A160812 -id:A160812 - cp's OEIS frontend

A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

1, 1, 2, 1, 2, 3, 4, 1, 3, 5, 6, 1, 2, 4, 7, 8, 9, 10, 1, 5, 11, 12, 1, 2, 3, 6, 13, 14, 15, 16, 1, 7, 17, 18, 1, 2, 4, 8, 19, 20, 21, 22, 1, 3, 9, 23, 24, 25, 26, 27, 28, 1, 2, 5, 10, 29, 30, 1, 11, 31, 32, 33, 34, 35, 36, 1, 2, 3, 4, 6, 12, 37, 38, 39, 40

Offset: 0

Omar E. Pol, Jun 30 2009

			Triangle begins:
1;
1,(2);
1,.2,(3),4;
1,....3,...(5),6;
1,.2,....4,......(7),8,.9,10;
1,..........5,..............(11),12;
1,.2,.3,.......6,..................(13),14,15,16;
1,................7,............................(17),18;
1,.2,....4,..........8,................................(19),20,21,22;

Omar E. Pol, Illustration of initial terms: Divisors and pi(x)
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835.

A236627 Number of positive integers <= sqrt(n) not dividing n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 3, 0, 3, 3, 3, 2, 4, 1, 4, 2, 3, 3, 3, 1, 5, 4, 4, 2, 5, 2, 5, 3, 3, 4, 5, 1, 5, 4, 5, 4, 6, 3, 5, 3, 5, 5, 6, 1, 6, 5, 4, 4, 6, 4, 7, 5, 6, 4, 7, 2, 7, 6, 5, 5, 6, 4, 7, 3, 6, 7, 8, 3, 7, 7, 7, 5, 8, 3, 7, 6, 7, 7, 7, 3, 8, 6, 6, 5, 9

Offset: 1

Omar E. Pol, Feb 05 2014

nonn

It appears that the indices of the zeros in the sequence are in A018253.

Omar E. Pol, , Three illustrations of A000005 giving the structure of this sequence, (2013)

Cf. A000005, A000196, A010052, A018253, A038548, A007491, A160812, A161205, A212119, A228813, A228814, A229940, A229942, A229944, A229950, A229951.

a(n) = sum(i=1, sqrtint(n), (n % i) != 0); \\ Michel Marcus, Mar 16 2014

a(n) = A000196(n) - A038548(n) = A160812(n)/2.

A161416 Partial sums of A056737.

Original entry on oeis.org

0, 1, 3, 3, 7, 8, 14, 16, 16, 19, 29, 30, 42, 47, 49, 49, 65, 68, 86, 87, 91, 100, 122, 124, 124, 135, 141, 144, 172, 173, 203, 207, 215, 230, 232, 232, 268, 285, 295, 298, 338, 339, 381, 388, 392, 413, 459, 461, 461, 466, 480, 489, 541, 544, 550, 551, 567, 594

Offset: 1

Omar E. Pol, Jun 21 2009

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Divisors and pi(x)

Cf. A056737, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161828, A161835, A219729, A219730.

a(n) = A219730(n) - A219729(n). - Tamas Sandor Nagy, Jan 20 2024

Extended beyond a(16) by R. J. Mathar, Aug 01 2009

A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 3, 5, 6, 7, 1, 2, 4, 7, 8, 9, 10, 11, 1, 5, 11, 12, 13, 1, 2, 3, 6, 13, 14, 15, 16, 17, 1, 7, 17, 18, 19, 1, 2, 4, 8, 19, 20, 21, 22, 23, 1, 3, 9, 23, 24, 25, 26, 27, 28, 29, 1, 2, 5, 10, 29, 30, 31

Offset: 0

Omar E. Pol, Jun 30 2009

See also A162190, a sequence with a similar structure.

			Triangle begins:
1,(2);
1,(2),(3);
1,.2.,(3),4,(5);
1,.....3,...(5),6,(7);
1,.2,.....4,......(7),8,.9,10,(11);
1,...........5,...............(11),12,(13);
1,.2,..3,.......6,....................(13),14,15,16,(17);
1,.................7,...............................(17),18,(19);
1,.2,.....4,..........8,....................................(19),20,21,22,(23);

Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835, A162190.

cp's OEIS Frontend

A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

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A236627 Number of positive integers <= sqrt(n) not dividing n.

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A161416 Partial sums of A056737.

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A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

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