A080035 -id:A080035 - cp's OEIS frontend

A252483 Primes among "orderly" Friedman numbers A080035.

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

A036057 Friedman numbers: can be written in a nontrivial way using their digits and the operations + - * / ^ and concatenation of digits (but not of results).

Original entry on oeis.org

25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3159

Offset: 1

Erich Friedman

Mitchell's and Wilson's lists both lack two terms, 16387 = (1-6/8)^(-7)+3 and 41665 = 641*65. - Giovanni Resta, Dec 14 2013

Primes in this sequence are listed in A112419. See also the subsequence A080035 of "orderly" terms, and its subset A156954. - M. F. Hasler, Jan 04 2015

			E.g., 153=51*3, 736=3^6+7. Not 26 = 2 6 (concatenated), that's trivial.

M. F. Hasler, Table of n, a(n) for n = 1..844 (data from E. Friedman's page as collected by K. Mitchell, completed by the two missing terms found by G. Resta).
M. 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
Erich Friedman, Friedman Numbers
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Giovanni Resta, Friedman numbers Friedman numbers and expressions up to 10^6
Robert G. Wilson v, Table of n, a(n) with factorizations for n=1..844
Index entries for Four 4's problem

Cf. A080035, A156954, A046469.

a(n) ~ n, see Brand. - Charles R Greathouse IV, Jun 04 2013

Edited by Michel Marcus and M. F. Hasler, Jan 04 2015

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

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

A119710 Radical narcissistic numbers: numbers n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ and the radical symbol, but which are not already 'Good' Friedman numbers (i.e., the radical is needed for the solution to exist). Concatenation is allowed.

Original entry on oeis.org

729, 1296, 1764, 2378, 2744, 2746, 3645, 4372, 4374, 4913, 5184, 6495, 6859, 8192

Offset: 1

Colin Rose, Jun 10 2006

There are only 14 radical narcissistic integers n < 10000.

			729 = (7 + 2)^sqrt(9);
2378 = -23 + sqrt(7^8).

J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons, (1966), pp. 163 - 175.
Colin Rose, "Radical Narcissistic numbers", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 250-254.

C. Rose, Pretty Wild Narcissistic Numbers - numbers that pwn (2006)
Wikipedia, Friedman number

Cf. A080035.

Mathematica code to be available at: http://www.numq.com/pwn/

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.

A193069 Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.

Original entry on oeis.org

24, 36, 71, 119, 120, 127, 143, 144, 145, 216, 240, 343, 354, 355, 360, 384, 456, 595, 660, 693, 713, 715, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 733, 736, 744, 799, 936

Offset: 1

Colin Rose, Aug 08 2011

Unlike 'radical narcissistic numbers', the pretty wild variety are allowed to use factorials.

Pretty wild narcissistic numbers nest both radical narcissistic numbers and ordered (nice) Friedman numbers.

			24 = (2 Sqrt[4])! , 36 = 3! 6,  127 = -1 + 2^7

J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons, (1966), pp. 163 - 175.
Colin Rose, "Radical Narcissistic numbers", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 250-254.

C. Rose, Pretty Wild Narcissistic Numbers - numbers that pwn
Inder J. Taneja, Selfie Numbers-IV: Addition, Subtraction and Factorial, RGMIA Research Report Collection, 19 (2016), #163, pp. 1-42.
Inder J. Taneja, Selfie Numbers, Fibonacci Sequence and Selfie Fractions, RGMIA Collection, 19 (2016), #167, pp. 1-24.
Wikipedia, Friedman number

Cf. A080035, A119710

cp's OEIS Frontend

A252483 Primes among "orderly" Friedman numbers A080035.

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A036057 Friedman numbers: can be written in a nontrivial way using their digits and the operations + - * / ^ and concatenation of digits (but not of results).

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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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A112419 Prime Friedman numbers.

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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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A193069 Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.

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