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.

Previous Showing 11-15 of 15 results.

A068633 a(1) = 11 by convention; for n > 1, if n = p^a*q^b... then a(n) = concatenate(p,a,q,b,...).

Original entry on oeis.org

11, 21, 31, 22, 51, 2131, 71, 23, 32, 2151, 111, 2231, 131, 2171, 3151, 24, 171, 2132, 191, 2251, 3171, 21111, 231, 2331, 52, 21131, 33, 2271, 291, 213151, 311, 25, 31111, 21171, 5171, 2232, 371, 21191, 31131, 2351, 411, 213171, 431, 22111
Offset: 1

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Author

Amarnath Murthy, Feb 27 2002

Keywords

Comments

Apart from the conventional initial term, a duplicate of A067599. See there for further comments, examples, references and other information.

Extensions

Edited by M. F. Hasler, Oct 06 2013

A067600 Numbers n such that f(n) and f(f(n)) are prime, where f(k) = decimal encoding of the prime factorization of k.

Original entry on oeis.org

3, 20, 69, 171, 174, 267, 333, 360, 372, 448, 537, 665, 666, 776, 820, 824, 855, 873, 1016, 1125, 1330, 1413, 1450, 1532, 1604, 1689, 1796, 1860, 1899, 1959, 2048, 2068, 2184, 2319, 2449, 2620, 2658, 2670, 2804, 2823, 3139, 3210, 3342, 3464, 3552, 3589
Offset: 1

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Author

Joseph L. Pe, Jan 31 2002

Keywords

Comments

If n = p_1^e_1 * ... * p_r^e_r with p_1 < ... < p_r, then its decimal encoding is p_1 e_1...p_r e_r. For example, 15 = 3^1 * 5^1, so has decimal encoding 3151.

Examples

			The prime factorization of 20 = 2^2 * 5^1 with corresponding encoding 2251, which is a prime. 2251 = 2251^1 has encoding 22511, which is also prime. So 20 is a term of the sequence.

Crossrefs

Cf. A067599.

Programs

  • Mathematica
    f[n_] := FromDigits[Flatten[IntegerDigits[FactorInteger[n]]]]; Select[ Range[4000], Union[ PrimeQ[ Drop[ NestList[f, #, 2], 1]]] == {True} & ]

Extensions

Edited by Robert G. Wilson v and N. J. A. Sloane, Feb 02 2002

A241007 Decimal encoding of the prime factorization of a(n) is a prime.

Original entry on oeis.org

3, 6, 7, 8, 13, 19, 20, 31, 35, 38, 40, 43, 44, 45, 56, 60, 63, 65, 68, 69, 96, 97, 103, 104, 109, 116, 119, 122, 125, 133, 138, 151, 157, 159, 160, 171, 174, 181, 189, 193, 204, 209, 211, 212, 215, 216, 220, 221, 232, 241, 248, 250, 254, 267, 271, 273, 280
Offset: 1

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Author

Paolo P. Lava, Aug 07 2014

Keywords

Comments

This sequence gives all arguments for which A067599 is prime: A067599(a(n)) = prime. - Wolfdieter Lang, Aug 16 2014

Examples

			Prime factorization of a(4) = 8 is 2^3 and its decimal encoding is A067599(8) = 23 that is prime.
Prime factorization of a(188) = 994 is 2^1*7^1*71^1 and its decimal encoding is  A067599(994) = 2171711 that is prime.

Links

Crossrefs

Cf. A067598, A067599.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,b,c,d,k,j,n;
for n from 1 to q do a:=(ifactors(n)[2]); b:=[];
for k from 1 to nops(a) do b:=[op(b),a[k,1]]; od; b:=sort(b); c:=0;
for k from 1 to nops(b) do d:=1; while b[k]<>a[d,1] do d:=d+1; od;
j:=b[k]*10^(ilog10(a[d,2])+1)+a[d,2];
c:=c*10^(ilog10(j)+1)+j; od; if isprime(c) then print(n);
fi; od; end: P(1000);

Formula

Sequence of the members of the set {k >= 2: A067599(k) is a prime} in increasing order. - Wolfdieter Lang, Aug 16 2014

Extensions

Edited: Name n-> a(n), cf. A067599, examples specified. - Wolfdieter Lang, Aug 16 2014

A299400 a(n) = concatenation of all (i, e_i) with e_i > 0, when n = Product_{i >= 1} prime(i)^e_i.

Original entry on oeis.org

0, 11, 21, 12, 31, 1121, 41, 13, 22, 1131, 51, 1221, 61, 1141, 2131, 14, 71, 1122, 81, 1231, 2141, 1151, 91, 1321, 32, 1161, 23, 1241, 101, 112131, 111, 15, 2151, 1171, 3141, 1222, 121, 1181, 2161, 1331, 131, 112141, 141, 1251, 2231, 1191, 151, 1421, 42, 1132, 2171
Offset: 1

Views

Author

M. F. Hasler, Mar 08 2018

Keywords

Comments

The conventional a(1) = 0 represents the empty concatenation.
Due to simple concatenation, this encoding of the positive integers becomes ambiguous from n = 613 = prime(112)^1 on, which has the same encoding a(n) = 1121 as 6 = prime(1)^1*prime(2)^1. To get a unique encoding, one could use, e.g., the digit 9 as delimiter to separate indices and exponents, written in base 9 as to use only digits 0..8, as soon as a term would be the duplicate of an earlier term (or for all n >= 613). Then one would have, e.g., a(613) = prime(134_9)^1 = 13491.
Sequence A067599 is based on the same idea, but uses the primes instead of their indices. In A037276 the prime factors are repeated, instead of giving the exponent. In A080670 exponents 1 are omitted. In A124010 only the prime signature is given. In A054841 the sum e_i*10^(i-1) is given, i.e., exponents are used as digits in base 10, while they are listed individually in the rows of A067255.

Examples

			2 = prime(1)^1 => a(2) = 11,
3 = prime(2)^1 => a(3) = 21,
4 = prime(1)^2 => a(4) = 12,
5 = prime(3)^1 => a(5) = 31,
6 = prime(1)^1*prime(2)^1 => a(1) = 1121,
7 = prime(3)^1 => a(7) = 41,
8 = prime(1)^3 => a(8) = 13, and so on.

Crossrefs

Cf. A067599 (decimal encoding of prime factorization).

Programs

  • Maple
    a:= n-> `if`(n=1, 0, parse(cat(seq([numtheory[pi]
       (i[1]), i[2]][], i=sort(ifactors(n)[2]))))):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2018
  • Mathematica
    Array[FromDigits@ Flatten@ Map[{PrimePi@ #1, #2} & @@ # &, FactorInteger@ #] &, 51] (* Michael De Vlieger, Mar 16 2018 *)
  • PARI
    A299400(n)=if(n=factor(n),eval(concat(apply(f->Str(primepi(f[1]),f[2]), Col(n)~))))

A124436 a(1)=1, a(n)=p_i^d_i where p_i is i-th prime and d_i is i-th digit of a(n-1).

Original entry on oeis.org

1, 2, 4, 16, 1458, 2918430506250, 7164640537512654203797788776525821310188011060
Offset: 1

Views

Author

Zak Seidov, Dec 16 2006

Keywords

Comments

Or a(n-1)= decimal encoding of the prime factorization of a(n). Cf. A068633 Let n = p^a*q^b... then a(n) = concatenation paqb..., A067599 Decimal encoding of the prime factorization of n.

Examples

			a(4)=16, a(5)=2^1 * 3^6 = 1458;
a(6)= 2^1 * 3^4 * 5^5 * 7^8 = 2918430506250.

Crossrefs

Cf. A067599, A068633.

Programs

  • Mathematica
    a[1]=1;id[n_]:=id[n]=IntegerDigits[a[n-1]]; a[n_]:=a[n]= Times@@Table[Prime[i]^id[n][[i]],{i,1,Length[id[n]]}]; {1,Table[a[n],{n,2,7}]}//Flatten
Previous Showing 11-15 of 15 results.

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