A046759 -id:A046759 - cp's OEIS frontend

A254321 Hyper Wasteful numbers.

6, 8, 9, 22, 26, 30, 33, 34, 38, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 62, 63, 65, 66, 68, 70, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 94, 95, 96, 99, 102, 110, 111, 112, 114, 115, 117, 118, 122, 130, 133, 136, 141, 144, 152, 153, 155, 161

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the Wasteful numbers (A046760) is that the distinct digits are counted only instead all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) = number of distinct digits in n; sequence gives n such that D(n) > nbd(n).

			88 is in the sequence because 88 = 2 ^ 3 * 11 => D(88)=3 > nbd(88)=1.

Cf. A043537, A046760, A046758, A046759, A254318, A254319.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]] > Length[IntegerDigits[n]]]

for(n=1,100,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)<#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A047738 Earliest sequence of 4 consecutive economical numbers.

Original entry on oeis.org

179210312, 179210313, 179210314, 179210315

Offset: 1

Carlos Rivera

			179210312 = 2^3* 4733^2; 9 digits in 179219312, 7 digits in its prime factorization, 7 < 9, so that number is economical.

Tanya Khovanova, Non Recursions
J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, Integers: Electronic Journal of Combinatorial Number Theory 5, (2005), #A5.
Eric Weisstein's World of Mathematics, Economical Number

Cf. A046759.

A241034 Start number of first run of exactly n consecutive economical numbers.

Original entry on oeis.org

125, 4374, 1097873, 179210312, 10546514090622

Offset: 1

Arkadiusz Wesolowski, Apr 15 2014

J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, Integers: Electronic Journal of Combinatorial Number Theory 5, (2005), #A5.
Richard G.E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998
Carlos Rivera, Puzzle 53
Eric Weisstein's World of Mathematics, Economical Number

Cf. A046759, A079323.

a(5) found by Jens Kruse Andersen

A377478 a(n) = first frugal number > 1 in base n.

Original entry on oeis.org

27, 32, 27, 27, 81, 64, 64, 81, 125, 125, 243, 243, 243, 243, 256, 343, 343, 512, 512, 512, 512, 625, 625, 625, 729, 729, 1024, 1024, 1024, 1024, 1024, 1331, 1331, 1331, 1331, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2187, 2401, 2401, 2401, 3125, 3125

Offset: 2

Paolo Xausa, Dec 28 2024

A frugal number in base n is a number with more digits (in its base n representation) than the total number of digits (in base n representation) of its prime factorization (including exponents > 1).

Following the definition by Pinch (1998), 1 is considered a frugal number.

			a(2) = 27 because 27 = 3^3 = 11_2^11_2; the total number of bits of (11_2, 11_2) = 4 < the number of bits of 27 = 11011_2 (5); and 27 is the first number > 1 with this property.
a(3) = 32 because 32 = 2^5 = 2_3^12_3; the total number of digits of (2_3, 12_3) = 3 < the number of digits in base 3 of 32 = 1012_3 (4); and 32 is the first number > 1 with this property.

Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Giovanni Resta, Frugal numbers, Numbers Aplenty, 2013.
Wikipedia, Frugal number.

Column k = 2 of A379538.

Cf. A046759, A379537.

Module[{m}, Table[m = 1; While[Total[IntegerLength[Select[Flatten[FactorInteger[++m]], # > 1 &], n]] >= IntegerLength[m, n]]; m, {n, 2, 50}]]

A047739 Economical palindromes.

Original entry on oeis.org

343, 1331, 10201, 14641, 31213, 94249, 1030301, 1332331, 1367631, 1478741, 2060602, 2513152, 2551552, 2570752, 2939392, 2977792, 3090903, 3498943, 3637363, 3735373, 3850583, 5221225, 5265625, 5734375, 6948496, 11288211, 11333311, 12100121, 12499421, 13577531, 14655641, 22666622, 27700772, 29399392, 31244213, 33999933, 38988983, 58344385, 71111117, 100020001, 100585001, 101000101, 104060401, 104555401, 106878601

Offset: 1

Carlos Rivera

			343 = 7^3; 3 digits in 343, 2 digits in 7^3, so 343 is economical.

Cf. A002113, A046759.

d[n_]:=IntegerDigits[n]; palQ[n_]:=Reverse[(x=d[n])]==x; ecQ[n_]:=Length[Flatten[d[{First/@(x=FactorInteger[n]), DeleteCases[Last/@x,1]}]]] < Length[d[n]]; t={}; Do[If[palQ[n] && ecQ[n], AppendTo[t,n]], {n,3860000}]; t (* Jayanta Basu, May 16 2013 *)

a(11)-a(45) from Donovan Johnson, Nov 24 2010

A172484 Partial sums of extravagant numbers, also called prodigal numbers, or wasteful numbers.

Original entry on oeis.org

4, 10, 18, 27, 39, 57, 77, 99, 123, 149, 177, 207, 240, 274, 310, 348, 387, 427, 469, 513, 558, 604, 652, 702, 753, 805, 859, 914, 970, 1027, 1085, 1145, 1207, 1270, 1335, 1401, 1469, 1538, 1608, 1680, 1754, 1829, 1905, 1982, 2060, 2140, 2222, 2306, 2391, 2477

Offset: 1

Jonathan Vos Post, Feb 04 2010

Every natural number, written in base 10, is either economical A046759 (also called frugal), or equidigital A046758, or extravagant (or prodigal or wasteful). An extravagant number is one for which the factorization requires more digits that the original number such as 30 = 2 * 3 * 5. The subsequence of economical partial sums of extravagant numbers begins: xxx, 18, 39, 57, 77, 99, 207, 240, 274, 310. The subsequence of equidigital partial sum of economical numbers begins: 10, 27, 123, 149, 177, 427, 469 (such as 1207 = 17 * 71). The subsequence of prime partial sums of economical numbers begins: xxx, 149, 859, 2477, 2833.

			a(1) = A046760(1) = 4. a(2) = 4 + 6 = 10. a(67) = 4 + 6 + 8 + 9 + 12 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 33 + 34 + 36 + 38 + 39 + 40 + 42 + 44 + 45 + 46 + 48 + 50 + 51 + 52 + 54 + 55 + 56 + 57 + 58 + 60 + 62 + 63 + 65 + 66 + 68 + 69 + 70 + 72 + 74 + 75 + 76 + 77 + 78 + 80 + 82 + 84 + 85 + 86 + 87 + 88 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 98 + 99 + 100 + 102 + 104 + 108 + 110 + 114 = 4138 = 2 * 2069 which is thus an economical number, with 4 digits but 5 in its prime factorization.

Cf. A046758, A046759, A046760.

SUM[i=1..n] A046760(i) = Partial sum of {Write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n)}.

27 inserted by R. J. Mathar, Feb 06 2010

A294077 Smallest integer k such that k + x is an economical number for every x with 0 <= x <= n-1.

Original entry on oeis.org

1, 1, 13, 13, 157, 157, 1169312, 10990399, 1016258233

Offset: 2

Felix Fröhlich, Feb 07 2018

The values of a(2)-a(10) are given in De Koninck, Luca, 2005.

J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, Integers: Electronic Journal of Combinatorial Number Theory 5, (2005), #A5.

Cf. A046759, A241034.

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A254321 Hyper Wasteful numbers.

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A047738 Earliest sequence of 4 consecutive economical numbers.

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A241034 Start number of first run of exactly n consecutive economical numbers.

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A377478 a(n) = first frugal number > 1 in base n.

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A047739 Economical palindromes.

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A172484 Partial sums of extravagant numbers, also called prodigal numbers, or wasteful numbers.

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A294077 Smallest integer k such that k + x is an economical number for every x with 0 <= x <= n-1.

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