AFriedman number is aninteger, whichrepresented in a givennumeral system, is the result of a non-trivial expression using all its owndigits in combination with any of the four basicarithmetic operators (+, −, ×, ÷),additive inverses, parentheses,exponentiation, andconcatenation. Here, non-trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in thedecimal numeral system, since 347 = 7³ + 4. The decimal Friedman numbers are:

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, ... (sequenceA036057 in theOEIS).

Friedman numbers are named after Erich Friedman, a now-retired mathematics professor atStetson University and recreational mathematics enthusiast.

AFriedman prime is a Friedman number that is alsoprime. The decimal Friedman primes are:

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, ... (sequenceA112419 in theOEIS).

The expressions of the first few Friedman numbers are:

Anice Friedman number is a Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 2⁷ − 1 as 127 = −1 + 2⁷. The first nice Friedman numbers are:

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, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 (sequenceA080035 in theOEIS).

Anice Friedman prime is anice Friedman number that's also prime. The first nice Friedman primes are:

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 (sequenceA252483 in theOEIS).

Michael Brand proved that the density of Friedman numbers among the naturals is 1,^[1] which is to say that the probability of a number chosen randomly and uniformly between 1 andn to be a Friedman number tends to 1 asn tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary nice Friedman numbers.^[2] The case of base-10 nice Friedman numbers is still open.

Vampire numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60.

There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number asmb +n, whereb is the base andm,n are integers from 0 tob−1, we need only check each possible combination ofm andn against the equalitiesmb +n =mⁿ, andmb +n =n^m to see which ones are true. We need not concern ourselves withm +n orm ×n, since these will always be smaller thanmb +n whenn <b. The same clearly holds form −n andm /n.

Friedman numbers also exist for bases other than base 10. For example, 11001₂ = 25 is a Friedman number in thebinary numeral system, since 11001 = 101¹⁰.

The first few known Friedman numbers in other small bases are shown below, written in their respective bases. Numbers shown in bold are nice Friedman numbers.^[3]

In base${\displaystyle b=mk-m}$,

${\displaystyle b^2+mb+k=(mk-m+m)b+k=mbk+k=k(mb+1)}$

is a Friedman number (written in base${\displaystyle b}$ as 1mk =k ×m1).^[4]

In base${\displaystyle b>2}$,

${\displaystyle {(b^n+1)}^2=b^{2n}+2b^n+1}$

is a Friedman number (written in base${\displaystyle b}$ as 100...00200...001 = 100..001², with${\displaystyle n-1}$ zeroes between each nonzero number).^[4]

In base${\displaystyle b=\frac{k(k-1)}{2}}$,

${\displaystyle 2b+k=2\left(\frac{k(k-1)}{2}\right)+k=k^2-k+k=k^2}$

is a Friedman number (written in base${\displaystyle b}$ as 2k =k²). From the observation that all numbers of the form 2k × b²ⁿ can be written ask000...000² withn 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for${\displaystyle k=5}$, or inbase 10, 250068 = 500² + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099 inbase 10.^[4]

Repdigit Friedman numbers:

• The smallest repdigit inbase 8 that is a Friedman number is 33 = 3³.
• The smallest repdigit inbase 10 that is thought to be a Friedman number is 99999999 = (9 + 9/9)^9−9/9 − 9/9.^[4]
• It has been proven thatrepdigits with at least 22 digits are nice Friedman numbers.^[4]

There are an infinite number of prime Friedman numbers in all bases, because for base${\displaystyle 2\le b\le 6}$ the numbers

${\displaystyle n\times {10}^{1111}+11111111=n\times {10}^{1111}+{10}^{1000}-1+0+0}$ in base 2

${\displaystyle n\times {10}^{102}+1101221=n\times {10}^{102}+2^{101}+0+0}$ in base 3

${\displaystyle n\times {10}^{20}+310233=n\times {10}^{20}+{33}^3+0}$ in base 4

${\displaystyle n\times {10}^{13}+2443111=n\times {10}^{4+4}+(2\times 3{)}^{11}}$ in base 5

${\displaystyle n\times {10}^{13}+25352411=n\times {10}^{2\times 5-1}+(5+2{)}^{(3+4)}}$ in base 6

for base${\displaystyle 7\le b\le 10}$ the numbers

${\displaystyle n\times {10}^{60}+164351=n\times {10}^{60}+(10+4-3{)}^5+0+0+\dots }$ in base 7,

${\displaystyle n\times {10}^{60}+163251=n\times {10}^{60}+(10+3-2{)}^5+0+0+\dots }$ in base 8,

${\displaystyle n\times {10}^{60}+162151=n\times {10}^{60}+(10+2-1{)}^5+0+0+\dots }$ in base 9,

${\displaystyle n\times {10}^{60}+161051=n\times {10}^{60}+(10+1-0{)}^5+0+0+\dots }$ in base 10,

and for base${\displaystyle b>10}$

${\displaystyle n\times {10}^{50}+\text{15AA51}=n\times {10}^{50}+(10+\text{A}/\text{A}{)}^5+0+0+\dots }$

are Friedman numbers for all${\displaystyle n}$. The numbers of this form are an arithmetic sequence${\displaystyle pn+q}$, where${\displaystyle p}$ and${\displaystyle q}$ are relatively prime regardless of base as${\displaystyle b}$ and${\displaystyle b+1}$ are always relatively prime, and therefore, byDirichlet's theorem on arithmetic progressions, the sequence contains an infinite number of primes.

In a trivial sense, allRoman numerals with more than one symbol are Friedman numbers. The expression is created by simply inserting + signs into the numeral, and occasionally the − sign with slight rearrangement of the order of the symbols.

Some research into Roman numeral Friedman numbers for which the expression uses some of the other operators has been done. The first such nice Roman numeral Friedman number discovered was 8, since VIII = (V - I) × II. Other such nontrivial examples have been found.

The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case withpositional notation numbering systems) but with the numbers of symbols it has. For example, it is much tougher to figure out whether 147 (CXLVII) is a Friedman number in Roman numerals than it is to make the same determination for 1001 (MI). With Roman numerals, one can at least derive quite a few Friedman expressions from any new expression one discovers. Since 8 is a nice nontrivial nice Roman numeral Friedman number, it follows that any number ending in VIII is also such a Friedman number.

• OEISsequence A036057 (Friedman number)
  • Other Friedman numbers inThe On-Line Encyclopedia of Integer Sequences
• "Friedman numbers".Github. Problem of the Month. Aug 2000.
• Brand, Michael (Nov 2013)."Friedman numbers have density 1".Discrete Applied Mathematics.161 (16–17):2389–2395.doi:10.1016/j.dam.2013.05.027.
• OEISsequence A119710 (Radical narcissistic numbers)
  • "Pretty wild narcissistic numbers - numbers that pwn".Theoretical Research Institute.Extension to Friedman numbers

• Base-dependent integer sequences

• Articles with short description
• Short description is different from Wikidata