A034953 -id:A034953 - cp's OEIS frontend

A117962 Partial sums of hexagonal numbers with prime indices.

6, 21, 66, 157, 388, 713, 1274, 1977, 3012, 4665, 6556, 9257, 12578, 16233, 20604, 26169, 33072, 40453, 49364, 59375, 69960, 82363, 96058, 111811, 130532, 150833, 171948, 194739, 218392, 243817, 275948, 310139, 347540, 386043, 430296, 475747

Offset: 1

Jonathan Vos Post, Apr 05 2006

There are no prime hexagonal numbers. The n-th hexagonal number A000384(n) = n*(2*n-1) is semiprime iff both n and 2*n-1 are prime iff A000384(n) is an element of A001358 iff n is an element of A005382.

			a(4) = hexagonal(2) + hexagonal(3) + hexagonal(5) + hexagonal(7) = 6 + 15 + 45 + 91 = 157 is prime.
a(12) = 6 + 15 + 45 + 91 + 231 + 325 + 561 + 703 + 1035 + 1653 + 1891 + 2701 = 9257 is prime.
a(26) = 150833 is prime.

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

See also: A034953 Triangular numbers (A000217) with prime indices. A001248 Squares of primes. A116995 Pentagonal numbers with prime indices. A000384 Hexagonal numbers: n(2n-1). A117961 Hexagonal numbers with prime indices. A117965 Prime partial sums of hexagonal numbers with prime indices.

Cf. A000005, A000040, A000384, A001021, A034953, A001248, A116995, A117961, A117965.

Accumulate[Table[n(2n-1),{n,Prime[Range[50]]}]] (* Harvey P. Dale, Jan 30 2014 *)

a(n) = SUM[i=1..n] A117961(i). a(n) = SUM[i=1..n] A000040(i)*(2*A000040(i)-1). a(n) = SUM[i=1..n] A000384(prime(n)). a(n) = Partial sum of number of divisors of 12^(prime(n)-1) = SUM[i=1..n] A000005(A001021(A000040(n)-1)).

A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.

Original entry on oeis.org

1, 3, 6, 9, 15, 18, 28, 27, 36, 45, 66, 54, 91, 84, 90, 81, 153, 108, 190, 135, 168, 198, 276, 162, 225, 273, 216, 252, 435, 270, 496, 243, 396, 459, 420, 324, 703, 570, 546, 405, 861, 504, 946, 594, 540, 828, 1128, 486, 784, 675, 918, 819, 1431, 648, 990, 756

Offset: 1

Jonathan Vos Post, Oct 10 2007

There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A003958, A003959, A003960, A003961, A003964, A034953, A061142, A167338.

f[p_, e_] := (p*(p + 1)/2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 24 2022 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],binomial(f[i,1]+1,2)^f[i,2]) /* Charles R Greathouse IV, Sep 09 2010 */

for(n=1, 100, print1(direuler(p=2, n, 1 + (p^2 + p) / (2/X - p^2 - p))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023

a((p_1)^(e_1)*(p_2)^(e_2)*...*(p_k)^(e_k)) = T(p_1)^(e_1)*T(p_2)^(e_2)*...*T(p_k)^(e_k), where T(i) = A000217(i). a(prime(i)) = A034953(i).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - 2/(p*(p+1)))^(-1) = 2.12007865309570462566... . - Amiram Eldar, Dec 24 2022
Dirichlet g.f.: Product_{p prime} (1 + (p^2 + p) / (2*p^s - p^2 - p)). - Vaclav Kotesovec, Apr 05 2023
a(n) = A167338(n)/A061142(n). - Vaclav Kotesovec, Jan 28 2025
Conjecture: Sum_{k=1..n} a(k) = O(n^3/log(n)). - Vaclav Kotesovec, Jan 28 2025

A144549 Triangular numbers p*(p+1)/2 with p prime such that 1+(number of prime factors of p+1) is prime.

Original entry on oeis.org

3, 6, 15, 91, 276, 703, 1431, 1770, 1891, 2701, 3486, 4005, 5356, 8646, 9730, 11175, 11476, 12403, 18721, 19503, 24976, 25878, 27261, 28680, 38503, 43071, 47278, 49141, 60378, 61075, 64620, 72010, 75855, 79003, 88831, 98346, 104653, 106491

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2008

nonn

Triangular numbers n(n+1)/2 such that n and A073093(n+1) are both prime.

			3 has one prime factor; 1+1 = 2 is prime, hence 2*3/2 = 3 is in the sequence.
14 = 2*7 has two prime factors; 1+2 = 3 is prime, hence 13*14/2 = 91 is in the sequence.
24 = 2*2*2*3 has four prime factors; 1+4 = 5 is prime, hence 23*24/2 = 276 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A000040 (prime numbers), A001222 (number of prime divisors of n), A073093.

Subsequence of A034953. - R. J. Mathar, Jan 03 2009

[ p*(p+1)/2: p in PrimesUpTo(490) | IsPrime(1 + &+[ f[2]: f in Factorization(p+1) ]) ];

aQ[n_] := PrimeQ[n] && PrimeQ[PrimeOmega[n + 1] + 1]; p = Select[Range[470], aQ]; p*(p + 1)/2 (* Amiram Eldar, Aug 31 2019 *)

Edited, corrected (3 inserted) and extended beyond a(16) by Klaus Brockhaus, Jan 05 2009

3 inserted and extended by R. J. Mathar, Jan 03 2009

A267144 Octagonal numbers with prime indices.

Original entry on oeis.org

8, 21, 65, 133, 341, 481, 833, 1045, 1541, 2465, 2821, 4033, 4961, 5461, 6533, 8321, 10325, 11041, 13333, 14981, 15841, 18565, 20501, 23585, 28033, 30401, 31621, 34133, 35425, 38081, 48133, 51221, 56033, 57685, 66305, 68101, 73633, 79381, 83333, 89441, 95765

Offset: 1

Ilya Gutkovskiy, Jan 11 2016

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

Cf. A000040, A000567, A001248, A034953, A116995, A117961.

[NthPrime(n)*(3*NthPrime(n)-2): n in [1..50]]; // Vincenzo Librandi, Jan 12 2016

Table[Prime[n] (3 Prime[n] - 2), {n, 1, 45}]

lista(nn) = forprime(p=2, nn, print1(p*(3*p-2), ", ")); \\ Altug Alkan, Jan 11 2016

from sympy import prime
def a(n): p = prime(n); return p*(3*p-2)
print([a(n) for n in range(1, 42)]) # Michael S. Branicky, Aug 21 2021

a(n) = prime(n)*(3*prime(n) - 2) = A000040(n)*(3*A000040(n) - 2).
a(n) = A000567(A000040(n)).
a(n) = sigma_0(24^(prime(n) - 1)) = A000005(A009968(A000040(n) - 1)).

A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

Original entry on oeis.org

1, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 30, 8, 136, 9, 171, 20, 63, 55, 253, 12, 50, 78, 27, 42, 406, 30, 465, 16, 165, 136, 210, 18, 666, 171, 234, 40, 820, 63, 903, 110, 90, 253, 1081, 24, 147, 50, 408, 156, 1378, 27, 550, 84, 513, 406, 1711, 60, 1830, 465, 189

Offset: 1

Benoit Cloitre, Feb 12 2004

a(2n+1) is the conjectured value of the length of period of sequence of Genocchi number of first kind read modulo (2n + 1) (cf. A001469).

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

Cf. A023900, A034444. - R. J. Mathar, Feb 08 2011

Cf. A299822, A000010, A007947, A001221.

A023900 := proc(n) add( d*numtheory[mobius](d),d=numtheory[divisors](n)) ; end proc:
A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A076479 := proc(n) (-1)^A001221(n) ; end proc:
A034444 := proc(n) 2^A001221(n) ;end proc:
A090780 := proc(n) n/A076479(n)/A034444(n) *A023900(n); end proc:
seq(A090780(n),n=1..20) ; # R. J. Mathar, Apr 14 2011

a[n_] := Module[{f, p, e}, fun[p_, e_] := (p - 1)*p^e/2;
If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* Amiram Eldar, Nov 23 2018 *)

a(n) = my(f=factor(n)[,1]); n*prod(k=1, #f, (f[k]-1)/2); \\ Michel Marcus, May 26 2019

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]/2) \\ Jianing Song, Aug 11 2023

a(n) = (n/(-2)^omega(n))*(Sum_{d|n} d*mu(d)) = n*A023900(n)/(A076479(n)*A034444(n)).
a(n) = n*A173557(n)/2. - R. J. Mathar, Apr 14 2011
From Jianing Song, Nov 22 2018: (Start)
Multiplicative with a(p^e) = (p - 1)*p^e/2 = A000217(p-1)*p^(e-1).
a(n) = A299822(n)/2^A001221(n).
a(prime(n)) = A034953(n).
a(n) is odd if and only if n = A004614(k) or 2*A004614(k). (End)
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2/(p-1)^2) = 5.72671092223951683002237367406848393189560038246828458038126468772919585... - Vaclav Kotesovec, Sep 20 2020
From Jianing Song, Aug 11 2023: (Start)
a(n) = phi(n) * Product_{p|n, p prime} (p/2), where phi = A000010.
Equals A000010(n)*A007947(n)/2^A001221(n). (End)

A116911 Prime partial sums of pentagonal numbers with prime indices.

Original entry on oeis.org

5, 17, 4957, 129277, 2826443, 3861083, 5126483, 9451573, 19811083, 53751743, 68136617, 98729003, 264616831, 388771421, 498157871, 608312141, 682548511, 779346653, 918754301, 1174179079, 1700023891, 2056298683, 2149703411

Offset: 1

Jonathan Vos Post, Apr 03 2006

See also: A116994 Prime partial sums of triangular numbers with prime indices. A116995 Pentagonal numbers with prime indices.

			a(1) = Sum_{i=1..1} prime(i)*(3*prime(i)-1)/2 = P(2) = 5.
a(2) = Sum_{i=1..2} prime(i)*(3*prime(i)-1)/2 = P(2) + P(3) = 17.
a(3) = Sum_{i=1..11} prime(i)*(3*prime(i)-1)/2 = P(2) + P(3) + P(5) + P(7) + P(11) + P(13) + P(17) + P(19) + P(23) + P(29) + P(31) = 4957.
a(4) = P(2) + ... + P(103) = 129277.

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

Cf. A000040, A000217, A034953, A085739, A116994, A116995.

P:=n->n*(3*n-1)/2: seq(P(n),n=0..10): a:=proc(n) if isprime(sum(P(ithprime(j)),j=1..n))=true then sum(P(ithprime(j)),j=1..n) else fi end: seq(a(n),n=1..600); # Emeric Deutsch, Apr 15 2006

Module[{nn=4000,pn,pr},pn=PolygonalNumber[5,Range[nn]];pr=Table[If[ PrimeQ[ n],1,0],{n,nn}];Select[Accumulate[Pick[pn,pr,1]],PrimeQ]] (* Harvey P. Dale, Jan 27 2020 *)

A000040 INTERSECTION {Partial sums of A116995(n)}. (Sum_{i=1..k} A000326(A000040(i))) iff in A000040. (Sum_{i=1..k} prime(i)*(3*prime(i)-1)/2) iff in A000040.

More terms from Emeric Deutsch, Apr 15 2006

A123907 a(n) = T(p(n)) - p(T(n)) = Commutator[triangular numbers, primes] at n.

Original entry on oeis.org

1, 1, 2, -1, 19, 18, 46, 39, 79, 178, 179, 306, 394, 375, 469, 662, 887, 872, 1127, 1265, 1248, 1553, 1703, 2018, 2600, 2780, 2763, 2987, 2958, 3134, 4587, 4849, 5380, 5373, 6518, 6503, 7100, 7725, 8089, 8750, 9431, 9452, 10859, 10892, 11260, 11219, 13275, 15485, 15947, 15908, 16358, 17257, 17222, 19189

Offset: 1

Jonathan Vos Post, Oct 28 2006

Asymptotically p(T(n)) ~ (n^2 + n)*(log n) and T(p(n)) ~ (1/2)(n log n)^2, hence asymptotically a(n) ~ (1/2)(n log n)^2 - (n^2 + n)*(log n) = O((n^2)(log n)^2). a(4) = -1 should be the only negative value.

			a(1) = T(p(1)) - p(T(1)) = T(2) - p(1) = 3 - 2 = 1.
a(2) = T(p(2)) - p(T(2)) = T(2) - p(1) = 6 - 5 = 1.
a(3) = T(p(3)) - p(T(3)) = T(2) - p(1) = 15 - 13 = 1.
a(4) = T(p(4)) - p(T(4)) = T(2) - p(1) = 28 - 29 = -1.
a(5) = T(p(5)) - p(T(5)) = T(2) - p(1) = 66 - 47 = 19.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000217, A011756, A034953.

P:=NthPrime; B:=Binomial; [B(P(n)+1,2) - P(B(n+1,2)): n in [1..60]]; // G. C. Greubel, Aug 06 2019

A000040 := proc(n) ithprime(n) ; end; A000217 := proc(n) n*(n+1)/2 ; end; A123907 := proc(n) A000217(A000040(n))-A000040(A000217(n)) ; end ; for n from 1 to 80 do printf("%d,",A123907(n)) ; end; # R. J. Mathar, Jan 13 2007

With[{B=Binomial, P=Prime}, Table[B[P[n]+1, 2] -P[B[n+1, 2]], {n, 60}]] (* G. C. Greubel, Aug 06 2019 *)

vector(60, n, p=prime; b=binomial; b(p(n)+1,2) - p(b(n+1,2)) ) \\ G. C. Greubel, Aug 06 2019

p=nth_prime; b=binomial; [b(p(n)+1,2) - p(b(n+1,2)) for n in (1..60)] # G. C. Greubel, Aug 06 2019

a(n) = T(p(n)) - p(T(n)) where T(i) = i*(i+1)/2, p(i) = prime(i).
a(n) = A000217(A000040(n)) - A000040(A000217(n)).
a(n) = p(n)*(p(n)+1)/2 - p(n*(n+1)/2) where p(i) = prime(i).
a(n) = A034953(n) - A011756(n).

More terms from R. J. Mathar, Jan 13 2007

A139117 Triangular numbers (A000217) with indices A000043.

Original entry on oeis.org

3, 6, 15, 28, 91, 153, 190, 496, 1891, 4005, 5778, 8128, 135981, 184528, 818560, 2427706, 2602621, 5176153, 9046131, 9783676, 46943205, 49416711, 62871291, 198751953, 235477551, 269340445, 990013753, 3718970646, 6105511756, 8718535225, 23347768186, 286403014380

Offset: 1

Omar E. Pol, May 08 2008

nonn

			a(4) = 28 because A000043(4) = 7 and the 7th triangular number A000217(7) is 28.

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A000217, A000396, A034953.

Map[#*(#+1)/2 &, MersennePrimeExponent[Range[48]]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A000217(A000043(n)).

A144519 Triangular numbers n*(n+1)/2 with n prime and n+1 nonprime.

Original entry on oeis.org

6, 15, 28, 66, 91, 153, 190, 276, 435, 496, 703, 861, 946, 1128, 1431, 1770, 1891, 2278, 2556, 2701, 3160, 3486, 4005, 4753, 5151, 5356, 5778, 5995, 6441, 8128, 8646, 9453, 9730, 11175, 11476, 12403, 13366, 14028, 15051, 16110, 16471, 18336, 18721, 19503, 19900, 22366, 24976, 25878

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2008

nonn

This is A034953 without the 3. [From R. J. Mathar, Feb 21 2009]

			If n=3(prime) and n=4(nonprime), then 3*4/2=6=a(1). If n=5(prime) and n=6(nonprime), then 5*6/2=15=a(2). If n=7(prime) and n=8(nonprime), then 7*8/2=28=a(3). If n=11(prime) and n=12(nonprime), then 11*12/2=66=a(4). If n=13(prime) and n=14(nonprime), then 13+14/2=91=a(5), etc.

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

Cf. A000040, A000217, A141468, A144291.

Table[(p(p+1))/2,{p,Prime[Range[2,50]]}] (* Harvey P. Dale, Dec 28 2023 *)

Corrected definition. Inserted 2701, extended beyond 11175. - R. J. Mathar, Dec 19 2008

A160973 a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1 <= k <= (n-1)/5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 1, 1, 0, 2, 1, 0, 0, 3, 2, 0, 1, 0, 0, 3, 2, 0, 2, 0, 2, 1, 0, 2, 1, 2, 0, 3, 0, 0, 5, 0, 0, 1, 0, 2, 3, 2, 1, 1, 2, 0, 1, 0, 2, 5, 0, 0, 1, 2, 2, 3, 0, 0, 3, 2, 0, 1, 2, 0, 5, 0, 1, 3, 0, 4, 1, 0, 0, 1

Offset: 0

Vladimir Shevelev, Jun 01 2009, Jun 07 2009

nonn

If n is different from 3, then a(n)=0 iff n is in A067076, i.e., 2n+3 is prime.

Cf. A067076, A034953, A054269, A082749, A006254, A161116.

a[n_] := Length[Select[Range[Floor[(n-1)/5]], IntegerQ[(n-3#)/(2#+1)] &]]; Array[a, 100, 0] (* Amiram Eldar, Dec 15 2018 *)

a(n) = sum(k=1, (n-1)/5, frac((n-3*k)/(2*k+1)) == 0); \\ Michel Marcus, Dec 15 2018

Edited by N. J. A. Sloane, Jun 07 2009

a(44) corrected and more terms from Michel Marcus, Dec 15 2018

cp's OEIS Frontend

A117962 Partial sums of hexagonal numbers with prime indices.

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A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.

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A144549 Triangular numbers p*(p+1)/2 with p prime such that 1+(number of prime factors of p+1) is prime.

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A267144 Octagonal numbers with prime indices.

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A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

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A116911 Prime partial sums of pentagonal numbers with prime indices.

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A123907 a(n) = T(p(n)) - p(T(n)) = Commutator[triangular numbers, primes] at n.

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