Amiram Eldar - cp's OEIS frontend

A387362 Cyclic numbers k such that k+2 is also a cyclic number.

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 35, 41, 51, 59, 65, 67, 69, 71, 77, 83, 85, 87, 89, 95, 101, 107, 113, 131, 137, 139, 141, 143, 149, 157, 159, 161, 177, 179, 185, 191, 197, 209, 211, 213, 215, 221, 227, 233, 239, 247, 249, 255, 257, 263, 265, 267, 269, 281, 293

Offset: 1

Amiram Eldar, Aug 27 2025

All the lesser members of twin primes (A001359) are terms since every prime is a cyclic number (A003277).

Cohen (2025) conjectured and Pomerance (2025) proved that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025.
Carl Pomerance, Patterns for cyclic numbers, 2025.

Cf. A001620, A005597, A073004, A387363.
Subsequence of A003277.
A001359 is a subsequence.

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; Select[Range[1, 300, 2], And @@ cyclicQ[{#, # + 2}] &]

iscyclic(k) = gcd(k, eulerphi(k)) == 1;
isok(k) = k % 2 && iscyclic(k) && iscyclic(k+2);

The number of terms <= x is ~ 2 * C_2 * x / (exp(gamma) * log(log(log(x))))^2, where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.

Original entry on oeis.org

1, 3, 3, 4, 3, 4, 5, 6, 8, 8, 7, 8, 7, 6, 9, 8, 11, 12, 11, 10, 12, 12, 13, 16, 12, 14, 16, 12, 13, 14, 13, 16, 19, 14, 19, 20, 19, 20, 20, 20, 21, 26, 19, 24, 26, 22, 25, 26, 24, 26, 33, 26, 27, 30, 26, 28, 32, 26, 29, 38, 25, 30, 34, 26, 33, 34, 29, 30, 41, 28

Offset: 1

Amiram Eldar, Aug 27 2025

Analogous to A002372 with cyclic numbers (A003277) instead of odd primes.

Pomerance (2025) proved that a(n) > 0 for every sufficiently large n.

			a(1) = 1 since 2*1 = 1 + 1.
a(2) = 3 since 2*2 = 1 + 3 = 2 + 2 = 3 + 1.
a(3) = 3 since 2*3 = 1 + 5 = 3 + 3 = 5 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025.
Carl Pomerance, Patterns for cyclic numbers, 2025.

Cf. A002372, A003277, A387362.

Cf. A001620, A005597, A073004.

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; a[n_] := Count[Range[2*n], _?(And @@ cyclicQ[{#, 2*n-#}] &)]; Array[a, 100]

iscyclic(k) = gcd(k, eulerphi(k)) == 1;
a(n) = sum(k = 1, 2*n, iscyclic(k) * iscyclic(2*n-k));

a(n) ~ C_2 * n / (exp(gamma) * log(log(log(n))))^2 * Product_{p | n, p odd prime < log(log(n/2))} (p-1)/(p-2), where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

Original entry on oeis.org

4, 9, 1, 6, 9, 9, 6, 7, 7, 6, 9, 3, 8, 2, 1, 1, 1, 7, 7, 1, 6, 5, 4, 6, 2, 5, 4, 1, 6, 8, 9, 0, 8, 1, 0, 0, 2, 2, 1, 5, 1, 0, 2, 7, 1, 2, 6, 8, 7, 5, 5, 0, 7, 7, 2, 5, 5, 9, 0, 4, 8, 1, 7, 9, 1, 4, 7, 4, 5, 0, 7, 2, 2, 3, 7, 5, 6, 2, 9, 6, 3, 8, 1, 0, 1, 9, 1, 1, 8, 9, 9, 8, 7, 5, 7, 6, 4, 6, 6, 2, 9, 0, 2, 1, 1

Offset: 0

Amiram Eldar, Aug 22 2025

			0.49169967769382111771654625416890810022151027126875...

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000, The Putnam Archive. See Problem A-4.
Editors of the Mathematics Magazine, The 61st Annual William Lowell Putnam Examination, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.
Jihyung Kang and others, Convergence of I = Integral_0^oo sinx sin(x2) dx, Mathematics StackExchange, 2017.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-first William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.
Eric Weisstein's World of Mathematics, Fresnel Integrals.
Wikipedia, Fresnel integral.

Cf. A069998, A217481, A231863.

RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
(* or *)
RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]

Equals sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.

Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.

A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590

Offset: 1

Amiram Eldar, Aug 19 2025

First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .

			30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A068403.
Subsequence of A087248 and A285615.
Cf. A000203, A001221, A002110, A013929, A048250, A088860.

q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]

isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}

A001221(a(n)) >= 6.

A387154 The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

401120980260, 360360, 55440, 110880, 100800, 120960, 241920, 483840, 967680, 1935360, 3870720, 7741440, 15482880, 30965760, 61931520, 123863040, 247726080, 495452160, 990904320, 1981808640, 3963617280, 7927234560, 15854469120, 31708938240, 63417876480, 126835752960

Offset: 2

Amiram Eldar, Aug 19 2025

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.
The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and 4, respectively.
All the terms are in A025487.

			For n = 2, the numbers k such that A048250(k) > 2*k include all the squarefree abundant numbers (A087248). The least nonsquarefree number (A013929) k such that A048250(k) > 2*k is 401120980260 = 2^2*3*5*7*11*13*17*19*23*29*31.
For n = 3, the numbers k such that A073185(k) > 2*k include all the cubefree abundant numbers (A357695). The least noncubefree number (A046099) k such that A073185(k) > 2*k is A357700(1) = 360360 = 2^3*3^2*5*7*11*13.

Amiram Eldar, Table of n, a(n) for n = 2..1000

Cf. A004709, A005117, A013929, A025487, A046099, A046100, A048250, A087248, A073185, A357695, A357700, A385006, A387155.

a[n_] := If[n < 7, {401120980260, 360360, 55440, 110880, 100800}[[n-1]], 945 * 2^n]; Array[a, 26, 2]

a(n) = if(n < 7, [401120980260, 360360, 55440, 110880, 100800][n-1], 945 * 2^n);

a(n) = 945 * 2^n for n >= 7.

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969

Offset: 2

Amiram Eldar, Aug 19 2025

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.

The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and, respectively.

			a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.

Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.

freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];

isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}

Let A_k(n) be the number of k-free abundant numbers that are not exceeding n. Then, a(n) = A_n(A387154(n)) - 1.

a(n) ~ c * 945 * 2^n, where c = A302991.

A387057 Numbers k that are infinitarily divisible by the number of infinitary divisors of k.

Original entry on oeis.org

1, 2, 8, 12, 20, 24, 28, 36, 40, 44, 52, 56, 64, 68, 72, 76, 88, 92, 100, 104, 116, 124, 128, 136, 148, 152, 164, 172, 184, 188, 196, 200, 212, 232, 236, 244, 248, 268, 284, 292, 296, 316, 324, 328, 332, 344, 356, 376, 384, 388, 392, 404, 412, 424, 428, 436, 452

Offset: 1

Amiram Eldar, Aug 15 2025

Numbers k such that A037445(k) is an infinitary divisor of k.

This sequence is infinite. For example, if p is an odd prime, then 8*p is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A387056.

Cf. A037445, A077609.

infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (fct = FactorInteger[d])], (e = Last /@ fct)] == e;
f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ f @@@ FactorInteger[n]; q[k_] := Module[{d = id[k]}, Divisible[k, d] && infDivQ[k, d]]; Select[Range[500], q]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
isok(k) = {my(f = factor(k), id = vecprod(apply(x -> 2^hammingweight(x), f[, 2]))); !(k % id) && isidiv(id, f);}

A387056 Numbers k that are divisible by the number of infinitary divisors of k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 100, 104, 112, 116, 124, 128, 136, 144, 148, 152, 160, 164, 172, 176, 184, 188, 192, 196, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 268, 272, 284, 288, 292, 296

Offset: 1

Amiram Eldar, Aug 15 2025

First differs from A048166 at n = 27: A048166(27) = 108 is not a term of this sequence.

This sequence is infinite. For example, if p is a prime, then 8*p is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

A387057 is a subsequence.

Cf. A033950, A037445, A048166.

f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ f @@@ FactorInteger[n]; q[k_] := Divisible[k, id[k]]; Select[Range[300], q]

isok(k) = !(k % vecprod(apply(x -> 2^hammingweight(x), factor(k)[, 2])));

A387055 Numbers that are unitarily divisible by their number of divisors.

Original entry on oeis.org

1, 2, 24, 36, 40, 56, 60, 84, 88, 104, 132, 136, 152, 156, 184, 204, 225, 228, 232, 248, 276, 296, 328, 344, 348, 372, 376, 424, 441, 444, 450, 472, 488, 492, 516, 536, 564, 568, 584, 600, 632, 636, 664, 708, 712, 732, 776, 804, 808, 824, 852, 856, 872, 876, 882, 904, 948, 996

Offset: 1

Amiram Eldar, Aug 15 2025

Refactorable numbers (A033950) k whose number of divisors is a unitary divisor of k.

This sequence is infinite. For example, if p is an odd prime, then 8*p is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A033950.

Cf. A000005, A023534, A048166, A077610.

q[k_] := Module[{d = DivisorSigma[0, k]}, Divisible[k, d] && CoprimeQ[d, k/d]]; Select[Range[1000], q]

isok(k) = {my(d = numdiv(k)); !(k % d) && gcd(d, k/d) == 1;}

A386925 a(n) = numerator(Sum_{k=1..n} d(k+1)/d(k)), where d is the number of divisors function.

Original entry on oeis.org

2, 3, 9, 31, 43, 23, 29, 125, 47, 49, 61, 187, 211, 223, 119, 607, 697, 707, 797, 817, 847, 431, 491, 3973, 4133, 4253, 4433, 1491, 1651, 1661, 1781, 5423, 5543, 5663, 5933, 17879, 18599, 18959, 19679, 19769, 21209, 21299, 22379, 22739, 22979, 23159, 24959, 25067

Offset: 1

Amiram Eldar, Aug 08 2025

			Fractions begin with 2, 3, 9/2, 31/6, 43/6, 23/3, 29/3, 125/12, 47/4, 49/4, 61/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and Igor E. Shparlinski, On the values of the divisor function, Monatshefte für Mathematik, Vol. 154, No. 1 (2008), pp. 59-69.

Cf. A000005, A386926 (denominators).

With[{s = DivisorSigma[0, Range[100]]}, Numerator[Accumulate[Rest[s]/Most[s]]]]

list(nmax) = {my(s = 0, d1 = 1, d2); for(n = 2, nmax, d2 = numdiv(n); s += (d2/d1); print1(numerator(s), ", "); d1 = d2);}

a(n)/A386926(n) ≍ n * sqrt(log(n)) (Luca and Shparlinski, 2008).

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User: Amiram Eldar

A387362 Cyclic numbers k such that k+2 is also a cyclic number.

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A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.

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A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

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A387154 The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

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A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

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A387057 Numbers k that are infinitarily divisible by the number of infinitary divisors of k.

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