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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
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
Numbers which are not prime powers and their prime factors share a last digit in base 10.
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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
COMMENTS
Also called the one-sided numbers.
They can end only in either 1, 3, 7 or 9.
MAPLE
q:= n-> (l-> nops(l)>1 and nops({map(i-> irem(i[1], 10), l)[]})=1)(ifactors(n)[2]):
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
Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.
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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
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.
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
For length instead of product we have A046660. For factors instead of indices we have A066503, see A007947 (squarefree kernel). For quotient instead of difference we have A290106, for factors A003557.
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
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
Position of first appearance of n in A290106. For difference instead of quotient see A380986.
Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).
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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
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 difference instead of quotient see A380986.
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.
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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
COMMENTS
The case for P_n X C_2 is determined using a double edge for C_2.
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
Columns 2..12 are A123932(n-1), A003945(n-1), A003699, A003731, A180582, A180583, A180584, A180585, A180586, A180587, A180588.
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
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
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.
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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
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.
Expansion of e.g.f. exp( -LambertW(-2 * sin(x)) / 2 ).
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0
1, 1, 5, 48, 709, 14152, 356793, 10882648, 389790889, 16040853568, 745908722477, 38681745244032, 2213527304014189, 138556837227204736, 9417928265797994145, 690818806495197538816, 54391227913053881634001, 4575388875753714015748096, 409532433006878699321370197
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));
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