A364598, A364599, A364600.

a(n) = my(k=1); while (sumdiv(k, d, (d%5)==1) != n, k++); k; \\ Michel Marcus, Jul 29 2023

list(nmax) = {my(v = vector(nmax), c = 0, k = 1, i); while(c < nmax, i = sumdiv(k, d, d % 5 == 1); if(i <= nmax && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A364598 a(n) is the least number with exactly n divisors of the form 5*k+2.

Original entry on oeis.org

1, 2, 12, 42, 84, 252, 462, 672, 924, 2016, 2772, 4032, 5544, 9072, 7392, 17136, 14784, 26208, 22176, 34272, 33264, 52416, 44352, 119952, 66528, 117936, 99792, 183456, 125664, 222768, 188496, 235872, 199584, 487872, 288288, 616896, 399168, 1206576, 376992, 1097712, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 0..100

Cf. A001877, A005179, A038547, A364586, A364599, A364600.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 2) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A364599 a(n) is the least number with exactly n divisors of the form 5*k+3.

Original entry on oeis.org

1, 3, 18, 48, 144, 288, 504, 1248, 1008, 2016, 3024, 4368, 5544, 14112, 12096, 17136, 11088, 34272, 22176, 26208, 33264, 52416, 48048, 91728, 66528, 117936, 155232, 183456, 133056, 235872, 188496, 222768, 144144, 528528, 376992, 616896, 421344, 825552, 288288, 1707552, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 0..100

Cf. A001878, A005179, A038547, A364586, A364598, A364600.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 3) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A364600 a(n) is the least number with exactly n divisors of the form 5*k+4.

Original entry on oeis.org

1, 4, 24, 72, 144, 432, 504, 1008, 1512, 3528, 3024, 7056, 5544, 14112, 11088, 13104, 16632, 26208, 36288, 63504, 33264, 68544, 77616, 127008, 66528, 154224, 155232, 209664, 133056, 222768, 144144, 301392, 216216, 853776, 288288, 471744, 399168, 825552, 698544, 1707552, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 0..100

Cf. A001899, A005179, A038547, A364586, A364598, A364599.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 4) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A383402 Smallest number whose largest odd divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset: 1

Omar E. Pol, May 14 2025

nonn

From Peter Munn, May 15 2025 and May 20 2025: (Start)
A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
(End)
If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025
Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - David A. Corneth, May 17 2025

			The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.

David A. Corneth, Table of n, a(n) for n = 1..872
David A. Corneth, PARI program
David A. Corneth, Upper bounds on a(n) for n = 1..10000
Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Row 1 of A383961.
The range of terms is a subset of {1} U A355200.
Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 14 2025 *)

a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ Michel Marcus, May 14 2025

PARI
```
\\ See Corneth link
```

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

More terms from Amiram Eldar, May 14 2025

A078714 a(n) = smallest number m which can be obtained in n ways by subtracting twice a triangular number from a perfect square.

Original entry on oeis.org

1, 4, 16, 34, 142, 79, 1276, 289, 394, 709, 103336, 1024, 930022, 6379, 3544, 2599, 75331762, 5119, 677985856, 9214, 31894, 516679, 54916854316, 12994, 88594, 4650109, 30319, 82924, 40034386796182, 46069, 360309481165636, 33784, 2583394, 376658809, 797344

Offset: 1

R. L. Coffman, K. W. McLaughlin and R. J. Dawson (robert.l.coffman(AT)uwrf.edu), Dec 19 2002

nonn

The minimum number m (denoted by LSDT(n)) which can be represented in n different ways as a symmetric unimodal consecutive integer sequence (e.g., 6+7+8+7+6) that sums to the integer m. More precisely, n is the number of ways to arrange m objects into symmetrically-placed, congruent isosceles trapezoids adjoined at overlapping largest bases and m is the minimum number of objects that allows this number of arrangements.

a(23)-a(50) are ?, 12994, 88594, 4650109, 30319, 82924, ?, 46069, ?, 33784, 2583394, 376658809, 797344, 78829, ?, ?, 23250544, 148129, ?, 414619, ?, 6716824, 272869, ?, ?, 168919, 19933594, 1151719. - Robert G. Wilson v, Dec 24 2002

			Let SDT(n) = the number, k, of symmetric double trapezoidal arrangements of n objects, then SDT(34) = 4, since we have 34 or 11+12+11 or 6+7+8+7+6 or 2+3+4+5+6+5+4+3+2. For SDT(n) = 4, we have n = 34 or 49 or 58 or 64 ..., so that the least value of SDT(n)=4 is LSDT(4) = 34. Also 4*34 - 1 = 135 = (3^3)*(5^1) so that r1=3 and r2=1 (p1=3 and p2=5), resulting in SDT(34) = (3+1)*(1+1)/2 = 4 and 34 is the least value of n which satisfies 4*n-1 so that one half the number of odd divisors equals 4.

Ray Chandler, Table of n, a(n) for n = 1..2098 (a(2099) exceeds 1000 digits).
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A078703, A038547, A018782, A204046, A204086.

The following function determines the number of ways, SDT(n), of arranging n identical objects into symmetric double trapezoidal arrangements: SDT[n_] := (Times @@ Cases[FactorInteger[4 n - 1], {p_, r_} -> r + 1])/2 The program below computes the first few terms of the sequence LSDT(k)=min{n:SDT(n)=k}. The output is in the form {{1, LSDT(1)}, {2, LSDT(2)}, {3, LSDT(3)}, ...}: Union[Sort[{SDT[ # ], #} & /@ Range[1, 100000]], SameTest -> (#1[[1]] == #2[[1]] &)]

LSDT(k)={min n: SDT(n)=k}, where SDT(n)=((r1+1)*(r2+1)*...)/2 and ((p1^r1)*(p2^r2)*...) is the factorization of 4n-1 into (odd) primes.
a(n) = (A204086(n) + 1)/4. - Ray Chandler, Jan 10 2012
For odd prime p, a(p) = (3^(p-1)*7 + 1)/4.

Missing terms noted in Comments and b-file from Ray Chandler, Jan 10 2012

A358262 a(n) is the least number with exactly n noninfinitary square divisors.

Original entry on oeis.org

1, 16, 144, 256, 3600, 1296, 2304, 65536, 129600, 16777216, 32400, 20736, 57600, 331776, 589824, 4294967296, 6350400, 1099511627776, 150994944, 810000, 1587600, 1679616, 518400, 5308416, 2822400, 84934656, 8294400, 26873856, 14745600, 21743271936, 38654705664

Offset: 0

Amiram Eldar, Nov 06 2022

nonn

a(n) is the least number k such that A358261(k) = n.

Since A358261(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

			a(1) = 16 since 16 is the least number with exactly one noninfinitary divisor, 4.

Amiram Eldar, Table of n, a(n) for n = 0..50

Cf. A025487, A358261, A358263.

Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[15, 2*10^7]

s(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A364583 a(n) is the least number with exactly n divisors of the form 3*k+2.

Original entry on oeis.org

1, 2, 8, 20, 40, 80, 140, 320, 280, 800, 560, 5120, 1120, 6400, 2240, 3920, 3080, 40000, 5600, 102400, 6160, 15680, 35840, 20971520, 12320, 110000, 44800, 39200, 24640, 1342177280, 30800, 193600, 40040, 250880, 280000, 440000, 61600, 1210000, 716800, 313600, 80080, 5497558138880

Offset: 0

Ilya Gutkovskiy, Jul 28 2023

nonn

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A001822, A005179, A038547, A364582.

a(n) = my(k=1); while (sumdiv(k, d, (d%3)==2) != n, k++); k; \\ Michel Marcus, Jul 29 2023

a(n) <= 5*2^(n-1). - David A. Corneth, Jul 29 2023

More terms from David A. Corneth, Jul 29 2023

cp's OEIS Frontend

A338105 a(n) is the least integer that can be expressed as the difference of two n-gonal numbers in exactly n ways.

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A358252 a(n) is the least number with exactly n non-unitary square divisors.

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A364586 a(n) is the least number with exactly n divisors of the form 5*k+1.

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PARI

A364598 a(n) is the least number with exactly n divisors of the form 5*k+2.

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PARI

A364599 a(n) is the least number with exactly n divisors of the form 5*k+3.

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PARI

A364600 a(n) is the least number with exactly n divisors of the form 5*k+4.

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PARI

A383402 Smallest number whose largest odd divisor is its n-th divisor.

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A078714 a(n) = smallest number m which can be obtained in n ways by subtracting twice a triangular number from a perfect square.

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