A390495 - OEIS

A390495

Irregular triangle read by rows: T(n,k) is the sum of prime numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

0, 2, 0, 3, 2, 0, 5, 5, 0, 7, 2, 0, 3, 0, 2, 5, 0, 11, 5, 0, 13, 2, 7, 0, 8, 0, 2, 0, 17, 5, 0, 19, 7, 0, 3, 7, 0, 2, 11, 0, 23, 5, 0, 5, 0, 2, 13, 0, 3, 0, 0, 9, 0, 29, 10, 0, 31, 2, 0, 3, 11, 0, 2, 17, 0, 12, 0, 5, 0, 37, 2, 19, 0, 3, 13, 0, 7, 0, 41, 12, 0, 43, 2, 11, 0, 8, 0, 2, 23, 0, 47, 5, 0, 7, 0, 2, 5, 0

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OFFSET

1,2

COMMENTS

In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.

The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.

At least for the first 1000 rows the row lengths give A237271.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

FORMULA

T(n,k) = A384149(n,k) - A390494(n,k).

EXAMPLE

--------------------------------------------------------------------

--------------------------------------------------------------------

| 1 | 0; | [1]; | 1 |

| 2 | 2; | [1, 2]; | 1 |

| 3 | 0, 3; | [1], [3]; | 2 |

| 4 | 2; | [1, 2, 4]; | 1 |

| 5 | 0, 5; | [1], [5]; | 2 |

| 6 | 5; | [1, 2, 3, 6]; | 1 |

| 7 | 0, 7; | [1], [7]; | 2 |

| 8 | 2; | [1, 2, 4, 8]; | 1 |

| 9 | 0, 3, 0; | [1], [3], [9]; | 3 |

| 10 | 2, 5; | [1, 2], [5, 10]; | 2 |

| 11 | 0, 11; | [1], [11]; | 2 |

| 12 | 5; | [1, 2, 3, 4, 6, 12]; | 1 |

| 13 | 0, 13; | [1], [13]; | 2 |

| 14 | 2, 7; | [1, 2], [7, 14]; | 2 |

| 15 | 0, 8, 0; | [1], [3, 5], [15]; | 3 |

| 16 | 2; | [1, 2, 4, 8, 16]; | 1 |

| 17 | 0, 17; | [1], [17]; | 2 |

| 18 | 5; | [1, 2, 3, 6, 9, 18]; | 1 |

| 19 | 0, 19; | [1], [19]; | 2 |

| 20 | 7; | [1, 2, 4, 5, 10, 20]; | 1 |

| 21 | 0, 3, 7, 0; | [1], [3], [7], [21]; | 4 |

...

For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. In the first 2-dense sublist the sum of prime numbers is 2 + 3 = 5, so T(78,1) = 5. In the second 2-dense sublist the sum of prime numbers is 13, so T(78,2) = 13.

MATHEMATICA

A390495row[n_] := Map[Total[Select[#, PrimeQ]] &, Split[Divisors[n], #2 <= 2*# &]];

Array[A390495row, 50] (* Paolo Xausa, Nov 18 2025 *)

CROSSREFS

Row sums give A008472.

Cf. A000040, A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A386604, A386989, A386992, A387030, A389592, A389761, A389762, A390443, A390494.

Sequence in context: A287416 A097418 A362789 * A154752 A271868 A194354

Adjacent sequences: A390492 A390493 A390494 * A390496 A390497 A390498

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 10 2025

STATUS

approved