A018782 -id:A018782 - cp's OEIS frontend

A340624 Numbers k such that A340388(k) > A018782(k).

16, 32, 64, 128, 256, 512, 768, 1024, 1536, 2048, 2304, 3072, 4096, 4608, 6144, 8192, 9216, 12288, 16384, 18432, 24576, 32768, 36864, 49152, 65536, 73728, 98304, 110592, 131072, 147456, 165888, 196608, 221184, 248832, 262144, 294912, 331776, 373248, 393216, 442368

Offset: 1

Jianing Song, Apr 25 2021

Numbers k such that A340388(k) is not the smallest number whose prime factors are all congruent to 1 modulo 4 and with exactly k divisors.
Despite being an analog of A072066, this sequence seems to be considerably sparser than A072066. What's the reason for that?
All powers of 2 that are greater than or equal to 16 are here. All numbers of the form 3 * 2^e with e >= 8 are here.
All powers of 3 that are greater than or equal to 3^15 = 14348907 are here. For example, we have A340388(3^15) = (5 * 13 * 17 * 29 * ... * 113 * 137)^2, while a(3^15) <= (5^4 * 13 * 17 * 29 * .. * 113)^2, so 3^15 is a term. Apparently 3^15 is the smallest odd term in this sequence.
Similarly, let q be a prime, then all powers of q that are greater than or equal to q^(N+1) are here, where N is the number of primes congruent to 1 modulo 4 below 5^q. It seems that q^(N+1) is the smallest q-rough term in this sequence.

			16 is a term since A340388(16) = 5 * 13 * 17 * 29 > A018782(16) = 5^3 * 13 * 17.

Cf. A018782, A340388, A072066.

isA340624(n) = (A340388(n) > A018782(n)) \\ See A018782 and A340388 for their programs

A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

Original entry on oeis.org

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

Offset: 1

Robert G. Wilson v

			a(0) = 1 as 1 is the least positive integer not expressible as the sum of two squared positives.
a(1) = 2 from 2 = 1^2 + 1^2.
a(2) = 50 from 50 = 1^2 + 7^2 = 5^2 + 5^2.

A. Beiler, Recreations in the Theory of Numbers, Dover, pp. 140-141.

T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..2178 (a(2179) exceeds 1000 digits).
C. Rivera, Puzzle 62
Eric Weisstein's World of Mathematics, Square Number
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A018825, A048610, A025284-A025293 (first entries).

See A000446, A124980 and A093195 for other versions.

Array[Block[{k = 1}, While[Length@ DeleteCases[PowersRepresentations[k, 2, 2], ?(! FreeQ[#, 0] &)] != #, k++]; k] &, 6] (* _Michael De Vlieger, Mar 31 2019 *)

b(k)=my(c=0);for(i=1,sqrtint(k\2),if(issquare(k-i^2),c+=1));c \\ A025426
for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ Derek Orr, Mar 20 2019

a(n) = min(2*A018782(2n-1), A018782(2n), A018782(2n+1)).

Corrected and extended by Jud McCranie

Definition improved by several correspondents, Nov 12 2007

A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

Original entry on oeis.org

5, 65, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 126953125, 160225, 1221025, 3453125, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 30994415283203125, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Lekraj Beedassy, Apr 22 2004

nonn

An algorithm to compute the n-th term of this sequence: Write each of 2n and 2n+1 as products of their divisors in all possible ways and in decreasing order. For each product, equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1 >= a2 >= a3 >= ... >= ar, and solve for the ai. Evaluate A002144(1)^a1 * A002144(2)^a2 * ... * A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two distinct positive squares. [Ant King, Dec 14 2009; May 26 2010]

Ray Chandler, Table of n, a(n) for n = 1..1438 (a(1439) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A025302-A025311 (first entries). See A016032, A000446 and A124980 for other versions.

b(k)=my(c=0);for(i=1,sqrtint((k-1)\2),if(issquare(k-i^2),c+=1));c \\ A025441
for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ Derek Orr, Mar 20 2019

a(n) = min(A018782(2n), A018782(2n+1)).

More terms from Ant King, Dec 14 2009 and Feb 07 2010

A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

Original entry on oeis.org

0, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125

Offset: 1

N. J. A. Sloane

nonn

Could start with a(0) = 3: the smallest nonnegative integer that can be written as sum of two squares in 0 ways. - M. F. Hasler, Jul 05 2024

			a(1) = 0 because 0 is the smallest integer which is uniquely a unique sum of two squares, namely 0^2 + 0^2.
a(2) = 25 from 25 = 5^2 + 0^2 = 3^2 + 4^2.
a(3) = 325 from 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(4) = 1105 from 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000448 (similar, but "in at least n ways").
Cf. A002144, A018782, A054994.
See A016032, A093195 and A124980 for other versions.

A000446(n)=if(n>1, A124980(n), 0) \\ M. F. Hasler, Jul 06 2024

A000446=lambda n: A124980(n) if n>1 else 0 # M. F. Hasler, Jul 07 2024

An algorithm to compute the n-th term of this sequence for n>1: Write each of 2n and 2n-1 as products of their divisors, in decreasing order and in all possible ways. Equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1>=a2>=a3>=...>=ar, and solve for the ai. Evaluate A002144(1)^a1 x A002144(2)^a2 x ... x A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two squares. [Ant King, Oct 07 2010]
a(n) = min(A018782(2n-1), A018782(2n)) for n > 1.
a(n) = A124980(n) for n > 1. - M. F. Hasler, Jul 07 2024

Better description and more terms from David W. Wilson, Aug 15 1996

Definition improved by several correspondents, Nov 12 2007

A343771 Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

Original entry on oeis.org

1, 7, 49, 91, 2401, 637, 117649, 1729, 8281, 31213, 282475249, 12103, 13841287201, 1529437, 405769, 53599, 33232930569601, 157339, 1628413597910449, 593047, 19882681, 3672178237, 3909821048582988049, 375193, 68574961, 179936733613, 2989441, 29059303, 459986536544739960976801, 7709611

Offset: 1

Jianing Song, Apr 28 2021

nonn

a(n) is the smallest k such that A004016(k) = 6*n.
Also a(n) is the smallest index of n in A002324.
a(n) is the smallest term in A004611 that has exactly n divisors.

			91 = 7 * 13 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 4 divisors, so a(4) = 91.
8281 = 7^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 9 divisors, so a(9) = 8281.

Jianing Song, Table of n, a(n) for n = 1..1000
Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.

Cf. A004016, A002324, A004611, A230655, A355919.

Analog of A005179 and A018782.

primelist(d, r, l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p, d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A343771(n)=my(pf=prodR(n, n), a=1, b, v=primelist(3, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bA005179.

If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 3. The smallest n such that the equality does not hold is n = 128. [Those n such that the equality does not hold are listed in A355919. - Jianing Song, Jul 21 2022]
a(n) <= 7^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 7^(p-1) * 13^(q-1) for primes p >= q.

A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

Original entry on oeis.org

1, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Artur Jasinski, Nov 15 2006

nonn

The number must be strictly positive, but one of the squares may be zero, as we see from a(1) = 1 = 1^2 + 0^2 and a(2) = 25 = 3^2 + 4^2 = 5^0 + 0^2. - M. F. Hasler, Jul 07 2024

			a(3) = 325 is decomposable in 3 ways: 15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

A124980(n)={for(a=1, oo, A000161(a)==n && return(a))} \\ R. J. Mathar, Nov 29 2006, edited by M. F. Hasler, Jul 07 2024

PD(n, L=n, D=Vecrev(divisors(n)[^1])) = { if(n>1, concat(vector(#D, i, if(D[i] > L, [], D[i] < n, [concat(D[i], P) | P <- PD(n/D[i], D[i])], [[n]]))), [[]])}
apply( {A124980(n)=vecmin([prod(i=1, #a, A002144(i)^(a[i]-1)) | a<-concat([PD(n*2,n), PD(n*2-1)])])}, [1..44]) \\ M. F. Hasler, Jul 07 2024

from sympy import divisors, isprime, prod
def PD(n, L=None): return [[]] if n==1 else [
    [d]+P for d in divisors(n)[:0:-1] if d <= (L or n) for P in PD(n//d, d)]
A2144=lambda upto=999: filter(isprime, range(5, upto, 4))
def A124980(n):
    return min(prod(a**(f-1) for a,f in zip(A2144(), P))
               for P in PD(n*2, n)+PD(n*2-1)) # M. F. Hasler, Jul 07 2024

a(n) = A000446(n), n > 1. - R. J. Mathar, Jun 15 2008
a(n) = min(A018782(2n-1), A018782(2n)).
a(n) = min { k > 0 | A000161(k) = n }. - M. F. Hasler, Jul 07 2024

More terms from R. J. Mathar, Nov 29 2006

Edited and extended by Ray Chandler, Jan 07 2012

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

A204046 Least term of A094179 with exactly 2n divisors.

Original entry on oeis.org

3, 27, 63, 231, 567, 1323, 5103, 2079, 4851, 27783, 413343, 13167, 3720087, 250047, 43659, 100947, 301327047, 160083, 2711943423, 118503, 392931, 20253807, 219667417263, 276507, 2139291, 182284263, 1013859, 1066527, 160137547184727, 3361743, 1441237924662543

Offset: 1

Ray Chandler, Jan 10 2012

nonn

Ray Chandler, Table of n, a(n) for n = 1..2098 (a(2099) exceeds 1000 digits).

Cf. A018782, A078714, A094179, A204086.

For odd prime p, a(p) = 3^(p-1)*7.

A204086 Least term of A004767 with exactly 2n divisors.

Original entry on oeis.org

3, 15, 63, 135, 567, 315, 5103, 1155, 1575, 2835, 413343, 4095, 3720087, 25515, 14175, 10395, 301327047, 20475, 2711943423, 36855, 127575, 2066715, 219667417263, 51975, 354375, 18600435, 121275, 331695, 160137547184727, 184275, 1441237924662543, 135135

Offset: 1

Ray Chandler, Jan 10 2012

nonn

For odd prime p, a(p) = 3^(p-1)*7.

Ray Chandler, Table of n, a(n) for n = 1..2098 (a(2099) exceeds 1000 digits).

Cf. A004767, A018782, A204046, A078714.

a(n) = min{d|n: A018782(d) * A204046(n/d)}.

a(n) = 4*A078714(n)-1.

A338690 Inverse Moebius transform of A209615.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3

Offset: 1

Jianing Song, Apr 24 2021

Earliest occurrence of k is A018782(k).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A209615, A035184 (a similar sequence), A018782, A002654, A019673.

f[p_, e_] := If[Mod[p, 4] == 1, e + 1, (1 + (-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

a(n) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); if(p%4==1, r*=e+1, if(e%2, return(0)))); r

Multiplicative with a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p = 2 or p == 3 (mod 4).
a(n) = A002654(n) = A035184(n) for odd n. a(2^e * m) = a(m) for even m, 0 for odd m.
Dirichlet g.f.: zeta(s)*beta(s)/(1 + 2^(-s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 = 0.523598... (A019673). - Amiram Eldar, Oct 22 2022

cp's OEIS Frontend

A340624 Numbers k such that A340388(k) > A018782(k).

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A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

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A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

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A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

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A343771 Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

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A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

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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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