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 1-4 of 4 results.

A120319 RF(3): refactorable numbers with smallest prime factor 3.

Original entry on oeis.org

9, 225, 441, 1089, 1521, 2025, 2601, 3249, 4761, 5625, 6561, 7569, 8649, 12321, 15129, 16641, 19881, 25281, 31329, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 71289, 84681, 91809, 95481, 99225, 103041, 106929, 114921, 145161, 154449, 164025, 168921
Offset: 1

Views

Author

Walter Kehowski, Jun 20 2006

Keywords

Comments

Numbers that are odd squares, 3 is their smallest prime factor, and are refactorable.
See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 3^(3-1)=9 is the first element. Other elements would also be 3^2*17^2 or 3^16*17^2.

Links

Crossrefs

Intersection of A016945 and A033950.
Cf. A036896, A036897.
Subsequence of A016946.

Programs

  • Maple
    with(numtheory); RF3:=[]: p:=3: for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if k mod p = 0 then n:=k^2; t:=tau(n); if (n mod t = 0) then RF3:=[op(RF3),n]; print(ifactor(n)); fi fi; od od;
  • PARI
    lista(kmax) = forstep(k = 3, kmax, 6, if(!(k^2 % numdiv(k^2)), print1(k^2, ", "))); \\ Amiram Eldar, Aug 01 2024

Extensions

a(37)-a(39) from Amiram Eldar, Aug 01 2024

A120320 RF(5): refactorable numbers with smallest prime factor 5.

Original entry on oeis.org

625, 1500625, 9150625, 17850625, 37515625, 52200625, 73530625, 81450625, 174900625, 442050625, 577200625, 1171350625, 1766100625, 1838265625, 2136750625, 3049800625, 4931550625, 7573350625, 8653650625, 12594450625, 15882300625, 17748900625, 21970650625, 24343800625
Offset: 1

Views

Author

Walter Kehowski, Jun 20 2006

Keywords

Comments

Numbers that are odd squares, 5 is their smallest prime factor, and are refactorable.
See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 5^(5-1) = 625 is the first element. Other elements would also be 5^4*17^4 or 5^16*17^4.
All the terms are of the form 5^2 * A084967(k)^2 = 5^4 * A007310(k)^2. - Amiram Eldar, Aug 01 2024

Links

Crossrefs

Intersection of A033950 and A084967.
Cf. A007310, A036896, A036897.

Programs

  • Maple
    with(numtheory); RF5:=[]: p:=5: for w to 1 do for j from 1 to 12^5 do k:=2*j+1; if k mod 3 <> 0 and k mod p = 0 then n:=k^2; t:=tau(n); if (n mod t = 0) then RF5:=[op(RF5),n]; print(ifactor(n)); fi fi; od od;
  • PARI
    lista(kmax) = {my(m); for(k = 1, kmax, m = 25*(k\2*6-(-1)^k)^2; if(!(m % numdiv(m)), print1(m, ", ")));} \\ Amiram Eldar, Aug 01 2024

Extensions

a(37)-a(40) from Amiram Eldar, Aug 01 2024

A120321 RF(7): refactorable numbers with 7 as smallest prime factor.

Original entry on oeis.org

117649, 208422380089, 567869252041, 2839760855281, 5534900853769, 17416274304961, 69980368892329, 104413920565969, 301855146292441, 558845013849409, 743702041351801, 1268163904241521, 2607614922465721
Offset: 1

Views

Author

Walter Kehowski, Jun 21 2006

Keywords

Comments

Numbers that are odd squares, 7 is their smallest prime factor, and are refactorable.
See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 7^(7-1)=117649 is the first element. Other elements would also be 7^6*17^6 or 7^16*17^6. Here are the prime factorizations for the first 49 elements of RF7: (7^6), (7^6)*(11^6), (7^6)*(13^6), (7^6)*(17^6), (7^6)*(19^6), (7^6)*(23^6), (7^6)*(29^6), (7^6)*(31^6), (7^6)*(37^6), (7^6)*(41^6), (7^6)*(43^6), (7^6)*(47^6), (7^6)*(53^6), (7^6)*(59^6), (7^6)*(61^6), (7^6)*(67^6), (7^6)*(71^6), (7^6)*(73^6), (7^6)*(79^6), (7^6)*(83^6), (7^6)*(89^6), (7^12)*(13^6), (7^6)*(97^6), (7^6)*(101^6), (7^6)*(103^6), (7^6)*(107^6), (7^6)*(109^6), (7^6)*(113^6), (7^6)*(127^6), (7^6)*(131^6), (7^6)*(137^6), (7^6)*(139^6), (7^6)*(11^6)*(13^6), (7^6)*(149^6), (7^6)*(151^6), (7^6)*(157^6), (7^6)*(163^6), (7^6)*(167^6), (7^6)*(13^12), (7^6)*(173^6), (7^6)*(179^6), (7^6)*(181^6), (7^6)*(11^6)*(17^6), (7^6)*(191^6), (7^6)*(193^6), (7^6)*(197^6), (7^6)*(199^6), (7^6)*(11^6)*(19^6), (7^6)*(211^6).

Examples

			a(1) = 7^(7-1) = 117649.

Links

Crossrefs

Intersection of A033950 and A084968.
Cf. A036896, A036897.

Programs

  • Maple
    with(numtheory); p:=7: RF7:=[p^(p-1)]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=7^6*k^s; t:=tau(n); if not n in RF7 and (n mod t = 0) then RF7:=[op(RF7),n]; print(ifactor(n)); fi; od; fi; od od; RF7:=sort(RF7);

A120322 RF(11): refactorable numbers with 11 as smallest prime factor.

Original entry on oeis.org

25937424601, 3575694237941010577249, 52289689788971837545849, 159024068785448665562401, 604292326212030787555081, 1074497011086501939579049, 10912062142819279835644801
Offset: 1

Views

Author

Walter Kehowski, Jun 21 2006

Keywords

Comments

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 11^(11-1)=25937424601 is the first element. Other elements would also be 11^10*17^10 or 11^16*17^10. Here are the prime factorizations for the first 48 elements of RF11: (11^10), (11^10)*(13^10), (11^10)*(17^10), (11^10)*(19^10), (11^10)*(13^12), (11^10)*(23^10), (11^10)*(29^10), (11^10)*(31^10), (11^10)*(37^10), (11^10)*(41^10), (11^10)*(43^10), (11^10)*(47^10), (11^10)*(53^10), (11^10)*(59^10), (11^10)*(61^10), (11^10)*(67^10), (11^10)*(71^10), (11^10)*(73^10), (11^10)*(79^10), (11^10)*(83^10), (11^10)*(89^10), (11^10)*(17^16), (11^10)*(97^10), (11^10)*(101^10), (11^10)*(103^10), (11^10)*(107^10), (11^10)*(109^10), (11^10)*(113^10), (11^10)*(127^10), (11^10)*(131^10), (11^10)*(137^10), (11^10)*(139^10), (11^10)*(149^10), (11^10)*(151^10), (11^10)*(157^10), (11^10)*(163^10), (11^10)*(167^10), (11^10)*(173^10), (11^10)*(179^10), (11^10)*(181^10), (11^10)*(191^10), (11^10)*(193^10), (11^10)*(197^10), (11^10)*(199^10), (11^10)*(211^10), (11^10)*(13^10)*(17^10), (11^10)*(2 23^10), (11^10)*(227^10).

Examples

			a(1)=11^(11-1)=25937424601.

Crossrefs

Cf. A033950, A036896, A036897.

Programs

  • Maple
    with(numtheory); p:=11: a:=p^(p-1): RF11:=[a]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=a*k^s; t:=tau(n); if not n in RF11 and (n mod t = 0) then RF11:=[op(RF11),n]; print(ifactor(n)); fi; od; fi; od od; RF11:=sort(RF11);

Formula

a(n) = odd square, 11 is the smallest prime factor and refactorable.
Showing 1-4 of 4 results.

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