A342560 -id:A342560 - cp's OEIS frontend

A342555 2*a(n) is the start of 3 consecutive numbers (even-odd-even) that are sums of divisors, i.e., terms of A000203.

3, 6, 15, 19, 63, 153, 199, 255, 423, 480, 511, 546, 861, 1111, 1189, 1400, 1770, 1875, 1935, 1995, 2047, 2556, 3475, 3619, 4005, 4095, 4920, 5151, 5215, 6649, 8046, 8191, 8646, 8749, 9765, 11175, 11199, 14028, 14197, 15391, 15427, 15470, 16383, 19494, 25878, 26557, 26799

Offset: 1

Hugo Pfoertner, May 14 2021

nonn

There exists one exceptional case of 4 consecutive numbers 12, 13, 14, 15, where 13 would start an odd-even-odd progression.

			a(1) = 3, because 2*3 = 6 is the start of the first occurrence of a row of 3 consecutive numbers, all of which are in A000203. 6 = sigma(5), 7 = sigma(4), 8 = sigma(7).
a(2) = 6: 2*6 = 12 = sigma(6) = sigma(11), 13 = sigma(9), 14 = sigma(13). 15 = sigma(8), which would be at the end of the row 13, 14, 15, is excluded by the even-odd-even condition.
a(3) = 15: 2*15 = 30 = sigma(29), 31 = sigma(16) = sigma(25), 32 = sigma(21) = sigma(31).
See Jeppe Stig Nielsen's list for more examples.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005.
Jeppe Stig Nielsen, List of numbers with divisor sum k, k<=10000.

Cf. A000203, A085790, A342560.

a342555(nterms) = {my(N=vector(3, i, invsigmaNum(i+1)), n=0, k=4); while(n<=nterms, if(vecmin(N)>0 && !(k%2), print1((k-2)/2, ", "); n++); k++; N[1+k%3] = invsigmaNum(k))}; \\ see Alekseyev link for invsigmaNum()
a342555(46)

A379655 Numbers k such that k and k+1 are both possible values of the sum of divisors function (A000203).

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 30, 31, 38, 39, 56, 62, 90, 120, 126, 127, 132, 182, 194, 216, 255, 306, 307, 363, 380, 398, 399, 402, 464, 510, 511, 548, 552, 740, 780, 846, 847, 854, 920, 930, 960, 961, 992, 1022, 1023, 1092, 1093, 1280, 1407, 1650, 1658, 1722, 1723, 1728

Offset: 1

Amiram Eldar, Jan 03 2025

Numbers k such that k and k+1 are both in A002191.

			3 is a term since 3 = sigma(2) and 3 + 1 = 4 = sigma(3).
6 is a term since 6 = sigma(5) and 6 + 1 = 7 = sigma(4).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Cf. A000203, A342555, A342560.

Subsequence of A002191.

seq[lim_] := Module[{v = Select[Union[DivisorSigma[1, Range[lim]]], # <= lim &]}, v[[Position[Differences[v], 1] // Flatten]]]; seq[2000]

isA002191(n) = invsigmaNum(n) > 0; \\ using Max Alekseyev's invphi.gp
list(lim) = my(q1 = isA002191(1), q2); for(k = 2, lim, q2 = isA002191(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);

cp's OEIS Frontend

A342555 2*a(n) is the start of 3 consecutive numbers (even-odd-even) that are sums of divisors, i.e., terms of A000203.

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