A127639 -id:A127639 - cp's OEIS frontend

A127640 Triangle read by rows in which row n contains n-1 0's followed by prime(n).

2, 0, 3, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

A127641 = this sequence * A051731.

			First few rows of the triangle are:
2;
0, 3
0, 0, 5;
0, 0, 0, 7;
0, 0, 0, 0, 11;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127641, A127638, A127639, A051731.

A127640 := proc(n,m) if m < n then 0; else ithprime(n) ; fi ; end: for n from 1 to 15 do for m from 1 to n do printf("%d,",A127640(n,m)) ; od ; od ; # R. J. Mathar, May 19 2007

Table[PadLeft[{Prime[n]},n,0],{n,15}]//Flatten (* Harvey P. Dale, Feb 28 2025 *)

More terms from R. J. Mathar, May 19 2007

A127638 A054525 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Original entry on oeis.org

2, -2, 3, -2, 0, 5, 0, -3, 0, 7, -2, 0, 0, 0, 11, 2, -3, -5, 0, 0, 13, -2, 0, 0, 0, 0, 0, 17, 0, 0, 0, -7, 0, 0, 0, 19, 0, 0, -5, 0, 0, 0, 0, 0, 23, 2, -3, 0, 0, -11, 0, 0, 0, 0, 29, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 3, 0, -7, 0, -13, 0, 0, 0, 0, 0, 37, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 2, -3, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 43, 2, 0, -5, 0, -11, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

Right diagonal = primes: (2, 3, 5, 7, ...). Row sums = the Mobius transform of primes, A007444: (2, 1, 3, 4, 9, 7, ...).

			First few rows of the triangle:
   2;
  -2,  3;
  -2,  0,  5;
   0, -3,  0, 7;
  -2,  0,  0, 0, 11;
   2, -3, -5, 0,  0, 13;
  ...

Cf. A007444, A054525, A127639.

A054525 := proc(n,k) if n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi ; end: A127648 := proc(n,k) A054525(n,k)*ithprime(k) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d,", A127648(n,k)) ; od ; od ; # R. J. Mathar, Mar 14 2007

More terms from R. J. Mathar, Mar 14 2007

A127641 A127640 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

2, 3, 3, 5, 0, 5, 7, 7, 0, 7, 11, 0, 0, 0, 11, 13, 13, 13, 0, 0, 13, 17, 0, 0, 0, 0, 0, 17, 19, 19, 0, 19, 0, 0, 0, 19, 23, 0, 23, 0, 0, 0, 0, 0, 23, 29, 29, 0, 0, 29, 0, 0, 0, 0, 29, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 37, 37, 37, 37, 0, 37, 0, 0, 0, 0, 0, 37, 41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41

Offset: 1

Gary W. Adamson, Jan 21 2007

A prime transform of A051731.

			First few rows of the triangle are:
2;
3, 3;
5, 0, 5;
7, 7, 0, 7;
11, 0, 0, 0, 11;
13, 13, 13, 0, 0, 13;
...

Cf. A127640, A051731, A127638, A127639.

A127640 := proc(n,m) if m < n then 0; else ithprime(n) ; fi ; end: A051731 := proc(n,k) if n mod k = 0 then 1 ; else 0 ; fi ; end: A127641 := proc(n,m) add( A127640(n,k)*A051731(k,m),k=1..n) ; end: for n from 1 to 15 do for m from 1 to n do printf("%d,",A127641(n,m)) ; od ; od ; # R. J. Mathar, May 19 2007

More terms from R. J. Mathar, May 19 2007

A125518 a(n) = tau(n) * prime(n).

Original entry on oeis.org

2, 6, 10, 21, 22, 52, 34, 76, 69, 116, 62, 222, 82, 172, 188, 265, 118, 366, 134, 426, 292, 316, 166, 712, 291, 404, 412, 642, 218, 904, 254, 786, 548, 556, 596, 1359, 314, 652, 668, 1384, 358, 1448, 382, 1158, 1182, 796, 422, 2230, 681, 1374, 932, 1434, 482

Offset: 1

Gary W. Adamson, Jan 21 2007

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Row sums of A127641.
Cf. A000005 (tau), A000040 (prime).
Cf. A127640, A127638, A127639.

Table[DivisorSigma[0,n]Prime[n],{n,60}] (* Harvey P. Dale, Feb 19 2021 *)

a(n) = numdiv(n) * prime(n); \\ Andrew Howroyd, Aug 09 2018

Name changed and terms a(15) and beyond from Andrew Howroyd, Aug 09 2018

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A127640 Triangle read by rows in which row n contains n-1 0's followed by prime(n).

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A127638 A054525 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

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A127641 A127640 * A051731 as infinite lower triangular matrices.

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A125518 a(n) = tau(n) * prime(n).

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