cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A134621 Numbers such that the arithmetic mean of the 4th power of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 28, 39, 48, 51, 65, 68, 76, 77, 85, 87, 93, 111, 119, 133, 141, 143, 148, 155, 161, 175, 187, 189, 209, 212, 215, 221, 225, 235, 244, 275, 287, 295, 301, 315, 316, 320, 323, 329, 355, 393, 403, 404, 411, 428, 437, 447, 451, 455, 481, 505, 508, 515, 517
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(3)=39, since 39=3*13 and (3^4+13^4)/2=14321 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619.

Extensions

Minor edits by Hieronymus Fischer, May 06 2013

A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

20, 44, 188, 297, 336, 400, 425, 540, 575, 605, 704, 752, 764, 908, 912, 1025, 1053, 1124, 1172, 1183, 1365, 1380, 1412, 1420, 1452, 1475, 1484, 1519, 1604, 1625, 1809, 1844, 1856, 1936, 1953, 2107, 2192, 2205, 2255, 2320, 2325, 2348, 2368, 2372, 2468
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(10)=605, since 605=5*11*11 and (5^3+11^3+11^3)/3=929 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134621.

Programs

  • Mathematica
    amcpfQ[n_]:=PrimeQ[Mean[Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]^3]]; Select[ Range[ 2500],amcpfQ] (* Harvey P. Dale, Jun 06 2023 *)
  • PARI
    lista(m) = {for (i=2, m, f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && isprime(s), print1(i, ", ")););} \\ Michel Marcus, Apr 14 2013

Extensions

Minor edits by Hieronymus Fischer, May 06 2013

A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1512, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) are included. First term not included in A025475 is a(30) = 1512 = A134613(2) = A134613(1).
Most terms have a last digit of 1 or 9 (i.e., 8326 out of 9000 terms). Mainly, this comes from the fact that all squares of primes are included. Since each prime > 10 has a last digit of 1, 3, 7 or 9, its square has a last digit of 1 or 9. In addition, m-th powers of primes have a last digit of 1, if m == 0 (mod 4), and have a last digit of 1 or 9 if m == 2 (mod 4), and have a 50% chance, roughly, for a last digit of 1 or 9, if m == 1 (mod 4) or m == 3 (mod 4). Since the number of terms <= N which are squares of primes is PrimePi(sqrt(N)) = A000720(sqrt(N)), it follows that the number of terms <= N which have a last digit of 1 or 9 is greater than PrimePi(sqrt(N)). This can be estimated as 2*N^(1/2)/log(N), approximately.

Examples

			a(6) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
a(30) = 1512, since 1512 = 2*2*2*3*3*3*7 and ((3*2^3+3*3^3+7^3)/7)^(1/3) = 64^(1/3) = 4.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.

Programs

  • PARI
    lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Extensions

Edited by Hieronymus Fischer, May 30 2013

A134617 Numbers such that the arithmetic mean of the squares of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 20, 21, 28, 35, 39, 44, 48, 51, 52, 55, 65, 69, 85, 91, 92, 95, 108, 112, 115, 116, 129, 135, 141, 145, 159, 164, 172, 188, 189, 205, 208, 209, 215, 221, 225, 235, 236, 245, 249, 259, 268, 272, 295, 297, 299, 305, 309, 315, 316, 320, 325, 329, 339, 341, 365
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(2)=20, since 20=2*2*5 and (2^2+2^2+5^2)/3=33/3=11.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134619, A134621.

Programs

  • Mathematica
    amspQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]^2]]; Select[Range[400],amspQ] (* Harvey P. Dale, Jan 21 2017 *)

Extensions

Minor edits by the author, May 06 2013

A134615 Numbers (excluding primes and powers of primes) such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

707265, 1922816, 2284389, 12023505, 14689836, 21150800, 29444140, 30682000, 36533504, 39372480, 46309837, 52163097, 67303740, 73558065, 85751055, 107366283, 115291904, 161976045, 190384425, 204399585, 218317275, 231443940, 274960400, 286618640
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

The prime factors are taken with multiplicity.
Numbers included in A134612, but not in A025475.
a(1) = 707265 is the minimal number with this property. a(3) = 2284389 is the greatest such number < 10^7.

Examples

			a(1) = 707265, since 707265 = 3*3*3*5*13*13*31 and ((3*3^3+5^3+2*13^3+31^3)/7)^(1/3) = 4913^(1/3) = 17.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619, A134621.

Programs

  • PARI
    isok(n) = {if (omega(n) == 1, return (0)); f = factor(n); s = sum(i=1, #f~, f[i,2]*f[i,1]^3); s = s/bigomega(n); if (type(s) != "t_INT", return (0)); if (! ispower(s, 3, &p), return (0)); isprime(p);} \\ Michel Marcus, Nov 03 2013

Extensions

More terms and minor edits by Hieronymus Fischer, May 06 2013, May 30 2013

A134610 Composite numbers such that the cube root of the sum of cubes of their prime factors is a prime.

Original entry on oeis.org

14157, 141960, 466560, 1608575, 3097055, 5338710, 6235076, 16017300, 22353408, 24948000, 25073792, 25564544, 27843750, 29761408, 30570408, 31894350, 40837825, 44175248, 46120064, 49867818, 55814400, 56141963, 71214803, 77450890, 92682405
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

Numbers included in A134608, but not in A134609. a(1)=14157 is the minimal number with this property.
The prime factors are taken by multiplicity.

Examples

			a(2)=141960 since 141960=2*2*2*3*5*7*13*13 and (3*2^3+3^3+5^3+7^3+2*13^3)^(1/3)=4913^(1/3)=17.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134613, A134616, A134618, A134620.

Extensions

Minor edits and additional terms by the author, Apr 15 2013
Showing 1-6 of 6 results.

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