A348346 -id:A348346 - cp's OEIS frontend

A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

44, 75, 98, 116, 147, 171, 242, 243, 244, 332, 387, 507, 548, 603, 604, 724, 735, 819, 844, 908, 931, 963, 1035, 1075, 1083, 1196, 1251, 1274, 1275, 1324, 1412, 1449, 1467, 1556, 1587, 1665, 1675, 1772, 1924, 1925, 1952, 1988, 2324, 2331, 2511, 2523, 2524, 2540

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

First differs from A049103 at n=17.
Numbers k such that A348341(k) = A348341(k+1) > 0.
The terms are restricted to have a positive number of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

			44 is a term since A348341(44) = A348341(45) = 2 > 0.

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

Cf. A036537, A049103, A348341, A348346.
Subsequence of A162643.
Similar sequences: A005237, A006049, A343819, A344312, A344313, A344314.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Select[Range[2500],(nid1 = nid[#]) > 0 && nid1 == nid[# + 1] &]

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])));
isA348345(n) = { my(u=A348341(n)); (u>0&&(A348341(1+n)==u)); }; \\ Antti Karttunen, Oct 13 2021

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);}

A348628 Numbers k such that k and k+1 have the same sum of nonexponential divisors (A160135).

Original entry on oeis.org

1, 2, 3, 4, 15, 44, 674, 478899

Offset: 1

Amiram Eldar, Oct 26 2021

Numbers k such that A160135(k) = A160135(k+1).

a(9) > 1.6 * 10^11, if it exists.

			2 is a term since A160135(2) = A160135(3) = 1.
15 is a term since A160135(15) = A160135(16) = 9.

Cf. A160135.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A348346.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; Select[Range[500000], s[#] == s[# + 1] &]

cp's OEIS Frontend

A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

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

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A348628 Numbers k such that k and k+1 have the same sum of nonexponential divisors (A160135).

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Mathematica