A036057 -id:A036057 - cp's OEIS frontend

A080035 "Orderly" Friedman numbers (or "good" or "nice" Friedman numbers): Friedman numbers (A036057) where the construction digits are used in the proper order.

127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656

Offset: 1

David Rattner (david_rattner(AT)prusec.com), Mar 14 2003

Primes in this sequence are listed in A252483. The subsequence A156954 is a simpler variant where no parentheses, unary operations (negation) nor concatenation is allowed. - M. F. Hasler, Jan 07 2015

			127 = -1 + 2^7, 343 = (3 + 4) ^ 3, 736 = 7 + 3^6, etc.
The 4th "orderly" Friedman number is 1285 = (1 + 2^8) * 5.

Credit goes to Mike Reid (Brown University) and Eric Friedman (Stetson University).
Colin Rose, "Radical Narcissistic numbers", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 250-254. See page 251.

Michael Brand, Friedman numbers have density 1, Discrete Applied Mathematics, Volume 161, Issues 16-17, November 2013, Pages 2389-2395.
Ed Copeland and Brady Haran, Friedman numbers, Numberphile video, 2014
Eric Friedman, Friedman Numbers.
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Robert G. Wilson v, Table of n, a(n) for n = 1..108 .

Cf. A036057.

More terms from Alonso del Arte, Aug 25 2004

Edited by M. F. Hasler, Jan 07 2015

A106007 Coster numbers: similar to Friedman numbers (A036057), but here the operations + - * / are allowed. All digits of a number have to be used exactly twice.

Original entry on oeis.org

0, 1, 13, 14, 15, 18, 19, 24, 25, 26, 28, 29, 35, 36, 37, 39, 45, 46, 48, 49, 52, 55, 59, 64, 65, 66, 69, 73, 75, 78, 79, 81, 82, 85, 89, 91, 95, 99, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 143, 144, 145, 146, 147, 148, 149, 152, 153, 154, 155, 156, 157, 158

Offset: 0

Matthijs Coster, Dec 31 2006, Feb 02 2007

These numbers were introduced in 2006 in the yearly number puzzle in the Dutch Journal "Pythagoras" (see http://www.pythagoras.nu). It is known that there are infinitely many Coster numbers (result of David Kloet, Albert Hendriks and Arjen Stolk, unpublished).

			36 is in the sequence since 3*6+3*6 = 36

Matthijs Coster, Feb 02 2007, Table of n, a(n) for n = 0..89
Matthijs Coster, www.wiskundemeisjes.nl/20061030/coster-getallen/ (in Dutch)
Matthijs Coster, www.matcos.nl/sequences
Pythagoras, www.pythagoras.nu (in Dutch)

Cf. A036057.

A156954 Integers N such that by inserting + or - or * or / or ^ between each of their digits, without any grouping parentheses, you can get N (the ambiguous a^b^c is avoided).

Original entry on oeis.org

736, 2592, 11664, 15617, 15618, 15622, 15624, 15632, 15642, 15645, 15656, 15662, 15667, 15698, 17536, 27639, 32785, 39363, 39369, 45947, 46633, 46644, 46648, 46655, 46660, 46663, 117635, 117638, 117639, 117642, 117643, 117647, 117650

Offset: 1

Jean-Marc Falcoz, Feb 19 2009

The single-digit numbers 0, ..., 9 are here excluded by convention although they also ("voidly") satisfy the definition and therefore logically should be terms of this sequence. This is in contrast to the Friedman numbers A036057 where the construction also allows concatenation of digits but then of course has to exclude the case where only concatenation of the digits is used, which excludes the single-digit terms. - M. F. Hasler, Jan 07 2015

A subset of the orderly Friedman numbers A080035. - M. F. Hasler, Jan 04 2015

			736 = 7 + 3^6.
2592 = 2^5*9^2.
11664 = 1*1*6^6/4.
15617 = 1*5^6 - 1 - 7.
For more examples, see the link to "decompositions".

Giovanni Resta, Table of n, a(n) for n = 1..423 (terms < 10^8)
Giovanni Resta, Decompositions for terms < 10^8

Cf. A036057, A080035, A252484.

is(n,o=Vecsmall("*+-^/"))={v=Vecsmall(Str(n,n\10)); forstep(i=#v,3,-2,v[i]=v[i\2+1]); n>9 && forvec(s=vector(#v\2,i,[1,#o-(v[i*2+1]==48)]), for(i=1,#s,94==(v[2*i]=o[s[i]])&&i>1&&s[i-1]==4&&next(2));n==eval(Strchr(v))&&return(1))}

Edited by M. F. Hasler, Jan 04 2015

A075047 Numbers k whose prime factorization contains the same digits as k.

Original entry on oeis.org

25, 121, 471663, 931225, 4473225, 6953931, 7301441, 10713728, 13246317, 17332133, 19367424, 34706961, 36310761, 54363717, 68714219, 73553125, 73641071, 74390183, 93478133, 102712448, 102941361, 109502361, 113162997, 115521875, 120934784, 134179011, 134381673, 134472875, 135478125

Offset: 1

Amarnath Murthy, Sep 03 2002

From Robert G. Wilson v, Jun 06 2014, updated Jun 10 2014: (Start)
The number of terms < 10^n: 0, 1, 2, 2, 2, 4, 7, 19, 71, 289, ..., .
There are only two terms which have just one prime factor (excluding multiplicity), i.e., 25 and 121. By index, they are 1 and 2.
The least term with just two prime factors is 471663. By index, they are 3, 4, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, ..., .
The least term with just three prime factors is 4473225. By index, they are 5, 9, 10, 11, 23, 24, 26, 28, 29, 30, 32, 36, 38, 39, 44, 46, 47, 66, ..., .
The least term with just four prime factors is 1713131455. By index, they are 110, 115, 251, ..., .
The least term with k prime factors (including multiplicity), or 0 if impossible or -1 not yet found, are 0, 25, 0, 931225, 7301441, 73553125, 471663, 4473225, 141294375, 251317472, 134179011, 1931229184, -1, 19367424, ..., .
So far ( < 10000000000) the count of digits 1,2,...,9,0 is {520, 271, 388, 254, 216, 211, 371, 172, 262, 117}.
(End)

			25 = 5^2 and 121 = 11^2 are terms.
The term 1971753273 -> 1,9,7,1,7,5,3,2,7,3 -> 1,1,2,3,3,5,7,7,7,9 is in the sequence because its factorization is 3^7*7^1*37^1*59^2 -> 3,7,7,1,3,7,1,5,9,2 -> 1,1,2,3,3,5,7,7,7,9 and this coincides with the digits of the term itself. - _Robert G. Wilson v_, Jun 06 2014

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 311 terms from Robert G. Wilson v)

Cf. A025283, A036057, A064818.

fQ[n_] := Sort@ IntegerDigits@ n == Sort@ Flatten@ IntegerDigits@ FactorInteger@ n; k = 1; lst = {}; While[k < 100000001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Jun 05 2014 *)

isok(n, b=10) = {f = factor(n); v = []; for (i=1, #f~, v = concat(v, digits(f[i,1], b)); v = concat(v, digits(f[i,2], b));); vecsort(v) == vecsort(digits(n, b));} \\ Michel Marcus, Jul 14 2015

More terms from David Wasserman, Jan 02 2005
a(14)-a(23) from Donovan Johnson, Oct 10 2009
a(24)-a(29) from Robert G. Wilson v, Jun 06 2014

A112419 Prime Friedman numbers.

Original entry on oeis.org

127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 16879, 19739, 21943, 27653, 28547, 28559, 29527, 29531, 32771, 32783, 35933, 36457, 39313, 39343, 43691, 45361, 46619, 46633, 46643, 46649, 46663, 46691, 48751, 48757, 49277, 58921, 59051, 59053, 59263, 59273, 64513, 74353, 74897, 78163, 83357

Offset: 1

Lekraj Beedassy, Jan 23 2007

A Friedman number is one which is expressible in a nontrivial manner with the same digits by means of the arithmetic operations +, -, *, "divided by" along with ^ and digit concatenation.
Ron Kaminsky notes that, by Dirichlet's theorem, this sequence is infinite; see Friedman link. - Charles R Greathouse IV, Apr 30 2010
There are only 49 terms below 10^5, and there are less than 40 "orderly" terms (in A080035) below 10^6. - M. F. Hasler, Jan 03 2015

			Since the following primes have expressions 16381 = (1+1)^(6+8) - 3 ; 16447 = -1+64+4^7 ; 16759 = 7^5 - 6*(9-1), they are in the sequence.

Erich Friedman, Problem of the Month (August 2000).

Cf. A036057.

Intersection of A036057 and A000040. - M. F. Hasler, Jan 03 2015

Corrected and extended by Ray Chandler, Apr 24 2010

A252483 Primes among "orderly" Friedman numbers A080035.

Original entry on oeis.org

127, 15667, 16447, 19739, 28559, 32771, 39343, 46633, 46663, 117619, 117643, 117763, 125003, 131071, 137791, 147419, 156253, 156257, 156259, 229373, 248839, 262139, 262147, 279967, 294829, 295247, 326617, 466553, 466561, 466567, 585643, 592763, 649529, 728993, 759359, 786433, 937577

Offset: 1

M. F. Hasler, Jan 04 2015

Intersection of A080035 and A112419.

Cf. A036057, A080035.

A252483=select(isprime,A080035) \\ use, e.g., A080035=apply(s-> eval(concat(vecextract(Vec(s),"7..13"))),readvec("/tmp/a080035.txt")[8..115]); - M. F. Hasler, Jan 07 2015

A386936 Numbers that can be represented using their digits in the order of appearance, the operations +, -, *, /, ^, and any parentheses.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 343, 736, 1285, 2187, 2592, 2737, 3125, 3685, 3972, 4096, 6455, 11664, 14641, 15552, 15585, 15617, 15618, 15622, 15624, 15626, 15632, 15645, 15655, 15656, 15662, 15667, 15698, 16377, 16384, 17536, 19683, 23328, 24576, 27639

Offset: 1

Anuraag Pasula and Walter Robinson, Aug 09 2025

Each digit is its own operand (no concatenation of digits).
Real and imaginary intermediate values are allowed as long as the final value of the expression is an integer.
Unary minus is not allowed, otherwise we would have 127 = -1 + 2^7. - Sean A. Irvine, Aug 31 2025

			343 = (3+4)^3.
2737 = (2*7)^3-7.
46688 = (4 + 6^6/8)*8.

Walter Robinson, Expressions for Values up to 50000

Cf. A036057, A046469, A080035, A156954, A362769.

A083509 Friedman numbers that involve the "^" sign.

Original entry on oeis.org

25, 121, 125, 127, 128, 216, 289, 343, 347, 625, 736, 1022, 1024, 1285, 1296, 1792, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3281, 3375, 3378, 3685, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

			121 is a member as 121 = 11^2. but 126 is not a member as 126 = 21*6 does not involve "^" sign.

Cf. A036057.

Corrected and extended by Arkadiusz Wesolowski, Jun 02 2013

A104233 Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.

Original entry on oeis.org

125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1080, 1089, 1125, 1152, 1156, 1215, 1225, 1250, 1280, 1287, 1288, 1289, 1290, 1291, 1292, 1293, 1294

Offset: 1

Jack Brennen, Apr 01 2005

You are allowed to use the following symbols as well:
  ( ) grouping
  + addition
  - subtraction
  * multiplication
  / division
  ^ exponentiation
Note that 1015 to 1033 are all representable in the form 4^5-d or 4^5+d, where d is a single digit.
The complexity of a number has been defined in several different ways by different authors. See the Index to the OEIS for other definitions. - Jonathan Vos Post, Apr 02 2005
From Bernard Schott, Feb 10 2021: (Start)
These numbers are called "entiers compressibles" in French.
There are no 1-digit or 2-digit terms.
The 3-digit terms are all of the form m^q, with 2 <= m, q <= 9.
The 4-digit terms are of the form m^q with m, q > 1, or of the form m^q+-d or m^q*r with m, q, r > 1, d >= 0, and m, q, r, d are all digits (see examples where [...] is a corresponding "compact" representation). (End)

			From _Bernard Schott_, Feb 10 2021: (Start)
a(1) = 125 = [5^3] = 5*5*5 is the smallest cube.
a(5) = 256 = [2^8] = [4^4] = 16*16 is the smallest square.
a(6) = 343 = [7^3] is the smallest palindrome.
a(15) = 1019 = [4^5-5] is the smallest prime.
6555 = [3^8-5] = [35^2] = T(49) = 49*50/2 is the smallest triangular number.
362880 = 9! = [70*72^2] = [8*(6^6-6^4)] is the smallest factorial.
The smallest zeroless pandigital number 123456789 = [(10^10-91)/81] = [3*(6415^2+38)] is a term. (End)
The largest pandigital number 9876543210 = [(8*10^11+10)/81] = [(8*10^11+10)/9^2] = [5*(15^5+67)*51^2] is also a term. - _Bernard Schott_, Apr 20 2022

R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250. [Cached copy, with permission]
J. Arias de Reyna, Complexity of natural numbers, arXiv:2111.03345 [math.NT], 2021.
Diophante, A164, Les entiers compressibles (in French).
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Eric Weisstein's World of Mathematics, Integer Complexity
Index to sequences related to the complexity of n

Cf. A036057, A005245, A003313, A076142, A076091, A061373, A005421, A064097, A005520, A025280, A003037.

More terms from Bernard Schott, Feb 10 2021

Missing terms added by David A. Corneth, Feb 14 2021

A195420 Friedman numbers n such that n+1 is also a Friedman number.

Original entry on oeis.org

125, 126, 127, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 6144, 11663, 11664, 12100, 12101, 12102, 12103, 12104, 12105, 12106, 12107, 12108, 12768, 13822, 13823, 13824, 13825, 15378, 15567, 15585, 15612, 15617, 15620, 15621, 15622, 15623, 15624

Offset: 1

Kausthub Gudipati, Sep 18 2011

Cf. A036057.

Extended by T. D. Noe, Sep 20 2011

cp's OEIS Frontend

A080035 "Orderly" Friedman numbers (or "good" or "nice" Friedman numbers): Friedman numbers (A036057) where the construction digits are used in the proper order.

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A106007 Coster numbers: similar to Friedman numbers (A036057), but here the operations + - * / are allowed. All digits of a number have to be used exactly twice.

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A156954 Integers N such that by inserting + or - or * or / or ^ between each of their digits, without any grouping parentheses, you can get N (the ambiguous a^b^c is avoided).

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A075047 Numbers k whose prime factorization contains the same digits as k.

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Mathematica

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A112419 Prime Friedman numbers.

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A252483 Primes among "orderly" Friedman numbers A080035.

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A386936 Numbers that can be represented using their digits in the order of appearance, the operations +, -, *, /, ^, and any parentheses.

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A083509 Friedman numbers that involve the "^" sign.

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A104233 Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.

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