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Numbers k = rad(k) * p(k) * q(k) such that A008479(k) = 5, where rad = A007947, p = least prime factor of k, q = second smallest distinct prime factor of k.
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36, 225, 252, 396, 441, 468, 612, 684, 828, 1044, 1116, 1225, 1332, 1476, 1548, 1692, 1908, 2124, 2196, 2412, 2556, 2628, 2772, 2844, 2988, 3025, 3204, 3276, 3492, 3636, 3708, 3825, 3852, 3924, 4068, 4225, 4275, 4284, 4572, 4716, 4788, 4932, 5004, 5148, 5175, 5364
EXAMPLE
Let S = {m < n : rad(m) = rad(n)}. Let P(i..j) = Product_{h=i..j} prime(h).
Table of n, a(n) for select n:
n a(n) S(n)
-----------------------------------------------------------------------------
1 36 = 2^2 * 3^2 {6, 12, 18, 24}
2 225 = 3^2 * 5^2 {15, 45, 75, 135}
3 252 = 2^2 * 3^2 * 7 {42, 84, 126, 168}
4 396 = 2^2 * 3^2 * 11 {66, 132, 198, 264}
12 1225 = 5^2 * 7^2 {35, 175, 245, 875}
23 2772 = 2^2 * 3^2 * P(4..5) {462, 924, 1386, 1848}
51 5929 = 7^2 * 11^2 {77, 539, 847, 3773}
180 20449 = 11^2 * 13^2 {143, 1573, 1859, 17303}
313 36036 = 2^2 * 3^2 * P(4..6) {6006, 12012, 18018, 24024}
426 48841 = 13^2 * 17^2 {221, 2873, 3757, 37349}
902 104329 = 17^2 * 19^2 {323, 5491, 6137, 93347}
5248 612612 = 2^2 * 3^2 * P(4..7) {102102, 204204, 306306, 408408}
100102 11639628 = 2^2 * 3^2 * P(4..8) {1939938, 3879876, 5819814, 7759752}
MATHEMATICA
facs[x_] := facs[x] = FactorInteger[x][[All, 1]]; nn = 5400; Do[Set[r[n], Times @@ facs[n]], {n, nn}]; s = Select[Range[2, nn], Not @* PrimePowerQ]; t = Select[s, #1/Apply[Times, #2] == Times @@ #2[[1 ;; 2]] & @@ {#, facs[#]} &]; Select[t, Function[m, Count[Range[m], _?(r[#] == r[m] &)] == 5]]
Numbers k = rad(k) * q(k)^2 such that A008479(k) = 6, where rad = A007947 and q(k) = second smallest distinct prime factor of k.
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375, 1715, 6655, 9317, 10875, 11625, 13875, 15375, 15379, 16125, 17625, 19875, 22125, 22875, 24167, 25125, 26625, 27375, 29625, 31125, 33375, 34391, 36375, 37875, 38625, 40125, 40875, 42375, 47625, 49125, 51375, 52125, 54043, 55875, 56625, 58875, 61125, 62625
COMMENTS
Terms are odd.
Let P(i..j) = Product_{h=i..j} prime(h).
EXAMPLE
Let S = {m < n : rad(m) = rad(n)}.
Table of n, a(n) for select n:
n a(n) S(n)
-----------------------------------------------------------------------------
1 375 = 3 * 5^3 {15, 45, 75, 135, 225}
2 1715 = 5 * 7^3 {35, 175, 245, 875, 1225}
3 6655 = 5 * 11^3 {55, 275, 605, 1375, 3025}
4 9317 = 7 * 11^3 {77, 539, 847, 3773, 5929}
5 10875 = 3 * 5^3 * 29 {435, 1305, 2175, 3915, 6525}
15 24167 = 11 * 13^3 {143, 1573, 1859, 17303, 20449}
39 63869 = 13 * 17^3 {221, 2873, 3757, 37349, 48841}
70 116603 = 17 * 19^3 {323, 5491, 6137, 93347, 104329}
136 231173 = 19 * 23^3 {437, 8303, 10051, 157757, 190969}
194 337125 = 3 * 5^3 * P(10..11) {13485, 40455, 67425, 121365, 202275}
7101 12473625 = 3 * 5^3 * P(10..12) {498945, 1496835, 2494725, 4490505, 7484175}
291027 511418625 = 3 * 5^3 * P(10..13) {20456745, 61370235, 102283725, 184110705, 306851175}
MATHEMATICA
facs[x_] := facs[x] = FactorInteger[x][[All, 1]]; nn = 50000;
Table[Set[r[n], Times @@ facs[n]], {n, nn}];
s = Select[Range[2, nn], Not @* PrimePowerQ];
t = Select[s, #1/Apply[Times, #2] == #2[[2]]^2 & @@ {#, facs[#]} &];
Select[t, Function[m, Count[Range[m], _?(r[#] == r[m] &)] == 6] ]
a(n) is the smallest number m with exactly n divisors d whose first digit equals the last digit of m.
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10, 1, 11, 121, 252, 672, 2352, 1512, 4032, 6552, 11781, 17472, 24192, 46872, 16632, 44352, 99792, 72072, 148512, 185472, 175392, 182952, 301392, 417312, 235872, 552552, 376992, 497952, 598752, 432432, 891072, 1415232, 1818432, 1153152, 1264032, 1513512, 2677752
EXAMPLE
Of the 24 divisors of 672, five have their first digit equals to the last digit of 672: {2, 21, 24, 28, 224}, and there is no integer < 672 with 5 such divisors, hence a(5) = 672.
MATHEMATICA
f[n_] := f[n] = DivisorSum[n, 1 &, IntegerDigits[#][[1]] == Mod[n, 10] &]; seq[max_] := TakeWhile[FirstPosition[Array[f, max], #] & /@ Range[0, 50] // Flatten, NumberQ]; seq[10^6] (* Amiram Eldar, Sep 05 2026 *)
PROG
(PARI) a(n) = my(m=1); while(sumdiv(m, d, digits(d)[1] == m % 10) != n, m++); m; \\ Michel Marcus, Sep 06 2026 (Python)
from sympy import divisors
from itertools import count, islice
def f(n, t): return sum(1 for d in divisors(n, generator=True) if str(d)[0] == t)
def agen(): # generator of terms
adict, n = dict(), 0
for m in count(1):
v = f(m, str(m%10))
if v not in adict:
adict[v] = m
while n in adict:
yield adict[n]
n += 1
Squares k such that A161351(k) is a square.
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0, 144, 1444, 8649, 299825238969
COMMENTS
Squares k such that k + (product of digits of k) + (sum of digits of k) is a square.
a(6) > 10^15 if it exists.
EXAMPLE
0 = 0^2, 0 + 0 + 0 = 0 = 0^2.
144 = 12^2, 144 + (1+4+4) + (1*4*4) = 169 = 13^2.
1444 = 38^2, 1444 + (1+4+4+4) + (1*4*4*4) = 1521 = 39^2.
8649 = 93^2, 8649 + (8+6+4+9) + (8*6*4*9) = 10404 = 102^2.
299825238969 = 547563^2, 299825238969 + (2+9+9+8+2+5+2+3+8+9+6+9) + (2*9*9*8*2*5*2*3*8*9*6*9) = 300127569921 = 547839^2.
MAPLE
f:= proc(n) local L;
L:= convert(n, base, 10);
n + convert(L, `+`) + convert(L, `*`)
end proc:
select(t -> issqr(f(t)), [seq(i^2, i=0..10^7)]);
PROG
(Python)
from math import isqrt, prod
def issquare(n): return isqrt(n)**2 == n
def A161351(n): return n + sum(d:=list(map(int, str(n)))) + prod(d) def ok(n): return issquare(n) and issquare( A161351(n))
Number of ways to place n nonattacking rook-plus-range-2-diagonal pieces on an n X n chessboard with rotations and reflections counted separately.
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1, 1, 0, 0, 2, 10, 32, 164, 1290, 10404, 93566, 971256, 11230254, 141645552, 1935993604, 28498370740, 449077050122, 7538223190980, 134244525340518, 2527396958407380, 50148298072678244, 1045816256261329304, 22866942869988432582, 523066451213871050872, 12492044879543305073636
COMMENTS
For n>=2, a(n) is even by vertical reflection.
EXAMPLE
a(4)=2, represented by the column permutations 2413 and 3142.
PROG
(Python)
from functools import lru_cache
def a(n):
full = (1 << n) - 1
@lru_cache(maxsize=None)
def f(mask, prev2, prev1, depth):
if mask == full:
return 1
ans = 0
for x in range(n):
if mask >> x & 1:
continue
if depth >= 1 and abs(x-prev1) == 1:
continue
if depth >= 2 and abs(x-prev2) == 2:
continue
ans += f(mask | 1 << x, prev1, x, depth+1)
return ans
return f(0, -1, -1, 0)
Number of non-isomorphic perfect hierarchical matching (PHM) trees on 2^n vertices.
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1, 1, 3, 136, 2098176, 2251799847239680, 10384593717069655329118586696368128, 883423532389192164791648750371459258578355946330267415512706630786482176, 25573364124188608359478044506465618376692515984711443667838213813251045284524598172171364193389457267788849004759522024159549430084512430111856787456
FORMULA
a(n) = 2^(2^n-2*n-1)+2^(2^(n-1)-n-1).
PROG
(Python)
def a(n):
return 2**(2**n-2*n-1)+2**(2**(n-1)-n-1)
Number of ordered n-tuples of nonempty subsets of an n-element set (repetitions allowed) such that every two entries have nonempty intersection.
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1, 7, 175, 17887, 7803391, 14581420567, 115574021762935, 3839684983986874687, 528848032637299570027231, 299087653289072507323441921687, 688894536060315470902287271070641495
COMMENTS
Equivalently, a(n) is the number of n X n binary matrices such that every two distinct rows have a common 1; repeated rows are allowed.
FORMULA
For n >= 2, a(n) = Sum_{G subset K_n} (-1)^e(G) * i(G)^n, where the sum is over all labeled simple graphs G on n vertices, e(G) is the number of edges of G, and i(G) is the number of independent sets of G, including the empty set. Also a(1) = 1.
EXAMPLE
a(2) = 7, since the possible ordered pairs of nonempty subsets of {1,2} with nonempty intersection are ({1},{1}), ({1},{1,2}), ({2},{2}), ({2},{1,2}), ({1,2},{1}), ({1,2},{2}), and ({1,2},{1,2}).
CROSSREFS
Cf. A051185 (number of intersecting families of subsets of an n-element set).
Numbers k, such that Sum_{d|k} gpf(d) is a multiple of gpf(k), where gpf = A006530.
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1, 6, 18, 20, 48, 54, 56, 63, 84, 90, 100, 105, 130, 135, 144, 162, 238, 275, 352, 364, 378, 384, 392, 429, 432, 441, 450, 456, 486, 495, 500, 588, 640, 660, 675, 700, 735, 754, 828, 832, 836, 870, 969, 1050, 1053, 1150, 1152, 1232, 1258, 1296, 1360, 1458
EXAMPLE
18 = 2*3^2 is a term, since gpf(18) = 3, and sum of greatest prime factors of its divisors is gpf(1)+gpf(2)+gpf(3)+gpf(6)+gpf(9)+gpf(18) = 1+2+3+3+3+3 = 5*3.
PROG
(PARI)
gpf(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
isok(n) = sumdiv(n, d, gpf(d)) % gpf(n) == 0
Decimal expansion of 2^(1+1/2)^(1+1/2^2)^(1+1/2^3)^...
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3, 2, 1, 6, 9, 1, 1, 5, 0, 1, 7, 9, 9, 2, 7, 3, 2, 5, 8, 9, 7, 4, 0, 6, 7, 9, 0, 2, 5, 5, 9, 2, 6, 5, 8, 0, 3, 1, 8, 3, 8, 8, 5, 7, 5, 8, 4, 2, 9, 4, 4, 9, 1, 5, 8, 5, 5, 1, 1, 8, 3, 5, 0, 8, 8, 0, 9, 2, 2, 6, 4, 5, 3, 5, 3, 5, 7, 3, 9, 2, 6, 9, 7, 3, 0, 0, 5, 6, 9, 4, 1, 4, 0, 8, 1, 0, 8, 8, 2, 6, 8
EXAMPLE
3.216911501799273258974067902559265803183885758429...
MATHEMATICA
b[prec_] := Module[{x, n, wp}, n = Ceiling[3 + 3*Sqrt[prec]]; wp = prec + 10; x = N[1 + 2^-n, wp]; Do[x = (1 + 2^-j)^x, {j, n - 1, 0, -1}]; N[x, prec]]; b[100]
Numbers that can be obtained by concatenating the decimal representations of x, y, and x+y for x >= 1 and y >= 0.
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101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1910, 2810, 2911, 3710, 3811, 3912, 4610, 4711, 4812, 4913
COMMENTS
A term may arise from more than one pair (x, y); the smallest such is 11011, from x = 1, y = 10 and x = 11, y = 0.
For single-digit x and y the term is the concatenation of k = 10*x + y and its digit sum, so the 90 terms with 1 <= x <= 9 and 0 <= y <= 9 (all having 3 or 4 digits) form a subsequence of A108203; the two sequences first differ at a(91) = 10010, where x is no longer a single digit. The 45 three-digit terms are exactly 101*x + 11*y with 1 <= x <= 9 and 0 <= y <= 9 - x; each has digit sum 2*(x + y), which is always even, and none is divisible by 11.
When the terms are sorted, the family with two-digit x (values 1001*x + 101*y) interleaves with the family with two-digit y (values 10001*x + 101*y); in the five-digit range the sorted terms alternate between the two.
FORMULA
Let q be the number of decimal digits of y (with q = 1 for y = 0) and r the number of decimal digits of x + y; then the term formed from (x, y) is x*10^(q + r) + y*10^r + (x + y). Here q = max(1, floor(log_10(y)) + 1) and r = floor(log_10(x + y)) + 1.
EXAMPLE
101 is a term because it is the concatenation of x = 1, y = 0 and x + y = 1.
1910 is a term because it is the concatenation of x = 1, y = 9 and x + y = 10.
11011 is a term because it can be formed by x = 1, y = 10 or x = 11, y = 0.
MAPLE
f:= proc(a, b) local ab, c;
c:= a+b;
if b = 0 then ab:= 10*a else ab:= 10^(1+ilog10(b))*a+b fi;
ab*10^(1+ilog10(c))+c;
end proc:
N:= 5000: # for terms <= N
R:= {}:
for a from 1 while f(a, 0) <= N do
for b from 0 do
v:= f(a, b);
if v > N then break else R:= R union {v} fi
od od:
PROG
(Python)
def aupto(limit):
terms = set()
x = 1
while int(f"{x}0{x}") <= limit:
y = 0
while True:
t = int(f"{x}{y}{x+y}")
if t > limit:
break
terms.add(t)
y += 1
x += 1
return sorted(terms)
print(aupto(5000))
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