A373711 -id:A373711 - cp's OEIS frontend

A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

3, 3, 3, 3, 10, 14, 6, 8, 3, 4, 8, 3, 30, 11, 88, 14, 43, 50, 276, 17, 322, 20, 23, 26, 41, 29, 145, 32, 823, 35, 2378, 38, 41, 44, 47, 50, 53, 56, 59, 374, 62, 65, 2386, 68, 71, 74

Offset: 1

Pontus von Brömssen, Jan 08 2025

For n <= 46, there is a unique k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal. If this were true for all n, A027669 would be the sorted distinct terms of this sequence.

Cf. A027669, A057145, A080851, A373711, A379974, A379975.

A057145(a(n),A379974(n)) = A080851(a(n)-2,A379975(n)-1) = A373711(n).

A379974 A373711(n) is equal to the a(n)-th A379973(n)-gonal number.

Original entry on oeis.org

0, 1, 4, 15, 7, 9, 22, 19, 55, 70, 45, 119, 41, 73, 34, 181, 110, 115, 77, 361, 86, 631, 1009, 1513, 1683, 2161, 1191, 2971, 694, 3961, 604, 5149, 6553, 8191, 10081, 12241, 14689, 17443, 20521, 9000, 23941, 27721, 4970, 31879, 36433, 41401

Offset: 1

Pontus von Brömssen, Jan 08 2025

Indices to polygonal numbers are chosen so that the first k-gonal number is 1 (and the zeroth is 0).

Cf. A057145, A373711, A379973, A379975.

A057145(A379973(n),a(n)) = A373711(n).

A379975 A373711(n) is equal to the a(n)-th A379973(n)-gonal pyramidal number.

Original entry on oeis.org

0, 1, 3, 8, 5, 6, 11, 10, 20, 24, 18, 34, 17, 25, 15, 46, 33, 34, 26, 73, 28, 106, 145, 190, 204, 241, 162, 298, 113, 361, 103, 430, 505, 586, 673, 766, 865, 970, 1081, 624, 1198, 1321, 420, 1450, 1585, 1726

Offset: 1

Pontus von Brömssen, Jan 08 2025

Indices to pyramidal numbers are chosen so that the first k-gonal pyramidal number is 1 (and the zeroth is 0).

Cf. A080851, A373711, A379973, A379974.

A080851(A379973(n)-2,a(n)-1) = A373711(n).

A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

Original entry on oeis.org

3, 4, 6, 8, 10, 11, 14, 17, 30, 41, 43, 50, 60, 88, 145, 276, 322, 374, 823, 1152

Offset: 1

Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp)

The parametric solution: if k==2 (mod 3) and k >= 5, the sum of the first (k^2-4*k-2)/3 k-gonal numbers is the ((k^3-6*k^2+3*k+19)/9)-th k-gonal number A057145(k,(k^3-6*k^2+3*k+19)/9) = A344410((k-2)/3).
2378, 2386, and 31265 are also terms. See link "Cannon Ball Numbers". - Pontus von Brömssen, Jan 08 2025
Number k is a term iff the elliptic curve (3*k-6)*y^2 - (3*k-12)*y = (k-2)*x^3 + 3*x^2 - (k-5)*x has an integral point with x >= 2 different from (k^2-4*k-2)/3. The listed values may be incomplete. For example, I was not able to verify that k = 273 is not a term. - Max Alekseyev, Feb 27 2025

Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).

Cf. A027669, A057145, A344410, A373711.

More terms from Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp), Jan 05 1998

Name clarified by Max Alekseyev, Feb 27 2025

cp's OEIS Frontend

A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

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A379974 A373711(n) is equal to the a(n)-th A379973(n)-gonal number.

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A379975 A373711(n) is equal to the a(n)-th A379973(n)-gonal pyramidal number.

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A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

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