astonishing numbers

Richard Hoshino called astonishing those number $n=a|b$

whose representation can be decomposed into two parts, $a$

and

, such that $n$

is equal to the sum of the integers from $a$

Here we relax a little the requirement and also consider astonishing $n$ if it is the sum of the integers from $b$ to $a$ .

For example, $429 = 4+5+\cdots+29$ and $190=0+1+\cdots+19$ .

Every number greater than 473 can be written as the sum of astonishing numbers.

The first astonishing numbers are 15, 27, 190, 204, 216, 429, 1353, 1863, 3078, 3388, 3591, 7119, 19900, 20328, 21252, 21762, 23287, 23490, 78403, 133533, 178623, 1301613, 200207 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many astonishing numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links R. Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum 27(1), 2001, p. 39-44.
OEIS, Sequence A186074
OEIS, Sequence A070152
OEIS, Sequence A070153

Astonishing numbers can also be... (you may click on names or numbers and on + to get more values)

aban 15 27 204 216 429 abundant 204 216 3078 3388 20328 21252 21762 23490 2006118 2132532 2177622 2282148 17016076 32738728 alternating 27 216 3078 21252 23490 amenable 204 216 429 1353 3388 20328 21252 133533 1301613 2002077 + 203872812 213325332 227367888 793039833 apocalyptic 3078 3388 7119 20328 21252 21762 23287 23490 arithmetic 15 27 204 429 1353 3078 3591 7119 20328 21252 + 2282148 2732353 3882813 7103835 Bell 15 binomial 15 brilliant 15 cake 15 canyon 204 216 429 3078 Catalan 429 congruent 15 216 429 1863 3078 3388 3591 7119 21762 23287 + 2006118 2132532 2177622 3882813 constructible 15 204 cube 27 216 Cunningham 15 Curzon 429 1353 3882813 124015749 cyclic 15 1353 23287 78403 D-number 15 d-powerful 2132532 decagonal 27 deficient 15 27 429 1353 1863 3591 7119 23287 78403 133533 + 2077402 2732353 3882813 7103835 dig.balanced 15 204 216 3591 20328 23490 double fact. 15 Duffinian 27 3591 23287 78403 2732353 economical 15 27 1863 emirpimes 15 equidigital 15 27 1863 evil 15 27 204 216 429 3078 3591 7119 20328 23490 + 207093834 217006075 217776222 235629162 Friedman 216 23490 gapful 3078 20328 21252 2002077 3882813 17786223 202272147 203872812 213325332 217006075 + 2188066188 12643159518 15238175238 38788281213 happy 1353 21762 23287 2732353 Harshad 27 204 216 3078 3388 21252 21762 23490 2006118 2132532 + 124015749 202272147 203872812 4278092598 hexagonal 15 hoax 133533 178623 iban 27 204 idoneal 15 inconsummate 216 3591 20328 21762 interprime 15 21762 Jordan-Polya 216 junction 204 216 1301613 7103835 Lehmer 15 78403 lucky 15 429 133533 1301613 Lynch-Bell 15 216 magic 15 metadrome 15 27 Moran 27 mountain 1353 1863 3591 23490 nonagonal 204 nude 15 216 7119 O'Halloran 204 oban 15 27 odious 1353 1863 3388 21252 21762 23287 78403 178623 1301613 2132532 + 232728727 308725039 347826603 793039833 panconsummate 15 pandigital 15 216 partition 15 pernicious 1353 1863 3388 21252 21762 23287 78403 178623 1301613 2132532 7103835 plaindrome 15 27 3388 power 27 216 powerful 27 216 practical 204 216 3078 3388 20328 21252 21762 23490 2132532 2282148 pseudoperfect 204 216 3078 3388 20328 21252 21762 23490 repunit 15 Ruth-Aaron 15 1863 self 3078 2006118 202272147 227367888 232728727 semiprime 15 Smith 27 7119 sphenic 429 1353 23287 78403 1301613 2732353 super Niven 204 super-d 3388 7119 78403 178623 tau 204 2132532 tetranacci 15 triangular 15 uban 15 27 Ulam 429 3591 unprimeable 204 21252 untouchable 216 3078 3388 21762 23490 wasteful 204 216 429 1353 3078 3388 3591 7119 20328 21252 + 2282148 2732353 3882813 7103835 Zuckerman 15 216 7119 Zumkeller 204 216 3078 3388 20328 21252 21762 23490 zygodrome 3388