A034953 -id:A034953 - cp's OEIS frontend

A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 2, 0, 0, 4, 0, 1, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 0, 4, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2, 0, 4, 0, 0, 6, 0, 0, 2, 0, 2, 4, 2, 1, 2, 2, 0, 2, 0, 2, 6, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 2, 2, 0, 6, 0, 1, 4, 0, 4, 2, 0, 0, 2

Offset: 0

Vladimir Shevelev, Jun 02 2009

a(n)=0 iff n is in A067076, i.e., 2n+3 is prime; a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1<=k<=n/3.

			Since for n=3 we have 2n+3=9 and only nontrivial divisor of 9 is 3, then a(3)=1.

Cf. A067076, A034953, A054269, A082749, A006254.

Cf. A160973. - Vladimir Shevelev, Jun 07 2009

[NumberOfDivisors(2*n+3)-2: n in [0..100]]; // Vincenzo Librandi, Feb 08 2016

a(n) = numdiv(2*n+3) - 2; \\ Michel Marcus, Feb 08 2016

For n>=1, a(n)=A160973(n)+A079978(n). [Vladimir Shevelev, Jun 07 2009]

a(n) = A070824(2n+3).

Edited by Charles R Greathouse IV, Oct 12 2009

More terms from Michel Marcus, Feb 08 2016

A161624 Sum of all numbers from n to n-th prime.

Original entry on oeis.org

3, 5, 12, 22, 56, 76, 132, 162, 240, 390, 441, 637, 783, 855, 1023, 1311, 1634, 1738, 2107, 2366, 2491, 2929, 3233, 3729, 4453, 4826, 5005, 5400, 5589, 6006, 7663, 8150, 8925, 9169, 10580, 10846, 11737, 12663, 13287, 14271, 15290, 15610, 17433, 17775

Offset: 1

Juri-Stepan Gerasimov, Jun 15 2009

nonn

			First prime is 2, so a(1) = 1+2 = 3; fifth prime is 11, so a(5) = 5+6+7+8+9+10+11 = 56.

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

Cf. A000027, A000040, A000217, A034953, A161570.

[ &+[n..NthPrime(n)]: n in [1..44] ]; // Klaus Brockhaus, Jun 14 2009

snos[n_]:=Module[{pr=Prime[n]},((pr+n)(pr-n+1))/2]; Array[snos, 50] (* Harvey P. Dale, Jun 05 2012 *)

a(n) = Sum_{k=n..prime(n)} k.

a(n) = A034953(n) - A000217(n-1). - Michel Marcus, Feb 02 2018

Edited and extended by Klaus Brockhaus, Jun 15 2009

A231989 Least prime p such that f(0),...,f(n) are all primes, where f(0) = p, then f(i+1) = triangular(f(i))+1.

Original entry on oeis.org

3, 3, 43, 236367611, 31542795419

Offset: 1

Alex Ratushnyak, Nov 16 2013

Triangular(p) = p*(p+1)/2 (see A034953).

			a(2) = 3 because 3 is the least prime such that the following are two primes:
  p1 = 3 * 4 / 2 + 1 = 7.
  p2 = 7 * 8 / 2 + 1 = 29.
a(3) = 43 because 43 is the least prime such that the following are three primes:
  p1 = 43 * 44 / 2 + 1 = 947.
  p2 = 947 * 948 / 2 + 1 = 448879.
  p3 = 448879 * 448880 / 2 + 1 = 100746402761.

Cf. A000040, A000217, A034953, A231847, A231988.

A291199 Primes p such that phi(p*(p+1)/2) is a triangular number (A000217).

Original entry on oeis.org

2477, 44287823, 58192759, 110369351, 664009019, 2574106333, 6870260119, 7423240007, 60370077539, 188271042191, 235399729007, 236767359977, 305214702643, 717724689959

Offset: 1

Altug Alkan, Aug 20 2017

a(15) > 10^12. - Giovanni Resta, Aug 21 2017

			Prime number 2477 is a term since phi(2477*2478/2) = 1856*1857/2.

Cf. A000010, A000217, A034953, A086700.

isok(n) = isprime(n) && ispolygonal(eulerphi(n*(n+1)/2), 3);

is(n) = ispolygonal(eulerphi(n\2+1)*(n-1), 3) && isprime(n) \\ Charles R Greathouse IV, Aug 22 2017

from _future_ import division
from sympy.ntheory.primetest import is_square
from sympy import totient, nextprime
A291199_list, p = [], 3
while p < 10**8:
    if is_square(8*(p-1)*totient((p+1)//2)+1):
        A291199_list.append(p)
    p = nextprime(p) # Chai Wah Wu, Aug 22 2017

a(5)-a(14) from Giovanni Resta, Aug 21 2017

A307950 Primes that are the sum of two prime-indexed triangular numbers.

Original entry on oeis.org

31, 43, 97, 157, 181, 193, 281, 367, 463, 499, 587, 709, 769, 1051, 1381, 1459, 1621, 1831, 1861, 2081, 2281, 2293, 2377, 2473, 2647, 2707, 2713, 2729, 2767, 2837, 3019, 3163, 3251, 3259, 3313, 3709, 3863, 4021, 4447, 4591, 4759, 4943, 4951, 5051, 5179, 5647, 5791, 5861, 5869, 5881, 6217, 6271

Offset: 1

J. M. Bergot and Robert Israel, May 07 2019

nonn

Primes of the form A034953(i) + A034953(j).

There are primes with more than one expression of this form; e.g., 18749 = A034953(4) + A034953(44) = A034953(19) + A034953(42).

			a(3) = 97 is a term because 97 = 6 + 91 is prime where 6=A000217(3) and 91=A000217(13) are in A034953.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A034953, A117112.

A034953:= map(t ->t*(t+1)/2, [seq(ithprime(i),i=1..100)]):
A:= select(t -> t <= A034953[-1]+3 and isprime(t), {seq(seq(A034953[i]+A034953[j],j=i+1..100),i=1..99)}):
sort(convert(A,list));

A346494 Heptagonal numbers (A000566) with prime indices (A000040).

Original entry on oeis.org

7, 18, 55, 112, 286, 403, 697, 874, 1288, 2059, 2356, 3367, 4141, 4558, 5452, 6943, 8614, 9211, 11122, 12496, 13213, 15484, 17098, 19669, 23377, 25351, 26368, 28462, 29539, 31753, 40132, 42706, 46717, 48094, 55279, 56776, 61387, 66178, 69472, 74563, 79834

Offset: 1

Dumitru Damian, Aug 22 2021

			a(1) = Heptagonal(prime(1)) = A000566(2) = 2*(5*2-3)/2 = 7;
a(2) = Heptagonal(prime(2)) = A000566(3) = 3*(5*3-3)/2 = 18;
a(3) = Heptagonal(prime(3)) = A000566(5) = 5*(5*5-3)/2 = 55.

Cf. A000566, A034953, A001248, A116995, A117961, A267144.

A346494[n_] := PolygonalNumber[7, Prime[n]]; Table[A346494[n], {n, 1, 41}] (* Robert P. P. McKone, Aug 22 2021 *)

a(n) = my(p=prime(n)); p*(5*p-3)/2; \\ Michel Marcus, Sep 16 2021

from sympy import primerange
print([p*(5*p-3)//2 for p in primerange(1, 180)]) # Michael S. Branicky, Aug 22 2021

A = [int(p*(5*p-3)/2) for p in range(0,10^3) if p in Primes()]

a(n) = A000566(A000040(n)) = prime(n)*(5*prime(n)-3)/2.

A098997 (1/30)*(p(p+1)(2p+1)(3p^2+3p-1)) where p is prime.

Original entry on oeis.org

17, 98, 979, 4676, 39974, 89271, 327369, 562666, 1431244, 4463999, 6197520, 14822755, 24607093, 31137590, 48343448, 87633963, 149111998, 175917839, 280200834, 373671012, 428943109, 635050664, 811727882, 1148417997

Offset: 1

Parthasarathy Nambi, Nov 05 2004

nonn

			If p=2, (1/30)*(2*(2+1)*(2*2+1)*(3*2^2+3*2-1)) = 17

Similar to A034953

Table[(p(p+1)(2p+1)(3p^2+3p-1))/30,{p,Prime[Range[30]]}] (* Harvey P. Dale, Mar 10 2019 *)

More terms from Klaus Brockhaus, Nov 09 2004

A109923 a(n) = lcm(1,2,3,...,prime(n))/(1 + 2 + ... + prime(n)).

Original entry on oeis.org

1, 4, 15, 420, 3960, 80080, 1225224, 19399380, 5354228880, 145568097675, 7600186994400, 254425307479200, 9957281351799600, 392482839950100900, 114779426083185063200, 5474978624167927514640, 312603618218620377448800

Offset: 2

Amarnath Murthy, Jul 16 2005

			a(4)=15 because the 4th prime is 7 and lcm(1,2,3,4,5,6,7)/(1+2+3+4+5+6+7) = 420/28 = 15.

Cf. A006254, A034953, A056604, A099795, A109922.

a:=n->lcm(seq(i,i=1..ithprime(n)))/sum(j,j=1..ithprime(n)): seq(a(n),n=2..20); # Emeric Deutsch, Jul 16 2005

a(n) = lcm(vector(prime(n), k, k))/sum(k=1, prime(n), k); \\ Michel Marcus, Mar 07 2018

a(n) = A099795(n)/A006254(n-1). - Andrey Zabolotskiy, Mar 07 2018

a(n) = A056604(n)/A034953(n). - Michel Marcus, Mar 07 2018

More terms from Emeric Deutsch, Jul 16 2005

A122963 Triangular numbers with semiprime indices.

Original entry on oeis.org

10, 21, 45, 55, 105, 120, 231, 253, 325, 351, 561, 595, 630, 741, 780, 1081, 1225, 1326, 1540, 1653, 1711, 1953, 2145, 2415, 2775, 3003, 3403, 3655, 3741, 3828, 4186, 4371, 4465, 4560, 5671, 6216, 6670, 7021, 7140, 7381, 7503, 7626, 8385, 8911, 9045, 10011

Offset: 1

Jonathan Vos Post, Oct 26 2006

nonn

Semiprime analog of A034953 Triangular numbers (A000217) with prime indices.

			a(1) = semiprime(1)*(semiprime(1)+1)/2 = 4*5/2 = 10.
a(2) = semiprime(2)*(semiprime(2)+1)/2 = 6*7/2 = 21.
a(3) = semiprime(3)*(semiprime(3)+1)/2 = 9*10/2 = 45.

Cf. A000217, A001358, A034953.

a(n) = A000217(A001358(n)) = A001358(n)*(A001358(n)+1)/2.

a(41)-a(46) from Giovanni Resta, Jun 13 2016

A267217 10-gonal (or decagonal) numbers with prime indices.

Original entry on oeis.org

10, 27, 85, 175, 451, 637, 1105, 1387, 2047, 3277, 3751, 5365, 6601, 7267, 8695, 11077, 13747, 14701, 17755, 19951, 21097, 24727, 27307, 31417, 37345, 40501, 42127, 45475, 47197, 50737, 64135, 68251, 74665, 76867, 88357, 90751, 98125, 105787, 111055, 119197, 127627, 130501, 145351, 148417, 154645

Offset: 1

Ilya Gutkovskiy, Jan 12 2016

Eric Weisstein's World of Mathematics, Decagonal Number
Eric Weisstein's World of Mathematics, Prime Number

Cf. A000040, A001107, A001248, A034953, A116995, A117961, A267144.

Table[Prime[n] (4 Prime[n] - 3), {n, 1, 45}]
Module[{nn=200,pn},pn=PolygonalNumber[10,Range[nn]];Table[pn[[p]],{p,Prime[ Range[PrimePi[nn]]]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 27 2020 *)

lista(nn) = forprime(p=2, nn, print1(p*(4*p-3), ", ")); \\ Altug Alkan, Jan 12 2016

a(n) = prime(n)*(4*prime(n) - 3) = A000040(n)*(4*A000040(n) - 4).
a(n) = A001107(A000040(n)).
a(n) = sigma_0(48^(prime(n) - 1)) = A000005(A009992(A000040(n) - 1)).

cp's OEIS Frontend

A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

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A161624 Sum of all numbers from n to n-th prime.

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A231989 Least prime p such that f(0),...,f(n) are all primes, where f(0) = p, then f(i+1) = triangular(f(i))+1.

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A291199 Primes p such that phi(p*(p+1)/2) is a triangular number (A000217).

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A307950 Primes that are the sum of two prime-indexed triangular numbers.

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Maple

A346494 Heptagonal numbers (A000566) with prime indices (A000040).

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A098997 (1/30)*(p(p+1)(2p+1)(3p^2+3p-1)) where p is prime.

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A109923 a(n) = lcm(1,2,3,...,prime(n))/(1 + 2 + ... + prime(n)).

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