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Also numbers with odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321649 A321650 and ranked by A122111.

Also numbers with odd omega Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321649, and ranked by A122111. The version for even Omega is A345962, and the union is A345961.

Also numbers with odd omega (A001222) and exactly two odd conjugate prime indices. Conjugate prime indices are listed by A321649, ranked by A122111. The version for even omega Omega is A345962, and the union is A345961.

The negative (k = -2) version is A345962.

A001791/A345910/A345912 count/rank compositions with alternating sum -1.

A056239 adds up prime indices, row sums of A112798.

Cf. A000037 non_sqrs, A000070 ptns_altsum_1, ~A000346 comps_ev_ats_neq0, ~A008549 comps_ev_ats_less0, A027187 ptns_altsum_wkneg, A027193 ptns_sats_grtr0, A028260 h_altsum_wkneg, ~A032443 comps_oddsum_altsum_geq_0, `A034871 tri_comps_2n_ats_2k, A035363 ptns_use_ev, ~A114121 comps_evsum_altsum_geq_0, A239830 tri_ptns_altsum_ev_ev, A341446 only_odd_prix_least, A344607, A344608 ptns_sats_strneg, A344609 h_altsum_wkpos, A344610, `A344650 strptns_ev_revaltsum_wkpos, A344651 tri_ptns_altsum_mod2pos_ev_k, A344741, `A345197 tetr_comps_len_ats, `A345908 comps_len_ats_trace, ~A345917 stc_ats_grtr0, ~A345918 stc_sats_grtr0.

Cf. A000037, A000070, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.

Cf. A000037 non_sqrs, A000070 ptns_altsum_1, ~A000346 comps_ev_ats_neq0, ~A008549 comps_ev_ats_less0, A027187 ptns_altsum_wkneg, A027193 ptns_sats_grtr0, A028260 h_altsum_wkneg, ~A032443 comps_oddsum_altsum_geq_0, `A034871 tri_comps_2n_ats_2k, A035363 ptns_use_ev, ~A114121 comps_evsum_altsum_geq_0, A239830 tri_ptns_altsum_ev_ev, A341446 only_odd_prix_least, AA344607, 344608 A344607, A344608 ptns_sats_strneg, A344609 h_altsum_wkpos, A344610, `A344650 strptns_ev_revaltsum_wkpos, A344651 tri_ptns_altsum_mod2pos_ev_k, A344741, `A345197 tetr_comps_len_ats, `A345908 comps_len_ats_trace, ~A345917 stc_ats_grtr0, ~A345918 stc_sats_grtr0.

allocated for Gus WisemanNumbers whose prime indices have alternating sum 2.

3, 12, 27, 30, 48, 70, 75, 108, 120, 147, 154, 192, 243, 270, 280, 286, 300, 363, 432, 442, 480, 507, 588, 616, 630, 646, 675, 750, 768, 867, 874, 972, 1080, 1083, 1120, 1144, 1200, 1323, 1334, 1386, 1452, 1470, 1587, 1728, 1750, 1768, 1798, 1875, 1920, 2028

1,1

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.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.

Also numbers with odd omega and exactly two odd conjugate prime indices. The version for even omega is A345962, and the union is A345961.

The initial terms and their prime indices:

3: {2}

12: {1,1,2}

27: {2,2,2}

30: {1,2,3}

48: {1,1,1,1,2}

70: {1,3,4}

75: {2,3,3}

108: {1,1,2,2,2}

120: {1,1,1,2,3}

147: {2,4,4}

154: {1,4,5}

192: {1,1,1,1,1,1,2}

243: {2,2,2,2,2}

270: {1,2,2,2,3}

280: {1,1,1,3,4}

286: {1,5,6}

300: {1,1,2,3,3}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];

Select[Range[0, 100], ats[primeMS[#]]==2&]

These partitions are counted by A000097.

The k = 0 version is A000290, counted by A000041.

The k = 1 version is A001105 (reverse: A345958).

The k > 0 version is A026424.

These multisets are counted by A120452.

These are the positions of 2's in A316524 (reverse: A344616).

The k = -1 version is A345959.

The version for reversed alternating sum is A345961.

The negative (k = -2) version is A345962.

A000984/A345909/A345911 count/rank compositions with alternating sum 1.

A001791/A345910/A345912 count/rank compositions with alternating sum -1.

A002054/A345924/A345923 count/rank compositions with alternating sum -2.

A056239 adds up prime indices, row sums of A112798.

A088218/A345925/A345922 count/rank compositions with alternating sum 2.

A097805 counts compositions by alternating (or reverse-alternating) sum.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A325534 and A325535 count separable and inseparable partitions.

A344606 counts alternating permutations of prime indices.

Cf. A000037 non_sqrs, A000070 ptns_altsum_1, ~A000346 comps_ev_ats_neq0, ~A008549 comps_ev_ats_less0, A027187 ptns_altsum_wkneg, A027193 ptns_sats_grtr0, A028260 h_altsum_wkneg, ~A032443 comps_oddsum_altsum_geq_0, `A034871 tri_comps_2n_ats_2k, A035363 ptns_use_ev, ~A114121 comps_evsum_altsum_geq_0, A239830 tri_ptns_altsum_ev_ev, A341446 only_odd_prix_least, AA344607, 344608 ptns_sats_strneg, A344609 h_altsum_wkpos, A344610, `A344650 strptns_ev_revaltsum_wkpos, A344651 tri_ptns_altsum_mod2pos_ev_k, A344741, `A345197 tetr_comps_len_ats, `A345908 comps_len_ats_trace, ~A345917 stc_ats_grtr0, ~A345918 stc_sats_grtr0.

allocated

nonn

Gus Wiseman, Jul 12 2021

approved

editing

allocated for Gus Wiseman

allocated

approved