A393261 Irregular table T(n,k) = 2^(prime(n)-k) * prime(n)^k, n >= 1, k = 1..prime(n)-1.

4, 12, 18, 80, 200, 500, 1250, 448, 1568, 5488, 19208, 67228, 235298, 11264, 61952, 340736, 1874048, 10307264, 56689952, 311794736, 1714871048, 9431790764, 51874849202, 53248, 346112, 2249728, 14623232, 95051008, 617831552, 4015905088, 26103383072, 169671989968, 1102867934792, 7168641576148, 46596170244962

Offset

1,1

Comments

Let lpf = A020639, gpf = A006530, omega = A001221, and bigomega = A001222.

For m > 4, even numbers m such that lpf(m) = omega(m) and gpf(m) = bigomega(m); implies lpf(m) = omega(m) = 2 and odd bigomega(m) such that bigomega(m) > omega(m) implies m is nonsquarefree. Prime bigomega(m) implies m is not a perfect power.

T(1,1) = 4 is the only prime power.

For n > 1, proper subset of A303946, proper subset of A080256.

For n > 1, T(n,1) and T(n,prime(n)-1) are not powerful, thus in A386294 = A332785 intersect A080259, while T(n,k), k=2..prime(n)-2 are powerful (strictly Achilles numbers) and in A386434 = A052486 intersect A080259.

For n > 1, even numbers in A392728.

Squarefree kernels are in A100484 \ {4}, even squarefree semiprimes.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10817 (rows n = 1..70, flattened)

Formula

Length of row n = A006093(n) = prime(n)-1.

Example

Table begins:
  1:    4;
  2:   12,   18;
  3:   80,  200,  500,  1250;
  4:  448, 1568, 5488, 19208, 67228, 235298;
  ...
Prime decomposition of T(n,k) for n=2..4:
  1:  2^2;
  2:  2^2*3,   2*3^2;
  3:  2^4*5, 2^3*5^2, 2^2*5^3,   2*5^4;
  4:  2^6*7, 2^5*7^2, 2^4*7^3, 2^3*7^4, 2^2*7^5, 2*7^6;

Mathematica

Table[2^(Prime[n] - k) * Prime[n]^k, {n, 7}, {k, Prime[n] - 1}]

Crossrefs

Cf. A001221, A001222, A006093, A006530, A020639, A052486, A080259, A100484, A332785, A386294, A386434, A392728.

Supersets of T(n,k), n > 1: A007774, A007916, A013929, A024619, A126706, A303946.

Sequence in context: A307624 A177833 A166162 * A228166 A244433 A086109

Adjacent sequences: A393258 A393259 A393260 * A393262 A393263 A393264

Keyword

nonn,tabf,easy

Author

Michael De Vlieger, Feb 07 2026

Status

approved