A080670 -id:A080670 - cp's OEIS frontend

A291803 Partial sums of A230625.

1, 3, 6, 16, 21, 32, 39, 50, 64, 85, 96, 139, 152, 175, 204, 224, 241, 287, 306, 391, 422, 465, 488, 535, 557, 602, 617, 704, 733, 826, 857, 878, 937, 1018, 1065, 1239, 1276, 1359, 1420, 1513, 1554, 1649, 1692, 1863, 1980, 2067, 2114, 2197, 2227, 2313, 2426

Offset: 1

N. J. A. Sloane, Sep 06 2017

nonn

Although A230625 is a very irregular function, here the graph is much smoother, and it appears that a(n) approx= 1.2*n^2. It would be nice to have a more precise estimate.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A080670, A287881, A287882, A287874, A230625.

A230625 = Cases[Import["https://oeis.org/A230625/b230625.txt", "Table"], {, }][[All, 2]];
Accumulate[A230625] (* Robert Price, Mar 16 2020 *)

A067254 Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.

Original entry on oeis.org

11, 8571, 11371, 190911, 12711811, 14713491, 19090911, 71119711, 12531135391, 15311195711, 112717566411, 158318548011, 518914376931, 7292811659931

Offset: 1

Joseph L. Pe, Feb 20 2002

a(13) > 2*10^11. 518914376931, 7292811659931, 19090909090909090911 and prime repunits (A004022) are also terms. - Donovan Johnson, Dec 04 2012
Are there any terms not ending in 1? Equivalently, are any terms also in A070003? - Charles R Greathouse IV, Dec 05 2012
a(15) > 10^13. If exponents equal to 1 are not represented (as in A080670), the corresponding sequence starts with 113113, 31373137, and 533517177839 = 853 * 3517 * 177839. - Giovanni Resta, Jun 26 2017

			The prime factorization of 190911 is 3^1 * 7^1 * 9091^1 with decimal encoding 317190911, which ends in 190911. Hence 190911 is a term of the sequence.

Cf. A004022, A067599, A070003, A080670.

(*returns true if a ends with b, false o.w.*) f[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; e = StringLength[c]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[2]] == e, r = True]]; r]; (*gives the decimal encoding of the prime factorization of n*) g[n_] := FromDigits[Flatten[IntegerDigits[FactorInteger[n]]]]; Do[If[f[g[n], n], Print[n]], {n, 1, 10^6} ]

{a067254(a,b) = local(n,v,k,j); for(n=max(2,a),b,v=factor(n); if(eval(concat(vector(matsize(v)[1],k, concat(vector(matsize(v)[2],j,Str(v[k,j]))))))%(10^length(Str(n)))==n,print1(n,",")))}
a067254(2,2*10^7) \\ Klaus Brockhaus, Feb 22 2002

a(5)-a(7) from Klaus Brockhaus, Feb 22 2002
a(8)-a(10) from Donovan Johnson, Mar 26 2010
a(11)-a(12) from Donovan Johnson, Dec 04 2012
a(13) from Giovanni Resta, Jun 09 2017
a(14) from Giovanni Resta, Jun 26 2017

A288532 Literal reading of the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 23, 32, 25, 11, 223, 13, 27, 35, 222, 17, 232, 19, 225, 37, 211, 23, 233, 52, 213, 33, 227, 29, 235, 31, 25, 311, 217, 57, 2232, 37, 219, 313, 235, 41, 237, 43, 2211, 325, 223, 47, 2223, 72, 252, 317, 2213, 53, 233, 511, 237, 319, 229

Offset: 1

Rémy Sigrist, Jun 11 2017

The prime tower factorization of a number is defined in A182318.
The sequence is similar to A080670; however here we recursively factorize prime exponents.
a(1) = 1 by convention.
a(p) = p for any prime p.
As for A080670, 13532385396179 is a composite fixed point.

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A080670, A182318.

Array[FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[#, 1] /. {} -> {1}] &@ Flatten@ FixedPoint[Map[If[PrimeQ@ Last@ # || Last@ # == 1, #, {First@ #, FactorInteger@ Last@ #}] &, #, {Depth@ # - 2}] &, FactorInteger@ #] &, 58] (* or *)
Table[FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[ Flatten[ FactorInteger[n] //. {p_, e_} /; e > 1 :> {p, FactorInteger@ e}], 1] /. {} -> {1}], {n, 58}] (* Michael De Vlieger, Jun 11 2017 *)

a(n) = my (s="", f=factor(n)); for (i=1, #f~, s=concat(s,Str(f[i,1])); if (f[i,2]>1, s=concat(s,Str(a(f[i,2]))))); return (if(s=="", 1, eval(s)))

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

M. F. Hasler, Mar 08 2018

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.

			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.

Cf. A067599 (decimal encoding of prime factorization).

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

Array[FromDigits@ Flatten@ Map[{PrimePi@ #1, #2} & @@ # &, FactorInteger@ #] &, 51] (* Michael De Vlieger, Mar 16 2018 *)

A299400(n)=if(n=factor(n),eval(concat(apply(f->Str(primepi(f[1]),f[2]), Col(n)~))))

A384537 Composite numbers that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

Original entry on oeis.org

16, 27, 64, 256, 729, 1024, 3125, 4096, 4617, 16384, 19683, 29767, 65536, 255987, 262144, 395847, 531441, 631463, 823543, 1048576, 1332331, 4194304, 9765625, 14348907, 16777216, 25640947, 67108864

Offset: 1

Jianing Song, Jun 02 2025

Someone called James Davis found that 13532385396179 = 13 * 53^2 * 3853 * 96179, showing that a composite number can be equal the concatenation of the primes and exponents in its canonical prime factorization. In general, if a composite number is equal the concatenation in base b of the primes and exponents in its prime factorization, then let's call it a Davis number to base b.
Conjecture: a composite number can be a Davis number to at most one base.
Let (d_1,...,d_r) be the ordered tuple of prime factors and exponents > 1 in the prime factorization of n (e.g., 4617 = 3^5 * 19 -> (3,5,19), 13532385396179 = 13 * 53^2 * 3853 * 96179 -> (13,53,2,3853,96179)), then n is a Davis number to base b if and only if n = d_1*b^{s_1} + ... + d_{r-1}*b^{s_{r-1}} + d_r, where s_i = (Sum_{j=i+1..r} floor(log_b(d_j))) + r-i. In particular, we must have b dividing n - d_r.
Suppose that p^e is a Davis number to some base b, with e >= 2. We have p^e = p*b^(floor(log_b(e))+1) + e in base b, hence e is divisible by p. If b <= e, then we have p^e <= p*b^(log_b(e)+1) + e <= p*e^2 + e, which is impossible, and so we must have b > e. Conversely, when e is divisible by p and p^e > 4, p^e is a Davis number to base (p^e-e)/p > e.
No term can be squarefree: for primes p_1 < ... < p_r, the concatenation of p_1, ..., p_r in base b is p_1 * b^(Sum_{i=2..r} (floor(log_b(p_i))+1)) + ... >= p_1*...*p_r + ... > p_1*...*p_r.
Here are some examples that are near-miss of being Davis numbers to base 10. Each is equal to the concatenation of the factors and exponents in its generalized factorization (we call n = (q_1)^(e_1) * ... * (q_k)^(e_k) a generalized factorization of n, where 1 < q_1 < ... < q_k, (q_1,...,q_k) are pairwise coprime but are not necessarily primes, and exponents 1 are omitted; the number of such factorizations is A327399(n)):
  2592 = 2^5 * 9^2;
  34425 = 3^4 * 425;
  312325 = 31^2 * 325;
  492205 = 49^2 * 205;
  36233196159122085048010973936921313644799483579440006455257 = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257. (Note that in the last four examples, we can add as many trailing zeros as we want).

			In base 6: 24 = 2^4 (in decimal: 16 = 2^4);
In base 8: 33 = 3^3 (in decimal: 27 = 3^3);
In base 29: 26 = 2^6 (in decimal: 64 = 2^6);
In base 124: 28 = 2^8 (in decimal: 256 = 2^8);
In base 241: 36 = 3^6 (in decimal: 729 = 3^6);
In base 507: 2A = 2^A (in decimal: 1024 = 2^10);
In base 624: 55 = 5^5 (in decimal: 3125 = 5^5);
In base 2042: 2C = 2^C (in decimal: 4096 = 2^12);
In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19).
See A384540 for more nontrivial examples.

Cf. A080670, A195264, A230625, A384540, A327399.

F(n,b) = my(f=factor(n), d=[]); for(i=1, #f~, d=concat(d, digits(f[i,1],b)); if(f[i,2]>1, d=concat(d, digits(f[i,2],b)))); fromdigits(d,b)
isA384537(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(n > 4 && f[1,2] % f[1,1] == 0));
dr = if(f[#f~,2] == 1, f[#f~,1], f[#f~,2]);
fordiv(n - dr, b, if(b>=2 && F(n,b)==n, return(b))); return(0);
} \\ returns the (smallest) base to which n is a Davis number whenever possible

A384540 Numbers in A384537 that are not prime powers: composite numbers, not being prime powers, that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

Original entry on oeis.org

4617, 29767, 255987, 395847, 631463, 1332331, 25640947

Offset: 1

Jianing Song, Jun 02 2025

See A384537 for more information.

			In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19);
In base 12: 15287 = 15^2 * 87 (in decimal: 29767 = 17^2 * 103);
In base 2: 111110011111110011 = 11^11 * 10011 * 111110011 (in decimal: 255987 = 3^3 * 19 * 499);
In base 362: (3,7,181)_362 = 3^7 * 181.
In base 300: (7,4,263)_300 = 7^4 * 263.
In base 57: 7B4D = 7 * B^4 * D (in decimal: 1332331 = 7 * 11^4 * 13).
In base 1228: (17,4,307)_1228 = 17^4*307.

Cf. A080670, A195264, A230625, A384537.

F(n, b) = my(f=factor(n), d=[]); for(i=1, #f~, d=concat(d, digits(f[i, 1], b)); if(f[i, 2]>1, d=concat(d, digits(f[i, 2], b)))); fromdigits(d, b)
isA384540(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(0));
dr = if(f[#f~, 2] == 1, f[#f~, 1], f[#f~, 2]);
fordiv(n - dr, b, if(b>=2 && F(n, b)==n, return(b))); return(0);
} \\ returns the (smallest) base to which n is a Davis number whenever possible

A287763 Numbers n for which A230625(n) < n, where A230625 is the concatenation of the prime factorization written in binary, converted back to decimal.

Original entry on oeis.org

25, 27, 32, 49, 54, 64, 81, 96, 98, 121, 125, 128, 135, 147, 162, 169, 189, 192, 216, 224, 242, 243, 245, 250, 256, 289, 297, 320, 338, 343, 351, 352, 361, 363, 375, 384, 392, 405, 416, 432, 448, 486, 500, 507, 512, 513, 529, 539, 567, 576, 578, 605, 621, 625, 637, 640, 648, 675, 686, 704, 722, 729, 736, 750, 768, 775, 783, 784

Offset: 1

M. F. Hasler, Jun 21 2017

Relevant for the study of closed orbits. (This is to A230625 the analog of A195330 for A080670.) Up to a certain limit, the trajectory of all numbers, under iteration of A230625, end either in a prime (fixed point) or in one of the orbits {1007, 1269} or {1503,3751}.

See A288985 for the analog when A287874 is used instead of A230625, i.e., without converting back the concatenation of the binary strings to decimal, or rather, reading it as a decimal number.

Cf. A230625, A195330, A080670, A287874.

Select[Range@ 800, Function[n, FromDigits[#, 2] < n &@ Flatten@ Map[IntegerDigits[#, 2] &, FactorInteger[n] /. {p_, 1} :> {p}]]] (* Michael De Vlieger, Jun 23 2017 *)

PARI
```
forcomposite(n=1,1e6,A230625(n)
				
```

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A291803 Partial sums of A230625.

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A067254 Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.

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A288532 Literal reading of the prime tower factorization of n.

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

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A384537 Composite numbers that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

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A384540 Numbers in A384537 that are not prime powers: composite numbers, not being prime powers, that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

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A287763 Numbers n for which A230625(n) < n, where A230625 is the concatenation of the prime factorization written in binary, converted back to decimal.

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