A018782 -id:A018782 - cp's OEIS frontend

A340388 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)13^(p_2 - 1)17^(p_3 - 1)...*A002144(k)^(p_k - 1).

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 32045, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125, 37252902984619140625

Offset: 1

Jianing Song, Apr 24 2021

Analog of A037019: this is an easy way to produce a number k such that A002654(k) = n, or equivalently, a number k whose prime factors are all congruent to 1 modulo 4 and with exactly n divisors.

			12 = 3 * 2 * 2, so a(12) = 5^(3-1) * 13^(2-1) * 17^(2-1) = 5525.
15 = 5 * 3, so a(15) = 5^(5-1) * 13^(3-1) = 105625.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A002144, A002654, A018782, A037019, A340624.

a(n) = my(f=factor(n), w=omega(n), p=1, product=1); forstep(i=w, 1, -1, for(j=1, f[i,2], p=nextprime(p+1); while(!(p%4==1), p=nextprime(p+1)); product *= p^(f[i,1]-1))); product

By definition a(n) >= A018782(n) for all n. Note that a(16) = 32045 is strictly larger than A018782(16) = 27625. The "exceptional" numbers k such that a(k) > A018782(k) are listed in A340624.

If n = p for prime p or n = pq for primes p >= q, then a(n) = A018782(n).

A344470 Record values in A002654.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 216, 256, 288, 320, 384, 432, 512, 576, 640, 768, 864, 960, 1024, 1152, 1280, 1536, 1728, 1920, 2048, 2304, 2560, 2880, 3072, 3456, 3840, 4096, 4608, 5120, 5760, 6144

Offset: 1

Jianing Song, May 20 2021

nonn

Also numbers k such that A018782(m) > A018782(k) for all m > k.

			9 is a term because the circle with radius sqrt(4225) centered at the origin hits exactly 4*9 = 36 integer points, and any circle with radius < sqrt(4225) centered at the origin hits fewer than 36 points.

Jianing Song, Table of n, a(n) for n = 1..424

Cf. A071383, A071385, A002654, A018782, A000005, A054994.

Records of Sum_{d|n} kronecker(m, d): A344472 (m=-3), this sequence (m=-4), A279542 (m=-6).

my(v=list(10^15), rec=0); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(rec, ", "))) \\ see program for A054994

a(n) = A071385(n+1)/4.

a(n) = A000005(A071383(n+1)) = A002654(A071383(n+1)).

A261176 Minimum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

Original entry on oeis.org

0, 9, 126, 802, 3158, 10040, 25464, 58837, 123422, 238203, 429467, 733923, 1200319, 1912928, 2945116, 4369570, 6338678, 9053512, 12622814, 17359779, 23503546, 31347788, 41161317

Offset: 1

Hugo Pfoertner, Aug 15 2015

In one of his programming contests, Al Zimmermann coined the term "Delacorte Numbers" (after G. T. Delacorte, Jr., a New York City philantropist and benefactor) for the sum of D(a,b) = gcd(a,b) * distance^2(a,b), taken over all distinct pairs of integers (a,b) in a rectangular matrix.
The challenge in the contest was to find two kinds of arrangements of 1 to n^2, one minimizing the combined sum (this sequence) and the other maximizing the combined sum (A261177).
All terms beyond a(5) are conjectured based on numerical results. Terms up to a(17) have at least 5 independent verifications.
Upper bounds for the next terms are a(24)<=53670478, a(25)<=68938808, a(26)<=87777189, a(27)<=110759499.

			a(2)=9, because the matrix ((1 2)(3 4)) has Delacorte Number
D(1,2) + D(1,3) + D(1,4) + D(2,3) + D(2,4) + D(3,4) =
gcd(1,2)*(1^2 + 0^2) +
gcd(1,3)*(0^2 + 1^2) +
gcd(1,4)*(1^2 + 1^2) +
gcd(2,3)*(1^2 + 1^2) +
gcd(2,4)*(0^2 + 1^2) +
gcd(3,4)*(1^2 + 0^2) = 1*1 + 1*1 + 1*2 + 1*2 + 2*1 + 1*1 = 9.
Putting (2,4) in a row or column gives the minimum value of the matrix, whereas putting this pair in one of the diagonals gives the maximum.
a(3)=126, because no arrangement of the matrix elements exists that produces a smaller Delacorte Number than e.g. ((1 2 4)(3 6 8)(5 9 7)).

Al Zimmermann's Programming Contests, Delacorte Numbers, Description, October 2014.
Al Zimmermann's Programming Contests, Delacorte Numbers, Final Report, January 2015.
The New York Community Trust: George T. Delacorte.
Arch D. Robison, Computing Delacorte Numbers with Julia, January 21, 2015.

Cf. A261177, A003989, A018782.

A261177 Maximum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

Original entry on oeis.org

0, 10, 180, 1392, 6149, 21350, 57192, 137617, 298864, 593378, 1101739, 1936342, 3216080

Offset: 1

Hugo Pfoertner, Aug 15 2015

Best results found in Al Zimmermann's Programming Contest "Delacorte Numbers". For more information see A261176. All terms beyond a(5) are conjectured based on numerical results. Terms up to a(11) have at least 5 independent verifications. Lower bounds for the next terms are a(14)>=5189492, a(15)>=8110781, a(16)>=12239616, a(17)>=18073562, a(18)>=26055061, a(19)>=36769303, a(20)>=51095165.

			a(3)=180, because no arrangement of the matrix elements exists that produces a larger Delacorte Number than e.g. ((2 3 4)(9 1 5)(8 7 6)).

Al Zimmermann's Programming Contests, Delacorte Numbers, Description, October 2014.
Al Zimmermann's Programming Contests, Delacorte Numbers, Final Report, January 2015.

Cf. A261176, A003989, A018782.

Lower bounds for a(18) and a(20) improved by Hugo Pfoertner, Nov 22 2015

cp's OEIS Frontend

A340388 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)13^(p_2 - 1)17^(p_3 - 1)...*A002144(k)^(p_k - 1).

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A344470 Record values in A002654.

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A261176 Minimum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

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A261177 Maximum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

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cp's OEIS Frontend

A340388 Let n = p_1*p_2*...*p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)*13^(p_2 - 1)*17^(p_3 - 1)*...*A002144(k)^(p_k - 1).

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A344470 Record values in A002654.

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A261176 Minimum value of (1/2)*Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) * ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

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A261177 Maximum value of (1/2)*Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) * ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

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A340388 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)13^(p_2 - 1)17^(p_3 - 1)...*A002144(k)^(p_k - 1).

A261176 Minimum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

A261177 Maximum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.