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a(n) is the smallest k such that tau(k^2 + 1) is equal to 2^n, where tau = A000005 and a(n) = -1 if no such k exists.
+0
0
0, 1, 3, 13, 47, 307, 2163, 17557, 191807, 1413443, 16485763, 169053487
OFFSET
0,3
FORMULA
a(n) = A353008(2^(n-1)) for n > 0.
MATHEMATICA
a[n_]:=Module[{k=0}, While[DivisorSigma[0, k^2+1]!=2^n, k++]; k]; Array[a, 9, 0] (* Stefano Spezia, Feb 10 2025 *)
PROG
(PARI) a(n) = my(k=0); while (numdiv(k^2+1) != 2^n, k++); k; \\ Michel Marcus, Feb 09 2025
CROSSREFS
KEYWORD
nonn,hard,more,new
AUTHOR
EXTENSIONS
a(10)-a(11) from Stefano Spezia, Feb 12 2025
STATUS
approved
Numbers which are not prime powers and their prime factors share a last digit in base 10.
+0
0
39, 69, 117, 119, 129, 159, 207, 219, 249, 259, 299, 309, 329, 339, 341, 351, 387, 451, 469, 477, 489, 507, 519, 551, 559, 579, 621, 629, 657, 669, 671, 679, 689, 699, 747, 749, 781, 789, 799, 833, 849, 879, 889, 897, 927, 939, 949, 959, 989, 1017, 1053, 1059
OFFSET
1,1
COMMENTS
Also called the one-sided numbers.
They can end only in either 1, 3, 7 or 9.
LINKS
VOS 2025 Math Region stage, Problem 10.9 (in Russian).
EXAMPLE
39 = 3*13.
MAPLE
q:= n-> (l-> nops(l)>1 and nops({map(i-> irem(i[1], 10), l)[]})=1)(ifactors(n)[2]):
select(q, [$1..2000])[]; # Alois P. Heinz, Feb 18 2025
PROG
(PARI) isok(k) = my(f=factor(k)); (#f~ != 1) && (#Set(vector(#f~, i, f[i, 1] % 10)) == 1); \\ Michel Marcus, Feb 18 2025
(Python)
from sympy import factorint
def ok(n): return len(f:=factorint(n)) > 1 and len(set(p%10 for p in f)) == 1
print([k for k in range(1, 1060) if ok(k)]) # Michael S. Branicky, Feb 18 2025
CROSSREFS
Cf. A004615.
KEYWORD
nonn,base,new
AUTHOR
Yaroslav Deryavko, Feb 01 2025
STATUS
approved
Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.
+0
2
0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 12, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0
OFFSET
1,9
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
FORMULA
a(n) = A003963(n) - A156061(n).
EXAMPLE
The prime indices of 300 are {1,1,2,3,3}, so a(300) = 18 - 6 = 12.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@prix[n]-Times@@Union[prix[n]], {n, 100}]
CROSSREFS
Positions of nonzeros are A038838.
For length instead of product we have A046660.
For factors instead of indices we have A066503, see A007947 (squarefree kernel).
For sum of factors instead of product of indices we have A280292, see A280286, A381075.
For quotient instead of difference we have A290106, for factors A003557.
For sum instead of product we have A380955 (firsts A380956, sorted A380957).
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
KEYWORD
nonn,new
AUTHOR
Gus Wiseman, Feb 14 2025
STATUS
approved
Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).
+0
1
1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
All terms are odd.
EXAMPLE
The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
1: {}
9: {2,2}
25: {3,3}
27: {2,2,2}
121: {5,5}
169: {6,6}
289: {7,7}
81: {2,2,2,2}
125: {3,3,3}
841: {10,10}
961: {11,11}
675: {2,2,2,3,3}
1681: {13,13}
1849: {14,14}
2209: {15,15}
243: {2,2,2,2,2}
3481: {17,17}
1125: {2,2,3,3,3}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]], {n, 10000}];
Table[Position[q, k][[1, 1]], {k, mnrm[q]}]
CROSSREFS
For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
KEYWORD
nonn,new
AUTHOR
Gus Wiseman, Feb 14 2025
STATUS
approved
Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).
+0
1
1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
All terms are odd.
EXAMPLE
The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
1: {}
9: {2,2}
25: {3,3}
27: {2,2,2}
81: {2,2,2,2}
121: {5,5}
125: {3,3,3}
169: {6,6}
243: {2,2,2,2,2}
289: {7,7}
625: {3,3,3,3}
675: {2,2,2,3,3}
729: {2,2,2,2,2,2}
841: {10,10}
961: {11,11}
1125: {2,2,3,3,3}
1331: {5,5,5}
1681: {13,13}
1849: {14,14}
2025: {2,2,2,2,3,3}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
q=Table[Times@@prix[n]/Times@@Union[prix[n]], {n, 1000}];
Select[Range[Length[q]], FreeQ[Take[q, #-1], q[[#]]]&]
CROSSREFS
For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
KEYWORD
nonn,new
AUTHOR
Gus Wiseman, Feb 18 2025
STATUS
approved
Array read by antidiagonals: T(n,k) is the number of Hamiltonian cycles in the stacked prism graph P_n X C_k, n >= 1, k >= 2.
+0
0
1, 1, 4, 1, 3, 4, 1, 6, 6, 4, 1, 5, 22, 12, 4, 1, 8, 30, 82, 24, 4, 1, 7, 86, 160, 306, 48, 4, 1, 10, 126, 776, 850, 1142, 96, 4, 1, 9, 318, 1484, 7010, 4520, 4262, 192, 4, 1, 12, 510, 6114, 18452, 63674, 24040, 15906, 384, 4, 1, 11, 1182, 12348, 126426, 229698, 578090, 127860, 59362, 768, 4
OFFSET
1,3
COMMENTS
The case for P_n X C_2 is determined using a double edge for C_2.
LINKS
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.
EXAMPLE
Array begins:
=========================================================
n\k | 2 3 4 5 6 7 8 ...
----+---------------------------------------------------
1 | 1 1 1 1 1 1 1 ...
2 | 4 3 6 5 8 7 10 ...
3 | 4 6 22 30 86 126 318 ...
4 | 4 12 82 160 776 1484 6114 ...
5 | 4 24 306 850 7010 18452 126426 ...
6 | 4 48 1142 4520 63674 229698 2588218 ...
7 | 4 96 4262 24040 578090 2861964 53055038 ...
8 | 4 192 15906 127860 5247824 35663964 1087362018 ...
...
CROSSREFS
Rows 1..2 are A000012, A103889(n+1).
Cf. A222196 (order of recurrences), A222197 (main diagonal), A270273, A321172.
KEYWORD
nonn,tabl,new
AUTHOR
Andrew Howroyd, Feb 18 2025
STATUS
approved
Partial sums of A381116.
+0
0
6, 9, 16, 23, 29, 40, 48, 56, 57, 72, 84, 93, 106, 116, 135, 142, 160, 171, 189, 194, 216, 228, 249, 252, 272, 283, 318, 323, 339, 357, 375, 398, 415, 435, 461, 478, 481, 523, 527, 534, 576, 598, 623, 624, 658, 667, 675, 722, 728, 783, 791, 840, 847, 855, 861, 903, 917, 949, 968, 974, 1025, 1027, 1077, 1088, 1115, 1157
OFFSET
1,1
LINKS
KEYWORD
nonn,new
AUTHOR
N. J. A. Sloane, Feb 18 2025
STATUS
approved
First differences of A381116.
+0
0
6, 3, 7, 7, 6, 11, 8, 8, 1, 15, 12, 9, 13, 10, 19, 7, 18, 11, 18, 5, 22, 12, 21, 3, 20, 11, 35, 5, 16, 18, 18, 23, 17, 20, 26, 17, 3, 42, 4, 7, 42, 22, 25, 1, 34, 9, 8, 47, 6, 55, 8, 49, 7, 8, 6, 42, 14, 32, 19, 6, 51, 2, 50, 11, 27, 42, 25, 46, 54, 21, 22, 45, 4, 16, 53, 4, 73, 10, 25, 39, 9, 70, 11, 46, 33, 49, 27, 11, 20, 17
OFFSET
1,1
LINKS
CROSSREFS
KEYWORD
nonn,new
AUTHOR
N. J. A. Sloane, Feb 18 2025
STATUS
approved
a(n) is the number of integers in base n such that all the integers given by their first k digits are divisible by k and which cannot be extended further.
+0
0
1, 3, 8, 21, 54, 145, 367, 1039, 2492, 6709, 16799, 46610, 95597, 368134, 831886, 2245056, 6084180, 15798495, 41456343, 119786906, 292818176, 788255058, 2061079489, 5753392327, 14984432350
OFFSET
2,2
EXAMPLE
a(10)=2492 because from all A271374(10)=20457 polydivisible numbers, only 2492 cannot be further expanded into a larger polydivisible number. One such number is 4836545640368400: 4 is divisible by 1, 48 is divisible by 2, 483 is divisible by 3, 4836 is divisible by 4, and so on until 4836545640368400 which is divisible by 16; but one cannot extend it further since no digit (0 to 9) appended to 4836545640368400 would result in a number divisible by k=17.
CROSSREFS
KEYWORD
nonn,base,more,new
AUTHOR
Inigo Quilez, Jan 22 2025
STATUS
approved
Expansion of e.g.f. exp( -LambertW(-2 * sin(x)) / 2 ).
+0
0
1, 1, 5, 48, 709, 14152, 356793, 10882648, 389790889, 16040853568, 745908722477, 38681745244032, 2213527304014189, 138556837227204736, 9417928265797994145, 690818806495197538816, 54391227913053881634001, 4575388875753714015748096, 409532433006878699321370197
OFFSET
0,3
FORMULA
E.g.f. A(x) satisfies A(x) = exp( sin(x) * A(x)^2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
PROG
(PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (2*k+1)^(k-1)*I^(n-k)*a136630(n, k));
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Feb 18 2025
STATUS
approved

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