A011757 -id:A011757 - cp's OEIS frontend

A092504 a(n) = prime(n) + prime(n^2).

4, 10, 28, 60, 108, 164, 244, 330, 442, 570, 692, 864, 1050, 1236, 1474, 1672, 1938, 2204, 2504, 2812, 3156, 3540, 3886, 4300, 4734, 5152, 5622, 6114, 6590, 7110, 7700, 8292, 8874, 9480, 10080, 10778, 11478, 12212, 12910, 13672, 14506, 15282, 16068

Offset: 1

Giovanni Teofilatto, Apr 05 2004

nonn

			a(1) = 4 because prime(1) = 2 and prime(1) = 2.
a(2) = 10 because prime(2) = 3 and prime(4) = 7.
a(3) = 28 because prime(3) = 5 and prime(9) = 23.

Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Cf. A000040, A011757.

[NthPrime(n^2)+ NthPrime(n): n in [1..40]]; // Vincenzo Librandi, Jul 25 2025

Table[Prime[n] + Prime[n^2], {n, 45}] (* Robert G. Wilson v, Apr 08 2004 *)

a(n) = A000040(n) + A011757(n). - Elmo R. Oliveira, Mar 22 2023

Corrected and extended by Robert G. Wilson v, Apr 08 2004

A243896 a(n) = prime(n^2+1).

Original entry on oeis.org

2, 3, 11, 29, 59, 101, 157, 229, 313, 421, 547, 673, 829, 1013, 1201, 1429, 1621, 1889, 2153, 2441, 2749, 3089, 3463, 3821, 4217, 4639, 5059, 5521, 6011, 6491, 7001, 7577, 8167, 8741, 9343, 9941, 10631, 11329, 12071, 12757, 13513, 14341, 15107, 15881

Offset: 0

Freimut Marschner, Jun 17 2014

nonn

For n>1, the numbers prime(n^2-1), prime(n^2) and prime(n^2+1), that is, A243895(n), A001248(n) and a(n), constitute a triple of successive prime numbers.

			n = 4, n^2 = 16, n^2 + 1 = 17, prime(17) = 59.

Freimut Marschner, Table of n, a(n) for n = 0..97

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2). A011757 (prime(n^2)), A055875 (prime(n^3)), A096327 (prime((prime(n)^2))), A096328 (prime(prime(n)^3)), A038580 (prime(prime(prime(n)))).

Table[Prime[n^2+1],{n,0,50}] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = prime(n^2 + 1) = prime(A000290(n) + 1) = prime(A002522(n)).

A296857 For any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; a(n) = g(f(n) * f(n)) (where i * j denotes the Dirichlet convolution of i and j).

Original entry on oeis.org

1, 2, 7, 16, 23, 126, 53, 512, 2401, 1150, 97, 9072, 151, 5194, 27209, 65536, 227, 388962, 311, 230000, 133931, 23474, 419, 2612736, 279841, 51038, 40353607, 2036048, 541, 12244050, 661, 33554432, 571039, 131206, 1668811, 252047376, 827, 224542, 1447033

Offset: 1

Rémy Sigrist, Dec 21 2017

nonn

This sequence is the main diagonal of A248601.
See A248601 for additional comments.
For any n > 0, gcd(2 * n, a(2 * n)) = 2 * n.

			For n = 12:
- f(12) = (2, 1, 0, 0, ...),
- f(12) * f(12) = (4, 4, 0, 1, 0, 0, ...),
- a(12) = prime(1)^4 * prime(2)^4 * prime(4) = 2^4 * 3^4 * 7 = 9072.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 50000 terms (where the color is function of A001222(n))
Wikipedia, Dirichlet convolution.

Cf. A000040, A001221, A001222, A055396, A061395, A248601.

a(n) = my (f=factor(n), p=apply(primepi, f[,1]~)); prod(i=1, #p, prod(j=1, #p, prime(p[i]*p[j])^(f[i,2]*f[j,2])))

For any n > 0 and k >= 0:
- a(n) = A248601(n, n),
- A001221(a(n)) <= A001221(n)^2,
- A001222(a(n)) = A001222(n)^2,
- A055396(a(n)) = A055396(n)^2,
- A061395(a(n)) = A061395(n)^2,
- a(A000040(n)) = A011757(n),
- a(A000040(n)^k) = A011757(n)^(k^2).

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A067852 Numbers k that divide prime(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 18, 20, 32, 90, 129, 188, 244, 678, 688, 756, 1081, 1322, 1477, 19758, 21351, 29518, 43187, 49978, 50342, 95708, 182326, 248253

Offset: 1

Joseph L. Pe, Feb 15 2002

a(29) > 3*10^5. - Donovan Johnson, Nov 15 2009

Any further terms are > 10^6. - Lucas A. Brown, Sep 27 2024

			6 divides prime(6^2)-1 = 150, so 6 is a term of the sequence.

Lucas A. Brown, Python program.

Cf. A000040, A011757, A067851.

Select[Range[10^4], Mod[Prime[ #^2] - 1, # ] == 0 &]

isok(k) = !((prime(k^2)-1) % k); \\ Michel Marcus, Aug 15 2021

a(20)-a(28) from Donovan Johnson, Nov 15 2009

A109789 a(n) = prime(1^3) + prime(2^3) + prime(3^3) + ... + prime(n^3).

Original entry on oeis.org

2, 21, 124, 435, 1126, 2447, 4756, 8427, 13946, 21865, 32822, 47575, 66978, 91787, 123106, 161979, 209636, 267195, 336226, 418025, 514162, 626453, 756526, 906243, 1077772, 1272815, 1493676, 1742527, 2021958, 2334541, 2682248, 3068341

Offset: 1

Jonathan Vos Post, Aug 14 2005

nonn

Analogy with prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724). If we take the cumulative sum of A055875 including the 0th value of 1, the 3rd value becomes prime(0^3) + prime(1^3) + prime(2^3) + prime(3^3) = 1 + 2 + 19 + 103 = 125 = 5^3.

			a(1) = 2 because prime(1^3) = prime(1) = 2;
a(2) = 21 because prime(1^3) + prime(2^3) = prime(1) + prime(8) = 2 + 19;
a(3) = 124 because prime(1^3) + prime(2^3) + prime(3^3) = prime(1) + prime(8) + prime(27) = 2 + 19 + 103;
a(4) = 435 because prime(1^3) + prime(2^3) = prime(1) + prime(8) + prime(27) + prime(64) = 2 + 19 + 103 + 311.
a(6) = 2 + 19 + 103 + 311 + 691 + 1321 = 2447 (which is prime).
a(28) = 2 + 19 + 103 + 311 + 691 + 1321 + 2309 + 3671 + 5519 + 7919 + 10957 + 14753 + 19403 + 24809 + 31319 + 38873 + 47657 + 57559 + 69031 + 81799 + 96137 + 112291 + 130073 + 149717 + 171529 + 195043 + 220861 + 248851 = 1742527 (which is prime).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A055875, A011757, A109724, A109770.

a(n) = sum(k=1, n, prime(k^3)); \\ Michel Marcus, Apr 17 2021

Cumulative sums of A055875(n) for n>0.

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

A161846 Numerator of the ratio (prime((n+1)^2) - prime(n^2))/prime(n).

Original entry on oeis.org

5, 16, 6, 44, 54, 76, 84, 108, 122, 120, 166, 182, 184, 234, 192, 260, 264, 294, 304, 342, 378, 342, 408, 426, 414, 468, 488, 474, 516, 576, 588, 576, 604, 590, 696, 694, 728, 694, 756, 828, 774, 776, 870, 862, 852, 1010, 922, 998, 916, 1020, 1032, 1110, 1104

Offset: 1

Daniel Tisdale, Jun 20 2009

Note that prime(n^2) = A011757(n) and prime(n) = A000040(n).

Conjecture: the sequence of fractions (prime((n+1)^2) - prime(n^2)) / prime(n) converges to 4. There are several "heuristic demonstrations" but no proofs.

			The first few fractions are 5/2, 16/3, 6/1, 44/7, 54/11, ...= A161846/A161847.

Cf. A000040, A011757, A161847 (denominators).

A161846 := proc(n) ( ithprime((n+1)^2)-ithprime(n^2))/ithprime(n) ; numer(%) ; end: seq(A161846(n),n=1..25) ; # R. J. Mathar, Jun 22 2009

Table[(Prime[(n+1)^2]-Prime[n^2])/Prime[n],{n,60}]//Numerator (* Harvey P. Dale, Oct 24 2017 *)

a(n) = numerator((prime((n+1)^2) - prime(n^2))/prime(n)); \\ Michel Marcus, May 14 2020

a(n) = numerator((A011757(n+1) - A011757(n))/A000040(n)). - Petros Hadjicostas, May 13 2020

Extended by Ray Chandler, May 06 2010

Various sections edited by Petros Hadjicostas, May 13 2020

A161847 Denominator of the ratio (prime((n+1)^2) - prime(n^2))/prime(n).

Original entry on oeis.org

2, 3, 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Daniel Tisdale, Jun 20 2009

Conjecture: the sequence (prime((n+1)^2) - prime(n^2))/prime(n) converges to 4.

			The first few fractions are 5/2, 16/3, 6/1, 44/7, 54/11, ... = A161846/A161847.

Cf. A000040, A011757, A161846 (numerators).

a(n) = denominator((prime((n+1)^2) - prime(n^2))/prime(n)); \\ Michel Marcus, May 14 2020

a(n) = denominator((A011757(n+1) - A011757(n))/A000040(n)). - Petros Hadjicostas, May 13 2020

Keyword:frac added by R. J. Mathar, Jun 30 2009
Extended by Ray Chandler, May 06 2010
Various sections edited by Petros Hadjicostas, May 13 2020

A243892 a(n) = prime(k) with k = n^2 + prime(n)^2.

Original entry on oeis.org

11, 41, 139, 313, 839, 1259, 2273, 2953, 4493, 7417, 8689, 12659, 15881, 17837, 21683, 28097, 35401, 38321, 46993, 53353, 56909, 67499, 75277, 87539, 105167, 115061, 120431, 130817, 136559, 147881, 189127, 202493

Offset: 1

Freimut Marschner, Jun 14 2014

nonn

			n = 1, prime(1^2+prime(1)^2) = prime(1 + 2^2) = prime(5) = 11.
n = 2, prime(2^2+prime(2)^2) = prime(4 + 3^2) = prime(13) = 41.

Freimut Marschner, Table of n, a(n) for n = 1..63

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).

Also A011757 is prime(n^2), A096327 is prime(prime(n)^2).

a(n) = prime((n^2 + prime(n)^2)) = prime(A106587(n)).

A243895 a(n) = prime(n^2-1).

Original entry on oeis.org

5, 19, 47, 89, 149, 223, 307, 409, 523, 659, 823, 997, 1187, 1423, 1613, 1877, 2141, 2423, 2731, 3079, 3457, 3797, 4201, 4621, 5039, 5507, 5987, 6473, 6991, 7561, 8147, 8731, 9337, 9929, 10613, 11317, 12043, 12739, 13487, 14323, 15091, 15859, 16741

Offset: 2

Freimut Marschner, Jun 17 2014

nonn

The prime numbers prime(k-1), prime(k) = A001248 and prime(k+1) = A243896 with k = n^2 are building a triple of successive prime numbers. Remark: prime(n^2-1) is not defined for n=1.

			n = 3, n^2 = 9, n^2-1 = 8, prime(8) = 19.

Freimut Marschner, Table of n, a(n) for n = 2..97

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A011757 (prime(n^2)), A055875 (prime(n^3)), A096327 (prime((prime(n)^2))), A096328 (prime(prime(n)^3)), A038580 (prime(prime(prime(n)))).

Table[Prime[n^2-1],{n,2,50}] (* Harvey P. Dale, Jul 16 2025 *)

a(n) = prime(n^2-1) = prime(A000290(n) - 1) = prime(A005563(n-1)).

cp's OEIS Frontend

A092504 a(n) = prime(n) + prime(n^2).

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A243896 a(n) = prime(n^2+1).

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A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

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A067852 Numbers k that divide prime(k^2) - 1.

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A109789 a(n) = prime(1^3) + prime(2^3) + prime(3^3) + ... + prime(n^3).

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A161846 Numerator of the ratio (prime((n+1)^2) - prime(n^2))/prime(n).

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