A306985 -id:A306985 - cp's OEIS frontend

A343819 Numbers k such that k and k+1 have the same number of Fermi-Dirac factors (A064547).

2, 3, 4, 14, 16, 20, 21, 26, 27, 32, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 63, 64, 68, 74, 75, 76, 85, 86, 91, 92, 93, 94, 98, 99, 104, 111, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 143, 144, 145, 146, 147, 158, 161, 171, 175, 176, 177, 187, 189

Offset: 1

Amiram Eldar, Apr 30 2021

nonn

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), this is also the sequence of numbers k such that k and k+1 have the same number of infinitary divisors.

			2 is a term since A064547(2) = A064547(3) = 1.

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

Cf. A064547, A306985, A343818.
Similar sequences: A005237, A006049.
Subsequence of A086263.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; Select[Range[200], fd[#] == fd[# + 1] &]

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 3, 4, 186, 318, 434, 473, 582, 730, 978, 1024, 1035, 1245, 1357, 1397, 1506, 1661, 1902, 2085, 2116, 2224, 2329, 2453, 2505, 2506, 2770, 2954, 3144, 3345, 3377, 3624, 3641, 3765, 3790, 3882, 4037, 4172, 4438, 4898, 4938, 4975, 5221, 6126, 6285, 6312, 6356

Offset: 1

Amiram Eldar, Sep 03 2019

nonn

			186 is in the sequence since A070039(186) = A070039(187) = 12.

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

Cf. A002961, A064115, A064125, A070039, A164522, A164557, A293183, A306985, A325175.

s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 6500}]; seq

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 4, 7, 20, 31, 57, 94, 98, 118, 122, 127, 201, 213, 218, 230, 242, 243, 244, 334, 384, 393, 423, 429, 481, 565, 603, 633, 694, 704, 729, 766, 844, 921, 1138, 1141, 1221, 1262, 1401, 1533, 1654, 1726, 1761, 1837, 1838, 1862, 1882, 1942, 2162, 2245, 2361, 2362

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1).

The Mersenne primes (A000668) are terms since if 2^p - 1 is a prime then A093653(2^p-1) = A093653(2^p) = p+1.

			3 is a term since A093653(3) = A093653(4) = 3.

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

Cf. A000120, A093653.
A000668 is a subsequence.
Similar sequences: A002961, A005237, A054004, A064125, A238380, A293183, A306985.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[s, n - 1]]; f1 = f2, {n, 2, 240}]; s

A164522 Numbers k such that sigma_odd(k) = sigma_odd(k+1), where sigma_odd(k) is the sum of the odd divisors of k (A000593).

Original entry on oeis.org

1, 27089, 115289, 233729, 2529090, 2880989, 14059709, 17192909, 17540250, 18693990, 34902630, 54722249, 58517910, 82200689, 83087730, 92991990, 93623250, 93862230, 96578369, 111681990, 112244369, 155120129, 206450369, 269626769, 293182469, 303206310, 324764910

Offset: 1

Amiram Eldar, Aug 12 2019

nonn

			27089 is in the sequence since A000593(27089) = A000593(27089 + 1) = 27456.

Amiram Eldar, Table of n, a(n) for n = 1..100
Daeyeoul Kim, Nazli Yildiz Ikikardes, Yan Li, and Lianrong Ma, On the Problem sigma_od(n) = sigma_od(n+ 1), Filomat, Vol. 33, No. 2 (2019), pp. 543-559.

Cf. A000593, A002961, A064115, A064125, A293183, A306985.

v:=[&+[d:d in Divisors(m)|IsOdd(d)] :m in [1..5000000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019

f[p_, e_] := If[p == 2, 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[ seq, n-1]]; s1 = s2, {n, 1, 10^6}]; seq

A164557 Numbers k such that s(k) = s(k+1), where s(k) is the sum of divisors d of k such that k/d is odd (A002131).

Original entry on oeis.org

3, 6, 7, 10, 22, 31, 46, 58, 69, 82, 106, 127, 140, 154, 160, 166, 178, 226, 262, 286, 346, 358, 382, 466, 478, 502, 562, 586, 718, 748, 781, 838, 862, 886, 982, 1001, 1018, 1066, 1186, 1282, 1299, 1306, 1318, 1366, 1438, 1486, 1522, 1614, 1618, 1672, 1704, 1822

Offset: 1

Amiram Eldar, Aug 12 2019

nonn

			3 is in the sequence since A002131(3) = A002131(3 + 1) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daeyeoul Kim and Abdelmejid Bayad, Convolution identities for twisted Eisenstein series and twisted divisor functions, Fixed Point Theory and Applications 2013, No. 1 (2013), Article 81, alternative link.
Daeyeoul Kim, Nazli Yildiz Ikikardes, Yan Li, and Lianrong Ma, On the Problem sigma_od(n) = sigma_od(n+ 1), Filomat, Vol. 33, No. 2 (2019), pp. 543-559.

Cf. A002131, A002961, A064115, A064125, A293183, A306985.

v:=[&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..2000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019

f[p_, e_] := If[p == 2, p^e, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[seq, n-1]]; s1 = s2, {n, 1, 2000}]; seq

A348346 Numbers k such that k and k+1 have the same positive sum of noninfinitary divisors (A348271).

Original entry on oeis.org

20150, 52767, 99296, 835515, 1241504, 2199392, 6294015, 11158496, 12770450, 17016416, 19127907, 20128544, 23686748, 24790688, 26580554, 33366015, 34385247, 39687651, 42106976, 44157087, 45466676, 59825349, 60832449, 73780244, 75268775, 81654650, 84696849, 111457213

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

Numbers k such that A348271(k) = A348271(k+1) > 0.

The terms are restricted to have a positive sum of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

			20150 is a term since A348271(20150) = A348271(20151) = 6720.

Amiram Eldar, Table of n, a(n) for n = 1..154

Cf. A036537, A348271, A348345.
Subsequence of A162643.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[10^5], (s1 = s[#]) > 0 && s1 == s[# + 1] &]

A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597, 274533, 288765

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			14 is a term since A333926(14) = A333926(15) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A333926.

Analogous sequences: A002961, A064115 (nonunitary), A064125 (unitary), A293183 (bi-unitary), A306985 (infinitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == recDivSum[# + 1] &]

A360358 Numbers k such that A360327(k) = A360327(k+1) > 1.

Original entry on oeis.org

714, 6603, 16115, 18920, 23154, 24530, 39984, 41360, 42789, 51204, 56814, 58190, 59619, 60995, 65229, 66605, 68034, 69410, 73644, 79304, 82059, 84249, 84864, 86240, 94655, 101375, 101694, 103070, 107304, 108680, 121374, 125510, 126125, 126939, 135128, 135354, 137329

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

Numbers k such that A360327(k) = A360327(k+1) = 1 are terms of A360357.

			714 is a term since A360327(714) = A360327(715) = 72 > 1.

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

Cf. A360327, A360357.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360359.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2 > 1, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s2 > 1 && s1 == s2, print1(n-1, ", ")); s1 = s2); }

A360359 Numbers k such that A360331(k) = A360331(k+1).

Original entry on oeis.org

69, 574, 713, 781, 2394, 2506, 5699, 5750, 6499, 6509, 8441, 19250, 26529, 32130, 36549, 38065, 41749, 41929, 43239, 48025, 50301, 53037, 53382, 59178, 59822, 61754, 66906, 67689, 70277, 71198, 81620, 94000, 100775, 119214, 124640, 127442, 134665, 153202, 154908

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

			69 is a term since A360331(69) = A360331(70) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A360331.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360358.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

A372693 Numbers k such that A372692(k) = A372692(k+1) > 1.

Original entry on oeis.org

7380, 18755, 24804, 25631, 26299, 27467, 32799, 44891, 49196, 49725, 50940, 53603, 59652, 64386, 71027, 79739, 85788, 89300, 94275, 103212, 105056, 105875, 124992, 129348, 132011, 138060, 141899, 147100, 149435, 155484, 158147, 164196, 170324, 175571, 181620, 184283

Offset: 1

Amiram Eldar, May 10 2024

nonn

The numbers k such that A372692(k) = A372692(k+1) = 1 are in A372690.

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

Cf. A036537, A372690, A372692.
Subsequence of A068781.
A372694 is a subsequence.
Similar sequences: Cf. A002961, A064125, A293183, A306985, A343819, A348346.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); s[1] = 1; s[n] := s[n] = Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1);
Select[Range[10^5], (s1 = s[#]) > 1 && s1 == s[# + 1] &]

s(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = apply(x -> 1 - x, binary(f[i, 2])); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > 1 && s1 == s2, print1(k - 1, ", ")); s1 = s2);}

cp's OEIS Frontend

A343819 Numbers k such that k and k+1 have the same number of Fermi-Dirac factors (A064547).

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A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

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A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

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A164522 Numbers k such that sigma_odd(k) = sigma_odd(k+1), where sigma_odd(k) is the sum of the odd divisors of k (A000593).

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A164557 Numbers k such that s(k) = s(k+1), where s(k) is the sum of divisors d of k such that k/d is odd (A002131).

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A348346 Numbers k such that k and k+1 have the same positive sum of noninfinitary divisors (A348271).

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A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

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A360358 Numbers k such that A360327(k) = A360327(k+1) > 1.

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