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 1-9 of 9 results.

A340306 Numbers k such that A065642(k) = A081761(k).

Original entry on oeis.org

12, 420, 540, 2268, 7020, 10692, 11340, 17640, 24948, 42750, 56700, 87120, 152460, 409500, 413100, 609840, 996072, 2478600, 3822000, 5287500, 9189180, 9447840, 14871600, 20241900, 20567520, 23510592, 23832800, 27766152, 28552500, 39358800, 41135040, 44783648, 49985100
Offset: 1

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Author

Amiram Eldar, Jan 03 2021

Keywords

Comments

Numbers k such that the least number that is larger than k and has the same prime signature as k and the least number that is larger than k and has the same set of distinct prime divisors as k are equal.

Examples

			12 is a term since the A065642(12) = A081761(12) = 18, i.e., 18 = 2 * 3^2 is the least number with the same set of prime divisors, {2, 3}, and the same prime signature as 12 = 2^2 * 3.

Links

Crossrefs

Intersection of A340302 and A340305.
Cf. A065642, A081761.

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; next[n_] := Module[{r = rad[n]}, SelectFirst[Range[n + 1, n^2], rad[#] == r &]]; sig[n_] := Sort@FactorInteger[n][[;; , 2]]; nextsig[n_] := Module[{sign = sig[n], k = n + 1}, While[sig[k] != sign, k++]; k]; Select[Range[2, 600], sig[#] == sig[next[#]] && rad[#] == rad[nextsig[#]] &]

A085072 Smallest k such that n and n+k have the same prime signature.

Original entry on oeis.org

1, 2, 5, 2, 4, 4, 19, 16, 4, 2, 6, 4, 1, 6, 65, 2, 2, 4, 8, 1, 4, 6, 16, 24, 7, 98, 16, 2, 12, 6, 211, 1, 1, 3, 64, 4, 1, 7, 14, 2, 24, 4, 1, 5, 5, 6, 32, 72, 2, 4, 11, 6, 2, 2, 32, 1, 4, 2, 24, 6, 3, 5, 665, 4, 4, 4, 7, 5, 8, 2, 36, 6, 3, 1, 16, 5, 24, 4, 32, 544, 3, 6, 6, 1, 1, 4, 16, 8, 36, 2
Offset: 2

Views

Author

Amarnath Murthy, Jul 01 2003

Keywords

Examples

			a(28) = 17 as 28 = 2^2*7 and 28+17 = 45 = 3^2*5, both have the prime signature p^2*q where p and q are primes.

Links

Crossrefs

Cf. A046523, A081761, A085073.

Programs

  • Maple
    s:= n-> sort(map(i-> i[2], ifactors(n)[2])):
a:= proc(n) option remember; local k;
      for k while s(n)<>s(n+k) do od; k
    end:
seq(a(n), n=2..100);  # Alois P. Heinz, Feb 28 2018
  • Mathematica
    s[n_] := Sort[FactorInteger[n][[All, 2]]];
a[n_] := Module[{sn = s[n], k}, For[k = 1, True, k++, If[sn == s[n+k], Return[k]]]];
a /@ Range[2, 100] (* Jean-François Alcover, Nov 02 2020 *)
  • PARI
    a(n) = {my(k=1, s = vecsort(factor(n)[,2]~)); while (vecsort(factor(n+k)[,2]~) != s, k++); k;} \\ Michel Marcus, Nov 02 2020

Formula

a(prime(k)^r) = prime(k+1)^r- prime(k)^r.
a(2^m*prime(k)) = 2^m*(prime(k+1) - prime(k)).
a(n) = A081761(n) - n. - Michel Marcus, Nov 02 2020

Extensions

More terms from David Wasserman, Jan 12 2005

A340302 Numbers k such that k and the least number that is larger than k and has the same prime signature as k also has the same set of distinct prime divisors as k.

Original entry on oeis.org

12, 72, 144, 420, 432, 540, 864, 1728, 1800, 2000, 2268, 2520, 2592, 5184, 5400, 6300, 7020, 10125, 10368, 10692, 10800, 11340, 12600, 15120, 15552, 16200, 17640, 20000, 20736, 21168, 21600, 24000, 24948, 25200, 26460, 31104, 37800, 40500, 41472, 42750, 43200
Offset: 1

Views

Author

Amiram Eldar, Jan 03 2021

Keywords

Comments

Numbers k such that A007947(k) = A007947(A081761(k)).
This sequence is infinite since it includes all the numbers of the form 2*6^k for k>=1.

Examples

			12 = 2^2 * 3 is a term since the least number that is larger than 12 and has the same prime signature as 12 is 18 = 2 * 3^2 which also has the same set of distinct prime divisors as 12, {2, 3}.

Links

Crossrefs

Cf. A007947, A081761, A085079, A089247, A118914, A167747.

Programs

  • Mathematica
    sig[n_] := Sort@FactorInteger[n][[;; , 2]]; nextsig[n_] := Module[{sign = sig[n], k = n + 1}, While[sig[k] != sign, k++]; k]; rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; Select[Range[2, 1000], rad[#] == rad[nextsig[#]] &]

A077560 Second smallest integer of prime signature corresponding to that of A025487.

Original entry on oeis.org

3, 9, 10, 27, 18, 81, 40, 42, 243, 100, 80, 84, 729, 108, 160, 168, 2187, 324, 252, 320, 330, 1000, 336, 6561, 800, 504, 640, 630, 648, 672, 19683, 1600, 1008, 1280, 1320, 1944, 1764, 1344, 59049, 1512, 3200, 1980, 10000, 2016, 2560, 2640, 5832, 2700
Offset: 2

Views

Author

Amarnath Murthy, Nov 10 2002

Keywords

Comments

In A025487 the terms are in increasing order but here there are ups and downs.

Links

Crossrefs

Cf. A025487, A081761.

Formula

a(n) = A081761(A025487(n)). - Amiram Eldar, Dec 29 2020

Extensions

Corrected and extended by Ray Chandler, Aug 24 2003

A137509 a(1)=2. For n >= 2, a(n) = the smallest integer > a(n-1) that has the same multiset of prime-factorization exponents as n has.

Original entry on oeis.org

2, 3, 5, 9, 11, 14, 17, 27, 49, 51, 53, 63, 67, 69, 74, 81, 83, 92, 97, 98, 106, 111, 113, 135, 169, 177, 343, 356, 359, 366, 367, 3125, 3127, 3131, 3133, 3249, 3251, 3254, 3261, 3272, 3299, 3302, 3307, 3308, 3316, 3317, 3319, 3321, 3481, 3501, 3503, 3508
Offset: 1

Views

Author

Leroy Quet, Apr 23 2008

Keywords

Comments

Starting the sequence at a(1)=1 instead leads to a(n) = n for every positive integer n.

Examples

			12 = 2^2 * 3^1. So the multiset of exponents in the prime factorization of 12 is {1,2}. For a(12), we want the smallest integer > a(11)=53 of the form p^1 * q^2, where p and q are distinct primes. Checking: 54 = 2^1 *3^3, so 54 fails. 55 = 5^1*11^1. 56 = 2^3*7^1. 57 = 3^1*19^1. 58 = 2^1*29^1. 59=59^1. 60 = 2^2*3^1*5^1. 61 = 61^1. 62 = 2^1 *31^1. So 54 through 62 all fail. But 63 = 3^2 * 7^1, which has the same multiset of prime exponents, {1,2}, as 12 has. Therefore a(12) = 63.

Links

Crossrefs

Cf. A081761.

Programs

  • Maple
    pmset := proc(n) local e,a ; a := [] ; for e in ifactors(n)[2] do a := [op(a),e[2]] ; od: sort(a) ; end: A137509 := proc(n) option remember ; local nset,a ; if n = 1 then RETURN(2) ; fi ; nset := pmset(n) ; for a from A137509(n-1)+1 do if pmset(a) = nset then RETURN(a) ; fi ; od: end: seq(A137509(n),n=1..120) ; # R. J. Mathar, May 23 2008
  • Mathematica
    s={2};Do[fe=Sort[Last/@FactorInteger[n]];k=s[[-1]]+1;While[Sort[Last/@FactorInteger[k]]!=fe,k++];AppendTo[s,k],{n,2,52}];s (* James C. McMahon, May 30 2025 *)
  • PARI
    lista(nn) = my(va = vector(nn)); va[1] = 2; for (n=2, nn, my(k=va[n-1]+1, f=vecsort(factor(n)[,2])); while (vecsort(factor(k)[,2]) != f, k++); va[n] = k;); va; \\ Michel Marcus, May 30 2025

Extensions

More terms from R. J. Mathar, May 23 2008

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

Original entry on oeis.org

3, 5, 27, 7, 10, 11, 9, 25, 14, 13, 20, 17, 15, 21, 7625597484987, 19, 24, 23, 28, 22, 26, 29, 50, 32, 33, 3125, 44, 31, 42, 37, 49, 34, 35, 38, 100, 41, 39, 46, 45, 43, 66, 47, 52, 56, 51, 53, 80, 121, 98, 55, 54, 59, 68, 57, 63, 58, 62, 61, 84, 67, 65, 75
Offset: 2

Views

Author

Rémy Sigrist, Feb 07 2017

Keywords

Comments

The prime tower factorization of a number is defined in A182318.
The prime tower factorization equivalence classes are described in A279686.
For any n>1, a(n)=least k>n such that A279690(n)=A279690(k).
This sequence is a permutation of the complement of A279686.
This sequence is to prime tower factorization what A081761 is to prime signature.

Links

Crossrefs

Cf. A000040, A006881, A007304, A046386, A046387, A051674, A067885, A081761, A182318, A279686, A279690

Programs

  • PARI
    a(n) = my (c=a279690(n)); my (k=n+1); while (c!=a279690(k), k++); k

Formula

a(A000040(n)) = A000040(n+1) for any n>0.
a(A006881(n)) = A006881(n+1) for any n>0.
a(A051674(n)) = A051674(n+1) for any n>0.
a(A007304(n)) = A007304(n+1) for any n>0.
a(A046386(n)) = A046386(n+1) for any n>0.
a(A046387(n)) = A046387(n+1) for any n>0.
a(A067885(n)) = A067885(n+1) for any n>0.

A340303 Numbers k such that k and the next two numbers after k with the same prime signature as k also have the same set of distinct prime divisors as k.

Original entry on oeis.org

10800, 12600, 54000, 75600, 148176, 162000, 264600, 324000, 609840, 453600, 529200, 793800, 444528, 648000, 1058400, 1323000, 1764000, 1620000, 2116800, 2646000, 2721600, 3704400, 3969000, 6403320, 4233600, 5292000, 4860000, 6223392, 7938000, 7408800, 8467200
Offset: 1

Views

Author

Amiram Eldar, Jan 03 2021

Keywords

Comments

Numbers k such that A007947(k) = A007947(A081761(k)) = A007947(A081761(A081761(k))).

Examples

			10800 = 2^4 * 3^3 * 5^2 is a term since 10800 and the next two numbers with the same prime signature as 10800, 16200 = 2^3 * 3^4 * 5^2 and 18000 = 2^4 * 3^2 * 5^3, have the same set of distinct prime divisors, {2, 3, 5}.

Links

Crossrefs

Subsequence of A340302.
Cf. A007947, A081761, A118914.

A357126 a(n) is the smallest positive integer k such that k > n and A071364(k) = A071364(n).

Original entry on oeis.org

3, 5, 9, 7, 10, 11, 27, 25, 14, 13, 20, 17, 15, 21, 81, 19, 50, 23, 28, 22, 26, 29, 40, 49, 33, 125, 44, 31, 42, 37, 243, 34, 35, 38, 100, 41, 39, 46, 56, 43, 66, 47, 45, 52, 51, 53, 80, 121, 75, 55, 63, 59, 250, 57, 88, 58, 62, 61, 84, 67, 65, 68, 729, 69, 70, 71, 76, 74, 78, 73, 200, 79, 77, 98
Offset: 2

Views

Author

Gleb Ivanov, Oct 26 2022

Keywords

Examples

			a(12) = 20 as 12 has (2, 1) sequence of exponents in canonical prime factorization via 12 = 2^2 * 3^1 and the smallest positive integer > 12 with the same sequence of exponents in canonical prime factorization being (2, 1) is 20 as 20 = 2^2 * 5^1. - _David A. Corneth_, Oct 26 2022

Crossrefs

Cf. A000961, A003961, A071364, A065642, A081382, A081761.

Programs

  • PARI
    f4(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); \\ A071364
a(n) = my(k=n+1, f=f4(n)); while (f4(k) != f, k++); k; \\ Michel Marcus, Oct 26 2022
  • PARI
    first(n) = { my(res = vector(n + 1), todo = n, m = Map(), u = precprime(n)); for(e = 2, logint(n, 2), u = max(u, nextprime(sqrtnint(n, e) + 2)^e) ); forfactored(i = 2, u, cs = i[2][,2]; if(mapisdefined(m, cs), ci = mapget(m, cs); if(ci <= n + 1, res[ci] = i[1]; mapput(m, cs, i[1]); todo--; if(todo <= 0, res = res[^1]; return(res) ) ) , if(i[1] <= n + 1, mapput(m, cs, i[1]) ) ) ) } \\ David A. Corneth, Oct 26 2022
  • Python
    from sympy import factorint
to_s_exp = lambda n: tuple(i[1] for i in sorted(factorint(n).items()))
terms = []
for i in range(2, 100):
    k = i+1;t = to_s_exp(i)
    while t != to_s_exp(k):k+=1
    terms.append(k)
print(terms)

Formula

a(A000961(k)) = a(A003961(A000961(k))) for k > 1. - David A. Corneth, Oct 26 2022
a(n) >= A081761(n). - Rémy Sigrist, Feb 16 2023

A340304 Numbers k such that k and the next three numbers after k with the same prime signature as k also have the same set of distinct prime divisors as k.

Original entry on oeis.org

529200, 1058400, 2646000, 3704400, 5292000, 9720000, 15876000, 22226400, 18522000, 31752000
Offset: 1

Views

Author

Amiram Eldar, Jan 03 2021

Keywords

Comments

Numbers k such that A007947(k) = A007947(A081761(k)) = A007947(A081761(A081761(k))) = A007947(A081761(A081761(A081761(k)))).
The analogous sequence with the next four numbers after k begins with 2646000, 15876000, ...

Examples

			529200 = 2^4 * 3^3 * 5^2 * 7^2 is a term since 529200 and the next three numbers with the same prime signature as 529200, 793800 = 2^3 * 3^4 * 5^2 * 7^2, 882000 = 2^4 * 3^2 * 5^3 * 7^2, and 1234800 = 2^4 * 3^2 * 5^2 * 7^3, have the same set of distinct prime divisors, {2, 3, 5, 7}.

Links

Crossrefs

Subsequence of A340302 and A340303.
Cf. A007947, A081761, A118914.
Showing 1-9 of 9 results.

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