cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

Original entry on oeis.org

0, 1, 10, 120, 175, 441, 946, 1045, 1540, 4900, 5985, 7140, 23001, 23725, 48280, 195661, 245905, 314755, 801801, 975061, 1169686, 3578401, 10680265, 27453385, 55202400, 63016921, 101337426, 132361021, 197427385, 258815701, 432684460, 477132085, 837244045
Offset: 1

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Author

Kelvin Voskuijl, Jun 14 2024

Keywords

Comments

Matt Parker calls these numbers cannonball numbers, after the cannonball problem involving finding a number both square and square pyramidal.
If m==2 (mod 3), the m-gonal number A057145(m,(m^3-6*m^2+3*m+19)/9) = (m^2-4*m-2)*(m^2-4*m+1)*(m^3-6*m^2+3*m+19)/162 = A344410((m-2)/3) is a term. See comment in A027696. - Pontus von Brömssen, Dec 09 2024

Examples

			4900 is a term because it is both the 70th square and the 24th square pyramidal number.

Links

Crossrefs

Subsequences: A027568, A344376, A344410.
Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

Formula

a(n) = A057145(A379973(n),A379974(n)) = A080851(A379973(n)-2,A379975(n)-1). - Pontus von Brömssen, Jan 09 2025

Extensions

a(13)-a(33) from Pontus von Brömssen, Dec 08 2024

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

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Jan 08 2025

Keywords

Comments

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.

Crossrefs

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

Formula

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

Views

Author

Pontus von Brömssen, Jan 08 2025

Keywords

Comments

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

Crossrefs

Cf. A057145, A373711, A379973, A379975.

Formula

A057145(A379973(n),a(n)) = A373711(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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