A046469 -id:A046469 - cp's OEIS frontend

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

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

A046471 Number of numbers k>1 such that k equals the sum of digits in k^n.

Original entry on oeis.org

8, 1, 5, 5, 4, 4, 8, 3, 3, 6, 3, 1, 11, 5, 7, 6, 4, 2, 9, 3, 3, 7, 3, 3, 13, 4, 2, 6, 5, 1, 10, 1, 7, 3, 5, 2, 8, 2, 2, 6, 1, 4, 9, 5, 3, 8, 8, 4, 11, 1, 3, 4, 4, 5, 2, 1, 6, 3, 4, 4, 5, 2, 3, 4, 4, 3, 8, 1, 5, 3, 2, 2, 5, 4, 5, 3, 3, 4, 8, 4, 2, 4, 4, 1, 5, 2, 6, 6, 3, 2, 7, 3, 3, 8, 5, 1, 7, 1, 4, 5, 2, 3, 9

Offset: 1

Patrick De Geest, Aug 15 1998

The number of digits in k^n is at most 1+n*log(k). Hence the maximum sum of digits of k^n is 9(1+n*log(k)). By solving k=9(1+n*log(k)), we can compute an upper bound on k for each n. Sequence A133509 lists the n for which a(n)=0.

			a(17)=4 -> sum-of-digits{x^17}=x for x=80,143,171 and 216 (x>1).

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A046459, A046469, A046000.
a(n) = A046019(n) - 1.
Cf. A152147 (table of k such that the sum of digits of k^n equals k)

Edited by T. D. Noe, Nov 25 2008

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.

cp's OEIS Frontend

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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A046471 Number of numbers k>1 such that k equals the sum of digits in k^n.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs