A038580 -id:A038580 - cp's OEIS frontend

A270794 The prime/nonprime compound sequence BAA.

6, 9, 18, 26, 45, 57, 81, 91, 112, 143, 165, 203, 228, 244, 267, 303, 345, 354, 411, 437, 454, 495, 530, 564, 623, 668, 687, 714, 728, 749, 856, 893, 931, 959, 1032, 1054, 1104, 1158, 1185, 1233, 1268, 1298, 1372, 1392, 1425, 1445, 1539, 1672, 1698, 1714, 1742, 1773, 1802, 1886, 1914, 1966, 2031, 2050, 2104

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A270796 The prime/nonprime compound sequence BBA.

Original entry on oeis.org

8, 10, 15, 20, 27, 32, 38, 40, 49, 58, 63, 72, 78, 82, 88, 99, 110, 114, 121, 125, 129, 140, 146, 155, 166, 172, 175, 183, 185, 189, 212, 217, 225, 230, 245, 248, 258, 265, 272, 279, 289, 292, 306, 309, 315, 319, 334, 355, 360, 362, 368, 375, 377, 393, 402, 408, 416, 420, 427, 435, 438, 452, 473, 478, 482, 486, 507

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 9, 2, 8, 10, 14, 22, 3, 9, 15, 19, 23, 29, 41, 39, 63, 69, 2, 6, 16, 16, 24, 42, 48, 52, 54, 52, 74, 84, 88, 102, 114, 122, 134, 152, 156, 166, 168, 1, 7, 13, 19, 23, 31, 71, 71, 73, 73, 65, 77, 91, 79, 91, 109, 115, 125, 137, 149, 155, 185, 197, 203, 197, 235

Offset: 1

Labos Elemer, Oct 08 2002

			a(4) = 3 since prime(prime(4)) (mod prime(4)) = prime(7) (mod 7) = 17 (mod 7) = 3. - _Michael De Vlieger_, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076241, A076242, A076243.

a:= n-> (p-> irem(ithprime(p), p))(ithprime(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 09 2015

Table[Mod @@ Map[Nest[Prime, n, #] &, {2, 1}], {n, 65}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = prime(prime(n)) % prime(n); \\ Michel Marcus, Mar 25 2017

a(n) = prime^2(n) mod prime(n) = A006450(n) mod A000040(n).

A102616 Nonprime numbers of order 3.

Original entry on oeis.org

1, 14, 16, 22, 24, 25, 30, 33, 35, 36, 39, 44, 46, 48, 50, 51, 54, 55, 56, 62, 64, 66, 68, 69, 70, 75, 76, 77, 80, 85, 86, 87, 90, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 115, 116, 117, 118, 120, 122, 123, 124, 126, 130, 132, 134, 136, 138, 142, 144, 145, 148, 150

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

nps(n,1) -> list nonprime(n) or the sequence of nonprime numbers. nps(n,2) -> list nonprime(nonprime(n)) or nps of order 2. nps(n,3) -> list nonprime(nonprime(nonprime(n))) or npcs of order 3 ..... The order is the number of nestings - 1.

			Nonprime(2) = 4.
Nonprime(4) = 8.
Nonprime(8) = 14, the 2nd entry.

Cf. A018252, A102615.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

nonPrime[n_] := FixedPoint[n + PrimePi[ # ] &, n]; Nest[ nonPrime, Range[62], 3] (* Robert G. Wilson v, Feb 04 2005 *)

\\ We perform nesting(s) with a loop.
cics(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=nonprime(z); ); print1(z",") ) }
nonprime(n) = { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

A270795 The prime/nonprime compound sequence BAB.

Original entry on oeis.org

4, 12, 21, 28, 34, 42, 52, 60, 65, 74, 84, 95, 98, 106, 119, 128, 133, 135, 141, 147, 170, 177, 180, 192, 195, 209, 214, 220, 231, 246, 250, 253, 284, 288, 290, 295, 301, 316, 323, 329, 336, 339, 351, 365, 382, 387, 390, 394, 417, 429, 432, 445, 462, 470, 474, 481, 490, 505, 516, 518, 532, 538, 543, 550, 559, 566

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 5, 3, 3, 2, 9, 6, 1, 10, 9, 1, 1, 5, 13, 8, 13, 10, 5, 17, 5, 9, 1, 23, 27, 19, 17, 27, 3, 14, 15, 19, 13, 31, 17, 16, 31, 38, 37, 35, 27, 31, 21, 28, 17, 12, 47, 43, 43, 39, 31, 26, 45, 13, 1, 17, 23, 17, 53, 11, 15, 1, 53, 10, 25, 64, 41, 38, 41, 68, 33, 59, 63, 65

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

[NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017

Table[Mod[Prime[Prime[n]], n], {n, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = prime(prime(n)) % n; \\ Michel Marcus, Jul 09 2017

a(n) = A006450(n) mod n.

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)).

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).

A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

Original entry on oeis.org

0, 3, 5, 9, 11, 8, 17, 27, 25, 14, 31, 14, 41, 20, 16, 81, 59, 28, 67, 20, 22, 34, 83, 32, 121, 44, 125, 26, 109, 19, 127, 243, 36, 62, 28, 34, 157, 70, 46, 38, 179, 25, 191, 40, 36, 86, 211, 86, 289, 124, 64, 50, 241, 128, 42, 44, 72, 112, 277, 25, 283, 130, 42, 729, 52, 39, 331, 68, 88, 31

Offset: 1

Ilya Gutkovskiy, May 09 2018

nonn

			a(12) = 14 because 12 = 2^2*3 and prime(2)^2 + prime(3) = 3^2 + 5 = 14.

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A006450, A008475, A038580, A064988, A083186, A222416, A304037, A304253 (fixed points).

f:= proc(n) local t;
   add(ithprime(t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2024

a[n_] := Plus @@ (Prime[#[[1]]]^#[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 70}]

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,1])^f[k,2]); \\ Michel Marcus, May 09 2018

a(prime(i)^k) = prime(prime(i))^k.
a(A000040(k)) = A006450(k).
a(A006450(k)) = A038580(k).
a(A002110(k)) = A083186(k).

A237687 Primes p with pi(p), pi(pi(p)) and pi(p^2) all prime, where pi(.) is given by A000720.

Original entry on oeis.org

59, 127, 709, 1153, 1787, 9319, 13709, 19577, 32797, 35023, 39239, 40819, 53353, 62921, 75269, 90023, 161159, 191551, 218233, 228451, 235891, 238339, 239087, 272999, 289213, 291619, 339601, 439357, 500741, 513683

Offset: 1

Zhi-Wei Sun, Feb 11 2014

nonn

This is a subsequence of A237659.

Conjecture: The sequence has infinitely many terms.

			a(1) = 59 with 59, pi(59) = 17, pi(pi(59)) = pi(17) = 7 and pi(59^2) = 487 all prime.

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..150 from Zhi-Wei Sun)

Cf. A000040, A000290, A000720, A006450, A038107, A038580, A237656, A237657, A237658, A237659.

p[m_]:=PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[Prime[Prime[Prime[k]]]],n=n+1;Print[n," ",Prime[Prime[Prime[k]]]]],{k,1,1000}]

cp's OEIS Frontend

A270794 The prime/nonprime compound sequence BAA.

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A270796 The prime/nonprime compound sequence BBA.

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A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

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A102616 Nonprime numbers of order 3.

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A270795 The prime/nonprime compound sequence BAB.

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A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

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Magma

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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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A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

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