A028870 -id:A028870 - cp's OEIS frontend

A164722 Numbers whose sum of distinct prime factors is a square.

1, 14, 28, 39, 46, 55, 56, 66, 92, 94, 98, 112, 117, 132, 155, 158, 183, 184, 186, 188, 196, 198, 203, 224, 255, 264, 275, 290, 291, 295, 299, 316, 323, 334, 351, 354, 368, 372, 376, 392, 396, 446, 448, 455, 506, 507, 528, 546, 549, 558, 579, 580, 583, 594

Offset: 1

Jonathan Vos Post, Aug 23 2009

This is to A008472 as A051448 is to A001414. It does seem that for any given k there should be a maximum n such that the sum of the prime factors of n = k^2, and a (perhaps different) maximum n such that the sum of distinct prime factors on n = k^2.

If k >= 3 and p = k^2 - 2 is prime (see A028870) then 2 * p is the term. - Marius A. Burtea, Jun 12 2019

			a(7) = 66 because 66 = 2 * 3 * 11 has sum of distinct prime factors 2 + 3 + 11 = 16 = 4^2. 8748 = 2^2 * 3^7 is the largest number whose prime factors (with multiplicity) add to 25 = 5^2, but it is not in this sequence because the sum of distinct prime factors of 8748 is 2 + 3 = 5, which is not a square.

Marius A. Burtea, Table of n, a(n) for n = 1..14587 (terms up to 10^6)

Cf. A000290, A001414, A008472, A028870, A051448.

[n:n in [1..600]| IsPower(&+PrimeDivisors(n), 2)]; // Marius A. Burtea, Jun 12 2019

Select[Range[600],IntegerQ[Sqrt[Total[Transpose[FactorInteger[#]] [[1]]]]]&] (* Harvey P. Dale, Mar 05 2014 *)

isOK(n) = local(fac, i); fac = factor(n); issquare(sum(i=1, matsize(fac)[1], fac[i, 1])); \\ Michel Marcus, Mar 19 2013

{n such that A008472(n) = k^2 for k an integer}.

{n such that A008472(n) is in A000290}.

More terms (including missing terms 56, 183, and 196) from Jon E. Schoenfield, May 27 2010

A088572 Numbers n such that (2n+1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 13, 14, 16, 17, 18, 21, 23, 24, 27, 30, 31, 34, 35, 37, 38, 44, 46, 51, 53, 58, 59, 60, 63, 65, 67, 69, 72, 77, 80, 84, 86, 88, 91, 95, 102, 105, 108, 111, 115, 116, 118, 119, 123, 126, 128, 129, 132, 133, 136, 139, 142, 146, 149, 150, 151, 154, 156, 157

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Nov 17 2003

nonn

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

Cf. A028870, A073577.

q:= n-> isprime((2*n+1)^2-2):
select(q, [$0..157])[];  # Alois P. Heinz, Nov 13 2024

is(n)=isprime((2*n+1)^2-2) \\ Charles R Greathouse IV, Jun 12 2017

A154933 Numbers k such that k^6 - 2 is prime.

Original entry on oeis.org

3, 11, 17, 35, 37, 47, 49, 59, 67, 77, 99, 123, 127, 133, 139, 155, 161, 169, 173, 187, 195, 213, 225, 231, 237, 241, 245, 247, 253, 275, 279, 297, 319, 325, 367, 373, 381, 383, 385, 399, 411, 425, 431, 469, 507, 511, 523, 541, 545, 553, 569, 585, 589, 609

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

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

Cf. A028870, A038599, A154831, A154832, A154833, A154834.

lst={};Do[p=n^6-2;If[PrimeQ[p],AppendTo[lst,n]],{n,1,7!}];lst

isA154933(n) = isprime(n^6-2) \\ Michael B. Porter, Oct 06 2009

a(1) = 0 removed by Amiram Eldar, Apr 04 2020

A154935 Numbers n such that n^7-2 is prime.

Original entry on oeis.org

7, 15, 25, 87, 91, 99, 199, 211, 265, 337, 357, 361, 367, 405, 501, 511, 537, 595, 627, 685, 697, 771, 805, 841, 847, 861, 889, 931, 939, 979, 1035, 1047, 1081, 1125, 1135, 1177, 1225, 1231, 1287, 1315, 1321, 1387, 1425, 1497, 1501, 1627, 1741, 1795, 1807

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A028870, A038599, A154831, A154832, A154833, A154834, A154933, A154934.

[n: n in [1..500]|IsPrime(n^7-2)]; // Vincenzo Librandi, Nov 26 2010

lst={};Do[p=n^7-2;If[PrimeQ[p],AppendTo[lst,n]],{n,0,7!}];lst
Select[Range[2*10^3], PrimeQ[#^7 - 2] &] (* Vincenzo Librandi, Mar 20 2014 *)

is(n)=isprime(n^7-2) \\ Charles R Greathouse IV, Feb 17 2017

A239413 Numbers n such that n^5-5 is prime.

Original entry on oeis.org

4, 12, 16, 42, 102, 124, 132, 144, 184, 232, 274, 288, 306, 316, 336, 352, 406, 438, 478, 582, 606, 622, 706, 742, 754, 762, 814, 832, 916, 922, 964, 984, 996, 1026, 1044, 1072, 1086, 1096, 1156, 1174, 1204, 1258, 1272, 1366, 1408, 1416, 1428, 1432, 1456

Offset: 1

Derek Orr, Mar 17 2014

Note that all the numbers in this sequence are even.

There is no sequence "Numbers n such that n^4-4 is prime." since n^4 - 4 = (n^2 + 2)(n^2 - 2). - Michael B. Porter, Mar 18 2014

			4^5-5 = 1019 is prime. Thus, 4 is a member of this sequence.

Cf. A028870, A153974, A239343.

Select[Range[1500],PrimeQ[#^5-5]&] (* Harvey P. Dale, Dec 30 2019 *)

is(n)=isprime(n^5-5) \\ Charles R Greathouse IV, Feb 17 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**5-5)}

A154934 Primes in A154933.

Original entry on oeis.org

3, 11, 17, 37, 47, 59, 67, 127, 139, 173, 241, 367, 373, 383, 431, 523, 541, 569, 613, 631, 673, 683, 691, 829, 967, 977, 1019, 1063, 1163, 1213, 1249, 1291, 1301, 1303, 1327, 1367, 1439, 1483, 1487, 1601, 1607, 1609, 1733, 1747, 1789, 1801, 1823, 1907

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

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

Cf. A028870, A038599, A154831, A154832, A154833, A154834, A154933.

lst={}; Do[p=n^6-2; If[PrimeQ[p], If[PrimeQ[n], AppendTo[lst,n]]], {n,0,3*7!}]; lst

A239415 Numbers n such that n^7-7 is prime.

Original entry on oeis.org

60, 66, 132, 212, 242, 246, 290, 296, 312, 326, 380, 384, 446, 516, 524, 554, 654, 704, 740, 782, 834, 1026, 1086, 1142, 1154, 1172, 1182, 1214, 1424, 1430, 1464, 1482, 1494, 1500, 1524, 1604, 1682, 1686, 1752, 1794, 1796, 1844, 1854, 1940, 1952, 1980, 2000, 2010

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are even.

			60^7-7 = 2799359999993 is prime. Thus, 60 is a member of this sequence.

Cf. A028870, A153974, A239344, A239413, A239414.

Select[Range[2500],PrimeQ[#^7-7]&] (* Harvey P. Dale, Dec 21 2022 *)

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**7-7)}

A239416 Numbers n such that n^8-8 is prime.

Original entry on oeis.org

3, 7, 19, 39, 73, 75, 101, 107, 145, 147, 171, 213, 235, 247, 263, 285, 319, 353, 359, 369, 399, 443, 445, 521, 523, 557, 613, 675, 693, 707, 733, 781, 791, 805, 815, 829, 837, 879, 927, 943, 961, 999, 1033, 1097, 1103, 1109, 1129, 1137, 1141, 1155, 1157

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are odd.

			3^8-8 = 6553 is prime. Thus, 3 is a member of this sequence.

Cf. A028870, A153974, A239413, A239414, A239415, A239417, A239418, A239345.

Select[Range[3,1200,2],PrimeQ[#^8-8]&] (* Harvey P. Dale, Jun 27 2014 *)

is(n)=isprime(n^8-8) \\ Charles R Greathouse IV, Feb 20 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**8-8)}

A239417 Numbers n such that n^9-9 is prime.

Original entry on oeis.org

2, 62, 86, 88, 116, 152, 266, 292, 310, 326, 338, 356, 406, 436, 466, 470, 518, 550, 568, 616, 626, 650, 688, 700, 722, 812, 850, 926, 956, 992, 1058, 1076, 1126, 1186, 1252, 1430, 1550, 1570, 1642, 1672, 1682, 1766, 1808, 1852, 1868, 1888, 2138, 2210, 2306

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all numbers in this sequence are even.

			2^9-9 = 503 is prime. Thus, 2 is a member of this sequence.

Cf. A028870, A153974, A239413, A239414, A239415, A239416, A239418, A239346.

is(n)=isprime(n^9-9) \\ Charles R Greathouse IV, Feb 20 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**9-9)}

A239418 Numbers n such that n^10 - 10 is prime.

Original entry on oeis.org

21, 201, 267, 321, 369, 459, 537, 651, 669, 699, 723, 753, 1071, 1113, 1197, 1203, 1209, 1323, 1401, 1503, 1587, 1647, 1773, 1791, 1797, 1917, 1941, 2007, 2139, 2223, 2427, 2493, 2613, 2733, 2769, 2787, 2847, 3147, 3249, 3267, 3297, 3399, 3423, 3441, 3771

Offset: 1

Derek Orr, Mar 17 2014

nonn

All of the numbers in this sequence are odd multiples of 3 and, thus, congruent to 3 (mod 6).

The tenth powers modulo 6 are 1, 4, 3, 4, 1, 0, ... (A070431). Subtracting 10 (still modulo 6), we get 3, 0, 5, 0, 3, 2, ... which means that only n = 3 mod 6 can produce a potential prime p = 5 mod 6.

			21^10 - 10 = 16679880978191 is prime. Thus, 21 is a member of this sequence.

Cf. A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239347.

Select[Range[1000], PrimeQ[#^10 - 10] &] (* Alonso del Arte, Mar 18 2014 *)

is(n)=isprime(n^10-10) \\ Charles R Greathouse IV, Feb 20 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**10-10)}

cp's OEIS Frontend

A164722 Numbers whose sum of distinct prime factors is a square.

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A088572 Numbers n such that (2n+1)^2 - 2 is prime.

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A154933 Numbers k such that k^6 - 2 is prime.

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A154935 Numbers n such that n^7-2 is prime.

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A239413 Numbers n such that n^5-5 is prime.

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A154934 Primes in A154933.

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A239415 Numbers n such that n^7-7 is prime.

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A239416 Numbers n such that n^8-8 is prime.

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